homotopy analysis method in nonlinear differential...

Shijun Liao

Homotopy Analysis Methodin Nonlinear DifferentialEquations

– Monograph –

March 31, 2011

Springer

To my mother, wife and daughter

Preface

It is well-known that perturbation and asymptotic approximations of nonlinear prob-lems often break down as nonlinearity becomes strong. Therefore, they are onlyvalid for weakly nonlinear ordinary differential equations (ODEs) and partial differ-ential equations (PDEs) in general.

The homotopy analysis method (HAM) is an analytic approximation method forhighly nonlinear problems, proposed by the author in 1992. Unlike perturbationtechniques, the HAM is independent of any small/large physical parameters at all.Secondly, different from all of other analytic techniques,the HAM provides us aconvenient way to guarantee the convergenceof solution series so that it is valid evenif nonlinearity becomes rather strong. Besides, based on the homotopy in topology, itprovides us extremely large freedom to choose base function, initial guess and so on,so that complicated nonlinear ODEs and PDEs can often be solved in a simple way.Finally, the HAM logically contains some traditional methods such as Lyapunov’ssmall artificial method, Adomian decomposition method, theδ -expansion method,and even the Euler transform, so that it has the great generality. Therefore, the HAMprovides us a useful tool to solve highly nonlinear problemsin science, finance andengineering.

This book consists of three parts. In Part I, the basic ideas of the HAM, espe-cially its theoretical modifications and developments, aredescribed, including theoptimal HAM approaches, the theorems about the so-called homotopy-derivativeoperator and the different types of deformation equations,the methods to controland accelerate convergence, the relationship to Euler transform, and so on.

In Part II, inspirited by so many successful applications ofthe HAM in differ-ent fields and also by the ability of “computing with functions instead of numbers”of computer algebra system like Mathematica and Maple, a Mathematica packageBVPh(version 1.0) is developed by the author in the frame of the HAM for nonlinearboundary-value problems. A dozen of examples are used to illustrate its validity forhighly nonlinear ODEs with singularity, multiple solutions and multipoint boundaryconditions in either a finite or an infinite interval, and evenfor some types of non-linear PDEs. As an open resource, theBVPh1.0 is given in this book with a simpleusers guide and free available online.

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In Part III, we illustrate that the HAM can be used to solve some complicatedhighly nonlinear PDEs so as to enrich and deepen our understandings about theseinteresting nonlinear problems. For example, By means of the HAM, an explicitanalytic approximation of the optimal exercise boundary ofAmerican put optionwas gained, which is often valid a couple of dozen years priorto expiry, whereasthe asymptotic and perturbation formulas are valid only a couple of days or weeksin general. A Mathematica code based on such kind of explicitformula is given inthis book for businessmen to gain accurate results in a few seconds. In addition, bymeans of the HAM, the wave-resonance criterion of arbitrarynumber of travelinggravity waves was found, for the first time, which logically contains the famousPhillips’ criterion for four waves with small amplitude.

All of these show the originality, validity and generality of the HAM for highlynonlinear problems in science, finance and engineering.

All Mathematica codes and their input data files are given in the appendixes ofthis book and available either athttp://numericaltank.sjtu.edu.cn/HAM.htm, or athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

This book is suitable for researchers and postgraduates in applied mathematics,physics, finance and engineering, who are interested in highly nonlinear ODEs andPDEs.

I would like to express my gratitude to my collaborators for their valuable discus-sions and communications, and to my postgraduates for theirhard working. Thanksto Natural Science Foundation of China for the financial support.

I would like to express my sincere thanks to my parents, wife and daughter fortheir love, encouragement and support in the past 20 years.

Shanghai, ChinaMarch 2011 Shijun Liao

Contents

Part I Basic Ideas and Theorems

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Motivation and purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 31.2 Characteristic of homotopy analysis method . . . . . . . . . .. . . . . . . . . . 51.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 8

2 Basic ideas of the homotopy analysis method. . . . . . . . . . . . . . . . . . . . . 132.1 Concept of homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 132.2 Example 2.1: generalized Newtonian iteration formula .. . . . . . . . . . 172.3 Example 2.2: nonlinear oscillation . . . . . . . . . . . . . . . . .. . . . . . . . . . . 24

2.3.1 Analysis of the solution characteristic . . . . . . . . . . .. . . . . . . . 252.3.2 Mathematical formulations . . . . . . . . . . . . . . . . . . . . . .. . . . . . 302.3.3 Convergence of homotopy-series solution . . . . . . . . . .. . . . . 372.3.4 Essence of the convergence-control parameterc0 . . . . . . . . . 472.3.5 Convergence acceleration by Homotopy-Pade technique . . . 552.3.6 Convergence acceleration by optimal initial approximation . 592.3.7 Convergence acceleration by iteration . . . . . . . . . . . .. . . . . . . 622.3.8 Flexibility on the choice of auxiliary linear operator . . . . . . . 69

2.4 Concluding remarks and discussions . . . . . . . . . . . . . . . . .. . . . . . . . . 74Appendix 2.1 Derivation ofδn in (2.57) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Appendix 2.2 Derivation of (2.55) by the 2nd approach . . . . . .. . . . . . . . . . 81Appendix 2.3 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 83Appendix 2.4 Mathematica code (without iteration) for Example 2.2 . . . . 85Appendix 2.5 Mathematica code (with iteration) for Example2.2 . . . . . . . 90Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 95References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 96

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3 Optimal homotopy analysis method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 993.2 An illustrative description . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 105

3.2.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1053.2.2 Different types of optimal methods . . . . . . . . . . . . . . . .. . . . . 108

3.3 Systematic description . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1213.4 Concluding remarks and discussions . . . . . . . . . . . . . . . . .. . . . . . . . . 125Appendix 3.1 Mathematica code for Blasius flow . . . . . . . . . . . .. . . . . . . . . 127Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 131References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 132

4 Systematic descriptions and related theorems. . . . . . . . . . . . . . . . . . . . . 1354.1 Brief frame of the homotopy analysis method . . . . . . . . . . .. . . . . . . . 1354.2 Properties of homotopy-derivative . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1374.3 Deformation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 153

4.3.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 1534.3.2 High-order deformation equations . . . . . . . . . . . . . . . .. . . . . . 1574.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 169

4.4 Convergence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 1724.5 Solution expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 177

4.5.1 Choice of initial approximation . . . . . . . . . . . . . . . . . .. . . . . . 1794.5.2 Choice of auxiliary linear operator . . . . . . . . . . . . . . .. . . . . . 181

4.6 Convergence control and acceleration . . . . . . . . . . . . . . .. . . . . . . . . . 1834.6.1 Optimal convergence-control parameter . . . . . . . . . . .. . . . . . 1844.6.2 Optimal initial approximation . . . . . . . . . . . . . . . . . . .. . . . . . 1854.6.3 Homotopy-iteration technique . . . . . . . . . . . . . . . . . . .. . . . . . 1864.6.4 Homotopy-Pade technique . . . . . . . . . . . . . . . . . . . . . . .. . . . . 186

4.7 Discussions and open questions . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 187References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 190

5 Relationship to Euler transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1935.2 Generalized Taylor series . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 1945.3 Homotopy transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 2155.4 Relation between homotopy analysis method and Euler transform . . 2195.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 223References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 226

6 Some methods based on the HAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.1 A brief history of the homotopy analysis method . . . . . . . .. . . . . . . . 2276.2 Homotopy perturbation method . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 2296.3 Optimal homotopy asymptotic method . . . . . . . . . . . . . . . . .. . . . . . . . 2326.4 Spectral homotopy analysis method . . . . . . . . . . . . . . . . . .. . . . . . . . . 2346.5 Generalized boundary element method . . . . . . . . . . . . . . . .. . . . . . . . . 2346.6 Generalized scaled boundary finite element method . . . . .. . . . . . . . . 235

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 237

Part II Mathematica packageBVPh and its applications

7 Mathematica packageBVPh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2417.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 241

7.1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 2447.1.2 Brief mathematical formulas . . . . . . . . . . . . . . . . . . . . .. . . . . . 2447.1.3 Choice of the base functions and initial guess . . . . . . .. . . . . 2497.1.4 Choice of the auxiliary linear operator . . . . . . . . . . . .. . . . . . 2527.1.5 Choice of the auxiliary function . . . . . . . . . . . . . . . . . .. . . . . . 2547.1.6 Choice of the convergence-control parameterc0 . . . . . . . . . . 255

7.2 Approximation of solutions and iteration . . . . . . . . . . . .. . . . . . . . . . . 2567.2.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 2577.2.2 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 2587.2.3 Hybrid-base functions . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 259

7.3 A simple users guide of theBVPh1.0 . . . . . . . . . . . . . . . . . . . . . . . . . 2627.3.1 Key modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 2627.3.2 Control parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 2637.3.3 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 2657.3.4 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 2667.3.5 Global variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 266

Appendix 7.1 Mathematica packageBVPh(version 1.0) . . . . . . . . . . . . 268References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 282

8 Nonlinear boundary-value problems with multiple solutions . . . . . . . . 2878.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2878.2 Brief mathematical formulas . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 2888.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 291

8.3.1 Nonlinear diffusion-reaction model . . . . . . . . . . . . . .. . . . . . . 2918.3.2 A three-point nonlinear boundary-value problem . . . .. . . . . 2988.3.3 Channel flows with multiple solutions . . . . . . . . . . . . . .. . . . 303

8.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 308Appendix 8.1 Input data ofBVPh for Example 8.3.1 . . . . . . . . . . . . . . . . . . 310Appendix 8.2 Input data ofBVPh for Example 8.3.2 . . . . . . . . . . . . . . . . . . 312Appendix 8.3 Input data ofBVPh for Example 8.3.3 . . . . . . . . . . . . . . . . . . 314Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 316References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 317

9 Nonlinear eigenvalue equations with varying coefficients. . . . . . . . . . . 3199.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3199.2 Brief mathematical formula . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 3219.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 326

9.3.1 Non-uniform beam acted by axial load . . . . . . . . . . . . . . .. . . 3269.3.2 Gelfand equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 3379.3.3 Equation with singularity and varying coefficient . . .. . . . . . 340

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9.3.4 Multipoint boundary-value problem with multiplesolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 345

9.3.5 Orr-Sommerfeld stability equation with complexcoefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 349

9.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 353Appendix 9.1 Input data ofBVPh for Example 9.3.1 . . . . . . . . . . . . . . . . . . 356Appendix 9.2 Input data ofBVPh for Example 9.3.2 . . . . . . . . . . . . . . . . . . 358Appendix 9.3 Input data ofBVPh for Example 9.3.3 . . . . . . . . . . . . . . . . . . 360Appendix 9.4 Input data ofBVPh for Example 9.3.4 . . . . . . . . . . . . . . . . . . 362Appendix 9.5 Input data ofBVPh for Example 9.3.5 . . . . . . . . . . . . . . . . . . 364Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 366References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 367

10 A boundary-layer flow with an infinite number of solutions . . . . . . . . . 36910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 36910.2 Exponentially decaying solutions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 37110.3 Algebraically decaying solutions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 37510.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 382Appendix 10.1 Input data ofBVPh for exponentially decaying solution . . 384Appendix 10.2 Input data ofBVPh for algebraically decaying solution . . . 385References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 387

11 Non-similarity boundary-layer flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38911.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 38911.2 Brief mathematical formulas . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 39311.3 Homotopy-series solution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 39711.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 402Appendix 11.1 Input data ofBVPh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 406

12 Unsteady boundary-layer flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 40912.2 Perturbation approximation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 41212.3 Homotopy-series solution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 414

12.3.1 Brief mathematical formulas . . . . . . . . . . . . . . . . . . . .. . . . . . . 41412.3.2 Homotopy-approximation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 418

12.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 422Appendix 12.1 Input data ofBVPh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 427

Part III Applications in Nonlinear Partial Differential Eq uations

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13 Applications in finance: American put options . . . . . . . . . . . . . . . . . . . . 43113.1 Mathematical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 43113.2 Brief mathematical formulas . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 43413.3 Validity of the explicit homotopy-approximations . . .. . . . . . . . . . . . . 44213.4 A practical code for businessmen . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 44813.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 450Appendix 13.1 Detailed derivation offn(τ) andgn(τ) . . . . . . . . . . . . . . . . 452Appendix 13.2 Mathematica code for American put option . . . .. . . . . . . . 454Appendix 13.3 Mathematica codeAPOhfor businessmen . . . . . . . . . . . . . 461References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 465

14 Two and three dimensional Gelfand equation. . . . . . . . . . . . . . . . . . . . . 46714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 46714.2 Homotopy-approximations of 2D Gelfand equation . . . . .. . . . . . . . . 468

14.2.1 Brief mathematical formulas . . . . . . . . . . . . . . . . . . . .. . . . . . . 46814.2.2 Homotopy-approximations . . . . . . . . . . . . . . . . . . . . . .. . . . . . 474

14.3 Homotopy-approximations of 3D Gelfand equation . . . . .. . . . . . . . . 48014.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 485Appendix 14.1 Mathematica code of 2D Gelfand equation . . . . .. . . . . 487Appendix 14.2 Mathematica code of 3D Gelfand equation . . . . .. . . . . 491References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 496

15 Interaction of nonlinear water wave and nonuniform currents . . . . . . 49915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 49915.2 Mathematical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 500

15.2.1 Original boundary-value equation . . . . . . . . . . . . . . .. . . . . . . 50015.2.2 Dubreil-Jacotin transformation . . . . . . . . . . . . . . . .. . . . . . . . . 502

15.3 Brief mathematical formulas . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 50315.3.1 Solution expression . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 50315.3.2 Zeroth-order deformation equation . . . . . . . . . . . . . .. . . . . . . . 50415.3.3 High-order deformation equation . . . . . . . . . . . . . . . .. . . . . . . 50615.3.4 Successive solution procedure . . . . . . . . . . . . . . . . . .. . . . . . . 508

15.4 Homotopy approximations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 51115.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 521Appendix 15.1 Mathematica code of wave-current interaction . . . . . . . 522References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 527

16 Resonance of arbitrary number of periodic traveling water waves . . . 52916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 52916.2 Resonance criterion of two small-amplitude primary waves . . . . . . . 531

16.2.1 Brief Mathematical formulas . . . . . . . . . . . . . . . . . . . .. . . . . . 53116.2.2 Non-resonant waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 53916.2.3 Resonant waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 544

16.3 Resonance criterion of arbitrary number of primary waves . . . . . . . . 55216.3.1 Resonance criterion of small-amplitude waves . . . . .. . . . . . 553

xiv Contents

16.3.2 Resonance criterion of large-amplitude waves . . . . .. . . . . . . 55516.4 Concluding remark and discussions . . . . . . . . . . . . . . . . .. . . . . . . . . . 559Appendix 16.1 Detailed derivation of high-order equation .. . . . . . . . . . . . 562References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 568

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 569

Acronyms

2D Two Dimensional3D Three DimensionalAPO American Put OptionBEM Boundary Element MethodBVP Boundary Value ProblemBVPs Boundary Value ProblemsCPU Central Processing UnitDNS Direct Numerical SimulationFDM Finite Difference MethodFEM Finite Element MethodGBEM Generalized Boundary Element MethodHAM Homotopy Analysis MethodIVP Initial Value ProblemIVPs Initial Value ProblemsODE Ordinary Differential EquationODEs Ordinary Differential EquationsOHAM Optimal Homotopy Analysis MethodPDE Partial Differential EquationPDEs Partial Differential Equations

xv

Part IBasic Ideas and Theorems

“ The essence of mathematics lies entirely in its freedom”

by Georg Cantor (1845-1918)

Chapter 1Introduction

1.1 Motivation and purpose

It is well-known that nonlinear ordinary differential equations (ODEs) and partialdifferential equations (PDEs) for boundary-value problems are much more difficultto solve than linear ODEs and PDEs, especially by means of analytic methods. Tra-ditionally, perturbation [69,94,95] and asymptotic techniques are widely applied toobtain analytic approximations of nonlinear problems in science, finance and en-gineering. Unfortunately, perturbation and asymptotic techniques are too stronglydependent upon small/large physical parameters in general, and thus are often validonly for weakly nonlinear problems. For example, the asymptotic/perturbation ap-proximations of the optimal exercise boundary of American put option are validonly a couple of days or weeksprior to expiry, as shown in Fig. 1.1. Another fa-mous example is the viscous flow past a sphere in fluid mechanics: the perturbationformulas of the drag coefficient are valid only for rather small Reynolds numberRe≪ 1. Thus, it is necessary to develop some analytic approximation methods,

Fig. 1.1 Asymptotic/perturbationapproximations of the optimalexercise boundary ofAmerican put option inthe case of Example 13.1:X = $100,r = 0.1, σ = 0.3and T = 1 (year). Dashedline A: by Kuske andKeller [34]; Dashed line B: byKnessl [33]; Dashed line C:by Bunch and Johnson [20];Circles: numerical result byZhu [106] .

time to expiry (year)

Opt

imal

exer

cise

boun

dary

($)

0 0.2 0.4 0.6 0.8 165

70

75

80

85

90

95

100A

B

C

Dashed line A: by Kuske & Keller [34]Dashed line B: by Knessl [33]Dashed line C: by Bunch & Johnson [20]Symbols: numerical integral by Zhu [106]

3

4 1 Introduction

which are independent of any small/large physical parameters at all and besidesvalid for strongly nonlinear problems.

In 1992, one of such kind of analytic approximation methods was proposed bythe author [42], namely the homotopy analysis method (HAM) [37, 43–52, 54–58,99]. Based on homotopy [29] in topology [75], the HAM is independent of anysmall/large physical parameters. More importantly, unlike all other analytic tech-niques, it provides us a convenient way to guarantee the convergence of series so-lution of nonlinear problems by means of introducing an auxiliary parameterc0,called the convergence-control parameter. In 2003, the basic ideas of the HAM andsome applications mostly related to nonlinear ODEs were described systematicallyby the author in the book “Beyond Perturbation” [48].

Thereafter, the HAM attracts attention of many researchersin about a dozen ofcountries, and has been successfully applied to solve a lotsof nonlinear problems inscience, finance and engineering. For example, Some new solutions [50, 57] havebeen found by means of the HAM, which had been neglected even by numeri-cal techniques and never reported. Some analytic approximations [22, 23, 105] forthe optimal exercise boundary of American put option were given, which are oftenvalid for a couple of yearsprior to expiry and thus much better than the asymp-totic/perturbation approximations [20, 33, 34] that are often valid onlya couple ofdays or weeks. Besides, the HAM has been successfully employed to solve somecomplicated nonlinear PDEs so as to enrich and deepen our physical understand-ings: the wave-resonance criterion forarbitrary number of traveling waves withlargeamplitudes was founded [56],for the first time, by means of the HAM, whichlogically contains the famous Phillips’ criterion forfour waves withsmall ampli-tudes. All of these applications show the originality, validity and generality of theHAM for nonlinear problems.

In addition, some theoretical studies and modifications have been done. For ex-ample, some mathematical theorems related to the equationsfor high-order approx-imations and the so-called homotopy-derivatives were proved [52,66,84,89]. Someoptimal HAM approaches [55,64,65,70,100] were developed,which greatly accel-erate the convergence of solution series. Besides, it was proved [54] that the HAMlogically contains the famous Euler transform [12], which reveals the reason why theHAM can guarantee the convergence of solution series. Furthermore, some HAM-based approaches [6,50,64,68,99] were developed to searchfor multiple solutionsof nonlinear problems, and/or to increase the computational efficiency. All of thesegreatly developed and modified the HAM in theory and providedthe HAM a soundbase.

Therefore, it is valuable to systematically describe the theoretical modificationsand new applications of the HAM as a whole, and besides to discuss some openquestions and its possible development in future.

It should be emphasized that our aim is to develop an analyticapproximationmethod valid foras manynonlinear problemsas possible, since it seems impossibleto develop an analytic approach valid forall nonlinear problems, especially thoserelated to chaos [53,61] and turbulence.

1.2 Characteristic of homotopy analysis method 5

1.2 Characteristic of homotopy analysis method

The HAM has the following characteristics which differ it from other traditionalanalytic techniques.

First of all, based on the homotopy of topology, the HAM isindependentofany small/large physical parameters at all. So, unlike asymptotic/perturbation tech-niques, the HAM can be applied to solve most of nonlinear problems in science,finance and engineering, especially those without small/large physical parameters.For example, an analytic approximation of the optimal exercise boundaryB(τ) inpolynomials of

√τ to ordero(τ48) was obtained by means of the HAM, which is

often valid fora couple of dozen yearsor even toa half of centuryprior to expiry,and thus is much better than the asymptotic/perturbation approximations [20,33,34]that are often valida could of days or weeks, as shown in Fig. 1.2. For details, pleaserefer to Chapter 13.

Fig. 1.2 The optimal exer-cise boundary of Americanput option in case of Exam-ple 13.2:X = $1, r = 0.08and σ = 0.4. Solid line:10th-order approximation byHAM in polynomial of

√τ to

o(τ48); Symbols: approxima-tion by Pade method; Dashedlines: asymptotic/perturbationresults; Dash-dotted line: per-petual optimal exercise price$0.5.

Time (year) prior to expiry

Opt

imal

exer

cise

pric

e($

)

0 5 10 15 200

0.2

0.4

0.6

0.8

1 A

B

C

Example 13.2: X= 1, r = 0.08,σ = 0.4Dashed line A: by Kuske & Keller [34]Dashed line B: by Knessl [33]Dashed line C: by Bunch & Johnson [20]Solid line: by the HAM (10th-order)

Secondly, unlike all other analytic techniques, the HAM provides us a convenientway to guarantee the convergence of solution series so that it is valid for highlynonlinear problems. For example, when perturbation results are valid only for asmallphysical parameter 0≤ ε ≤ 1, one can gain accurate approximations valid inthewholeinterval 0≤ ε <+∞ by means of the HAM, as shown by Abbasbandy [2].Besides, when results given by other methods aredivergent, one can gainconvergentseries solution by means of the HAM, as shown by Liang and Jeffrey [41].

Thirdly, the HAM provides us extremely largefreedomto choose the auxiliarylinear operator and base functions. Using such kind of freedom, some complicatednonlinear problems can be solved in a much easier way. For example, the two-dimensional2nd-orderGelfand equation was solved by means of the HAM in arather easy way by transferring it into an infinite number of linear PDEs governedby a very simple4th-order linear operator, as shown by Liao [58]. Such kind oftransformation hasneverbeen used by other analytic and numerical techniques. Thissuggests that we human being might have much larger freedom to solve nonlinear

6 1 Introduction

problems than we traditionally thought and therefore we should always keep anopen mind.

Finally, it has been proved [48, 54] that the HAM logically contains the Lya-punov’s small artificial parameter method [62], Adomian decomposition method[10,11], theδ -expansion method [32], and the Euler transform [12]. Thus,it has thegreatgenerality.

In summary, the HAM has the following advantages:

• Independent of small/large physical parameters;• Guarantee of convergence;• Flexibility on choice of base function and initial guess;• Great generality.

The HAM provides a useful analytic tool to investigate highly nonlinear prob-lems with multiple solutions and singularity in science, finance and engineering.

1.3 Outline

This book consists of three parts. In Part I, the basic ideas of the HAM, its modifi-cations and developments in theory are briefly described. InChapter 2, two simpleexamples are used to describe the basic ideas of the HAM, all related conceptsand approaches in details. For beginners of the HAM, it is strongly suggested toread Chapter 2 first. In Chapter 3, some optimal HAM approaches are described.In Chapter 4, the basic ideas of the HAM is systematically described, mathemati-cal theorems about the so-called homotopy-derivative and equations for high-orderapproximations are proved, and some open questions are discussed. The relation-ship between the HAM with the famous Euler transform is revealed in Chapter 5.Some other HAM-based methods, together with the history of the HAM, are brieflydescribed in Chapter 6, respectively.

In Part II, inspirited by so many successful applications ofthe HAM in so manydifferent fields of researches [1–8, 13–19, 21, 22, 24–28, 30, 31, 35, 36, 38–41, 59,60, 63–68, 70–74, 76–81, 83, 83–93, 97, 98, 100–108] and the ability of “comput-ing with functions instead of numbers” [82] provided by computer algebra systemsuch as Mathematica [9] and Maple, a Mathematica packageBVPh(version 1.0) isdeveloped by the author in the frame of the HAM, which is mainly for highly non-linear ODEs in a finite or an infinite interval with multiple solutions, singularity andmultipoint boundary conditions. In Chapter 7, the Mathematica packageBVPh1.0is briefly described with related mathematical formulas anda simple users guide.Then, we illustrate its validity and generality for nonlinear boundary-value problemswith multiple solutions in a finite interval (Chapter 8), nonlinear eigenvalue prob-lems in a finite interval with multipoint boundary conditions, singularity and highnonlinearity (Chapter 9), nonlinear boundary-value problems governed by an ODEin an infinite interval with exponentially or algebraicallydecaying solutions (Chap-ter 10), and even some nonlinear PDEs related to non-similarity boundary layer

1.3 Outline 7

flows (Chapter 11) and unsteady boundary-layer flows (Chapter 12), respectively.TheBVPh1.0 provides us a tool to solve some nonlinear boundary-value problemsgoverned by a nonlinear ODE or PDE. As an open resource, theBVPh1.0 is givenin Appendix 7.1 and free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.Besides, the input data files for all examples in Part II are free available at the samewebsite.

In Part III, we illustrate that the HAM can be applied to investigate some rathercomplicated nonlinear PDEs with high nonlinearity so as to deepen and enrich ourphysical understandings about these nonlinear phenomena.In Chapter 13, the HAMis used to give an analytic approximation of the optimal exercise boundary of Amer-ican put option, which is often valida could of dozen yearsprior to expiry andsometimes may be valid even fora half of century, and thus is much better than theperturbation/asymptotic results that are often valid onlya couple of days or weeks.Based on this kind of analytic results, a short Mathematica codeAPOhfor business-men is given in Appendix 13.3, which gives accurate results in a few seconds only!In Chapter 14, the two (three) dimensional2nd-orderGelfand equation is solved bymeans of the HAM in a rather easier way by transferring it intoan infinite numberof 4th-order(6th-order) linear PDEs. Such kind of transformation hasneverbe usedby other analytic and numerical methods, which suggests that we might have muchlarger freedom to solve nonlinear problems than we thought traditionally and thuswe mustalways keep an open mind. In Chapter 15, the HAM is applied to solvea complicated nonlinear PDE describing the nonlinear interaction between gravitywaves and exponential shear currents. It is foundfor the first timethat the criterionfor wave breaking is the same for waves on uniform and non-uniform currents. InChapter 16, the HAM is successfully applied to investigate the nonlinear interactionof arbitrary number of traveling gravity waves. And it is found,for the first time, thewave-resonance criterion forarbitrary number of traveling waves withlargeampli-tudes, which logically contains the famous Phillips’ criterion for four waves withsmallamplitudes.

All of these show theoriginality, validity andgeneralityof the HAM for somehighly nonlinear problems with multiple solutions, singularity and multipoint bound-ary conditions.

For readers’ convenience, the related Mathematica codes and their input date filesof nearly all examples are given in the appendixes of this book and available eitherat

http://numericaltank.sjtu.edu.cn/HAM.htm,or at

http://numericaltank.sjtu.edu.cn/BVPh.htm,respectively.

8 1 Introduction

References

1. Abbas, Z., Wang, Y., Hayat, T., Oberlack, M.: Hydromagnetic flow in a viscoelstic fluid dueto the oscillatory stretching surface. Int. J. Nonlin. Mech. 43, 783 – 793 (2008)

2. Abbasbandy, S.: The application of the homotopy analysismethod to nonlinear equationsarising in heat transfer. Phys. Lett. A.360, 109 – 113 (2006)

3. Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hiro-taSatsuma coupled KdV equation. Phys. Lett. A.361, 478 – 483 (2007)

4. Abbasbandy, S., Parkes, E.J.: Solitary smooth hump solutions of the Camassa-Holm equationby means of the homotopy analysis method. Chaos Soliton. Fract. 36, 581 – 591 (2008)

5. Abbasbandy, S.: Solitary wave equations to the Kuramoto-Sivashinsky equation by means ofthe homotopy analysis method. Nonlinear Dynam.52, 35 – 40 (2008)

6. Abbasbandy, S., Magyari, E., Shivanian, E.: The homotopyanalysis method for multiple so-lutions of nonlinear boundary value problems. Communications in Nonlinear Science andNumerical Simulation.14, 3530 – 3536 (2009)

7. Abbasbandy, S., Parkes, E.J.: Solitary-wave solutions of the DegasperisProcesi equation bymeans of the homotopy analysis method. Int. J. Comp. Math.87, 2303 – 2313 (2010)

8. Abbasbandy, S., Shivanian, E.: Predictor homotopy analysis method and its application tosome nonlinear problems. Commun. Nonlinear Sci. Numer. Simulat.16, 2456 – 2468 (2011)

9. Abell, M.L., Braselton, J.P.: Mathematica by Example (3rd Edition). Elsevier AcademicPress. Amsterdam (2004)

10. Adomian, G.: Nonlinear stochastic differential equations. J. Math. Anal. Applic.55, 441 –452 (1976)

11. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. KluwerAcademic Publishers, Boston (1994)

12. Agnew, R.P.: Euler Transformations. Journal of Mathematics.66, 313-338 (1944)13. Akyildiz, F.T., Vajravelu, K. Magnetohydrodynamic flowof a viscoelastic fluid. Phys. Lett.

A. 372, 3380 – 3384 (2008)14. Akyildiz, F.T., Vajravelu, K., Mohapatra, R.N., Sweet,E., Van Gorder, R.A.: Implicit differ-

ential equation arising in the steady flow of a Sisko fluid. Applied Mathematics and Compu-tation.210, 189 – 196 (2009)

15. Alizadeh-Pahlavan, A., Borjian-Boroujeni, S.: On the analytical solution of viscous fluid flowpast a flat plate. Physics Letters A.372, 3678 – 3682 (2008)

16. Alizadeh-Pahlavan, A., Aliakbar, V., Vakili-Farahani, F., Sadeghy, K.: MHD flows of UCMfluids above porous stretching sheets using two-auxiliary-parameter homotopy analysismethod. Commun. Nonlinear Sci. Numer. Simulat.14, 473 – 488 (2009)

17. Allan, F.M., Syam, M.I.: On the analytic solutions of thenonhomogeneous Blasius problem.J. Comp. Appl. Math.182, 362 – 371 (2005)

18. Allan, F.M.: Derivation of the Adomian decomposition method using the homotopy analysismethod. Appl. Math. Comput.190, 6 – 14 (2007)

19. Allan, F.M.: Construction of analytic solution to chaotic dynamical systems using the Homo-topy analysis method. Chaos, Solitons and Fractals.39, 1744 – 1752 (2009)

20. Bunch, D.S., Johnson, H.: The American put option and itscritical stock price. Journal ofFinance.5, 2333-2356 (2000)

21. Cai, .W.H.: Nonlinear Dynamics of Thermal-Hydraulic Networks. PhD dissertation, Univer-sity of Notre Dame (2006)

22. Cheng, J.: Application of The Homotopy Analysis Method in Nonlinear Mechanics and Fi-nance. PhD dissertation, Shanghai Jiao Tong University (2008)

23. Cheng, J., Zhu, S.P., Liao, S.J.: An explicit series approximation to the optimal exerciseboundary of American put options. Communications in Nonlinear Science and NumericalSimulation.15 1148 – 1158 (2010),

24. Gao, L.M.: Analysis of the Propagation of Surface Acoustic Waves in Functionally GradedMaterial Plate. PhD dissertation, Tong Ji University (2007)

References 9

25. Hayat, T., Khan,M., Asghar, S.: Magnetohydrodynamic flow of an Oldroyd 6-constant fluid.Applied Mathematics and Computation.155, 417 – 425 (2004)

26. Hayat, T., Khan,M., Ayub, M.: On non-linear flows with slip boundary condition. Z. angew.Math. Phys.56, 1012 – 1029 (2005)

27. Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down avertical cylinder. Phys. Lett. A.361, 316–322 (2007)

28. Hayat, T., Sajjad, R., Abbas, Z., Sajid, M., Hendi, A.A.:Radiation effects on MHD flow ofMaxwell fluid in a channel with porous medium. Int. J. heat Mass Transfer.54, 854 – 862(2011)

29. Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge(1953)

30. Jiao, X.Y.: Approximate Similarity Reduction and Approximate Homotopy Similarity Reduc-tion of Several Nonlinear Problems. PhD dissertation, Shanghai Jiao Tong University (2009)

31. Jiao, X.Y., Gao, Y., Lou, S.Y.: Approximate homotopy symmetry method – Homotopy seriessolutions to the sixth-order Boussinesq equation. Sciencein China (G).52, 1169 – 1178(2009)

32. Karmishin, A.V., Zhukov, A.T., Kolosov, V.G.: Methods of Dynamics Calculation and Testingfor Thin-walled Structures (in Russian). Mashinostroyenie, Moscow (1990)

33. Knessl, C.: A note on a moving boundary problem arising inthe American put option. Studiesin Applied Mathematics.107, 157-183 (2001)

34. Kuske, R.A., Keller, J.B.: Optional exercise boundary for an American put option. AppliedMathematical Finance.5, 107-116 (1998)

35. Kumari, M., Nath, G.: Unsteady MHD mixed convection flow over an impulsively stretchedpermeable vertical surface in a quiescent fluid. Int. J. Non-Linear Mech.45, 310 – 319 (2010)

36. Kumari, M., Pop, I., Nath, G.: Transient MHD stagnation flow of a non-Newtonian fluid dueto impulsive motion from rest. Int. J. Non-Linear Mech.45, 463 – 473 (2010)

37. Li, Y.J., Nohara, B.T., Liao, S.J.: Series solutions of coupled Van der Pol equationby means of homotopy analysis method. J. Mathematical Physics 51, 063517 (2010).doi:10.1063/1.3445770

38. Liang, S.X.: Symbolic Methods for Analyzing Polynomialand Differential Systems. PhDdissertation, University of Western Ontario (2010)

39. Liang, S.X., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy pertur-bation method through an evalution equation. Commun. Nonlinear Sci. Numer. Simulat.14,4057 – 4064 (2009)

40. Liang, S.X., Jeffrey, D.J.: An efficient analytical approach for solving fourth order boundaryvalue problems. Computer Physics Communications.180, 2034 – 2040 (2009)

41. Liang, S.X., Jeffrey, D.J.:Approximate solutions to a parameterized sixth order boundaryvalue problem. Computers and Mathematics with Applications.59, 247 – 253 (2010)

42. Liao, S.J.: The proposed Homotopy Analysis Technique for the Solution of Nonlinear Prob-lems. PhD dissertation, Shanghai Jiao Tong University (1992)

43. Liao, S.J.: A kind of approximate solution technique which does not depend upon small pa-rameters (II) – an application in fluid mechanics. Int. J. Nonlin. Mech.32, 815 – 822 (1997)

44. Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Int.J. Nonlin. Mech.34, 759 – 778 (1999)

45. Liao, S.J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flatplate. J. Fluid Mech.385, 101 – 128 (1999)

46. Liao, S.J., Campo, A.: Analytic solutions of the temperature distribution in Blasius viscousflow problems. J. Fluid Mech.453, 411 – 425 (2002)

47. Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluidsover a stretching sheet. J. Fluid Mech.488, 189-212 (2003)

48. Liao, S.J.: Beyond Perturbation – Introduction to the Homotopy Analysis Method. Chapman& Hall/ CRC Press, Boca Raton (2003)

49. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput.147, 499 – 513 (2004)

10 1 Introduction

50. Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretchedplate. Int. J. Heat Mass Tran.48, 2529 – 2539 (2005)

51. Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Stud.Appl. Math.117, 2529 – 2539 (2006)

52. Liao, S.J.: Notes on the homotopy analysis method - some definitions and theorems. Com-mun. Nonlinear Sci. Numer. Simulat.14, 983 – 997 (2009)

53. Liao, S.J.: On the reliability of computed chaotic solutions of non-linear differential equa-tions. Tellus.61A, 550 – 564 (2009)

54. Liao, S.J.: On the relationship between the homotopy analysis method and Euler transform.Commun. Nonlinear Sci. Numer. Simulat.15, 1421 – 1431 (2010).

55. Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equa-tions. Commun. Nonlinear Sci. Numer. Simulat.15, 2003 – 2016 (2010).

56. Liao, S.J.: On the homotopy multiple-variable method and its applications in the interactionsof nonlinear gravity waves. Commun. Nonlinear Sci. Numer. Simulat.16, 1274 – 1303 (2011)

57. Liao, S.J., Magyari, E.: Exponentially decaying boundary layers as limiting cases of familiesof algebraically decaying ones. Z. angew. Math. Phys.57, 777 – 792 (2006)

58. Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differentialequations. Stud. Appl. Math.119, 297 – 355 (2007)

59. Liu, Y.P.: Study on Analytic and Approximate Solution ofDifferential equations by SymbolicComputation. PhD Dissertation, East China Normal University (2008)

60. Liu, Y.P., Li, Z.B.: The homotopy analysis method for approximating the solution of themodified Korteweg-de Vries equation. Chaos Soliton. Fract.39, 1 – 8 (2009)

61. Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmos.Sci.20, 130 – 141 (1963).62. Lyapunov, A.M.: General Problem on Stability of Motion (English translation). Taylor &

Francis, London (1992)63. Mahapatra, T. R., Nandy, S.K., Gupta, A.S.: Analytical solution of magnetohydrodynamic

stagnation-point flow of a power-law fluid towards a stretching surface. Applied Mathematicsand Computation.215, 1696 – 1710 (2009)

64. Marinca, V., Herisanu, N.: Application of optimal homotopy asymptotic method for solvingnonlinear equations arising in heat transfer. Int. Commun.Heat Mass.35, 710 – 715 (2008)

65. Marinca, V., Herisanu, N.: An optimal homotopy asymptotic method applied to the steadyflow of a fourth-grade fluid past a porous plat. Appl. Math. Lett. 22, 245 – 251 (2009)

66. Molabahrami, A., Khani, F. : The homotopy analysis method to solve the Burgers-Huxleyequation. Nonlin. Anal. B.10, 589-600 (2009)

67. Motsa, S.S., Sibanda, P., Shateyi, S.: A new spectral homotopy analysis method for solvinga nonlinear second order BVP. Commun. Nonlinear Sci. Numer.Simulat.15, 2293-2302,20010.

68. Motsa, S.S., Sibanda, P., Auad, F.G., Shateyi, S.: A new spectral homotopy analysis methodfor the MHD Jeffery-Hamel problem. Computer & Fluids.39, 1219 – 1225 (2010)

69. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (2000)70. Niu, Z., Wang, C.: A one-step optimal homotopy analysis method for nonlinear differential

equations. Commun. Nonlinear Sci. Numer. Simulat.15, 2026 – 2036 (2010).71. Pandey, R.K. , Singh, O.P., Baranwal, V.K.: An analytic algorithm for the space – time frac-

tional advection – dispersion equation. Computer Physics Communications.182, 1134 – 1144(2011)

72. Pirbodaghi, T., Ahmadian, M.T., Fesanghary, M.: On the homotopy analysis method for non-linear vibration of beams. Mechanics Research Communications.36, 143 – 148 (2009)

73. Sajid, M.: Similar and Non-Similar Analytic Solutions for Steady Flows of Differential TypeFluids. PhD dissertation, Quaid-I-Azam University (2006)

74. Sajid, M., Hayat, T.: Comparison of HAM and HPM methods innonlinear heat conductionand convection equations. Nonlinear Anal. B.9, 2296-2301 (2008)

75. Sen, S.: Topology and Geometry for Physicists. AcademicPress, Florida (1983)76. Shidfar, A., Babaei, A., Molabahrami, A.: Solving the inverse problem of identifying an un-

known source term in a parabolic equation. Computers and Mathematics with Applications.60, 1209 – 1213 (2010)

References 11

77. Shidfar, A., Molabahrami, A.: A weighted algorithm based on the homotopy analysis method- application to inverse heat conduction problems. Commun.Nonlinear Sci. Numer. Simulat.15, 2908 – 2915 (2010)

78. Siddheshwar, P.G.: A series solution for the Ginzburg-Landau equation with a time-periodic coefficient. Applied Mathematics.3, 542 – 554 ( 2010). Online available athttp://www.SciRP.org/journal/am

79. Singh, O.P., Pandey, R.K., Singh, V.K.: An analytic algorithm of LaneEmden type equationsarising in astrophysics using modified Homotopy analysis method. Computer Physics Com-munications.180, 1116 – 1124 (2009)

80. Song, H., Tao, L.: Homotopy analysis of 1D unsteady, nonlinear groundwater flow throughporous media. J. Coastal Res.50, 292 – 295 (2007)

81. Tao, L., Song, H., Chakrabarti, S.: Nonlinear progressive waves in water of finite depth – ananalytic approximation. Coastal Engineering.54, 825 – 834 (2007)

82. Trefethen, L.N.: Computing numerically with functionsinstead of numbers. Math. in Comp.Sci.1, 9 – 19 (2007).

83. Turkyilmazoglu, M.: Purely analytic solutions of the compressible boundary layer flow dueto a porous rotating disk with heat transfer. Physics of Fluids.21, 106104 (2009)

84. Turkyilmazoglu, M.: A note on the homotopy analysis method. Appl. Math. Lett.23, 1226 –1230 (2010)

85. Turkyilmazoglu, M.: Series solution of nonlinear two-point singularly perturbed boundarylayer problems. Computers and Mathematics with Applications.60, 2109 – 2114 (2010)

86. Turkyilmazoglu, M.: An optimal analytic approximate solution for the limit cycle of Duffing- van der Pol equation. ASME J. Appl. Mech.78, 021005 (2011).

87. Turkyilmazoglu, M.: Numerical and analytical solutions for the flow and heat transfer near theequator of an MHD boundary layer over a porous rotating sphere. Int. J. Thermal Sciences.50 831 – 842 (2011)

88. Turkyilmazoglu, M.: An analytic shooting-like approach for the solution of nonlinear bound-ary value problems. Math. Comp. Modelling.53, 1748 – 1755 (2011)

89. Turkyilmazoglu, M.: Some issues on HPM and HAM methods - Aconvergence scheme.Math. Compu. Modelling.53, 1929 – 1936 (2011)

90. Van Gorder, R.A., Vajravelu, K.: Analytic and numericalsolutions to the Lane-Emden equa-tion. Phys. Lett. A.372, 6060 – 6065 (2008)

91. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Analytical solutions of a coupled nonlinear sys-tem arising in a flow between stretching disks. Applied Mathematics and Computation.216,1513 – 1523 (2010)

92. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Nano boundary layers over stretching surfaces.Commun. Nonlinear Sci. Numer. Simulat.15, 1494 – 1500 (2010)

93. Van Gorder, R.A., Vajravelu, K.: Convective heat transfer in a conducting fluid over a perme-able stretching surface with suction and internal heat generation/absorption. Applied Mathe-matics and Computation.217, 5810 – 5821 (2011)

94. Van del Pol: On oscillation hysteresis in a simple triodegenerator. Phil. Mag.43, 700 – 719(1926)

95. Von Dyke, M.: Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford(1975)

96. Wu, Y.Y.: Analytic Solutions for Nonlinear Long Wave Propagation. PhD dissertation, Uni-versity of Hawaii (2009)

97. Wu, Y.Y., Cheung, K.F.: Explicit solution to the exact Riemann problems and application innonlinear shallow water equations. Int. J. Numer. Meth. Fl.57, 1649 – 1668 (2008)

98. Wu, Y.Y., Cheung, K.F.: Homotopy solution for nonlineardifferential equations in wave prop-agation problems. Wave Motion.46, 1 – 14 (2009)

99. Xu, H., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy-based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls. Physics of Fluids.22,053601 (2010). doi:10.1063/1.3392770

12 1 Introduction

100. Yabushita, K., Yamashita, M., Tsuboi, K.: An analytic solution of projectile motion with thequadratic resistance law using the homotopy analysis method. J. Phys. A – Math. Theor.40,8403 – 8416 (2007)

101. Zand, M.M., Ahmadian, M.T., Rashidian, B.: Semi-analytic solutions to nonlinear vibrationsof microbeams under suddenly applied voltages. J. Sound andVibration. 325, 382 – 396(2009)

102. Zand, M.M., Ahmadian, M.T: Application of homotopy analysis method in studying dynamicpull-in instability of microsystems. Mechanics Research Communications.36, 851 – 858(2009)

103. Zhao, J., Wong, H.Y.: A closed-form solution to American options under general diffusions(2008). Available at SSRN: http://ssrn.com/abstract=1158223.

104. Zhu, J.: Linear and Non-linear Dynamical Analysis of Beams and Cables and Their Combi-nations. PhD dissertation, Zhe Jiang University (2008)

105. Zhu, S.P.: A closed-form analytical solution for the valuation of convertible bonds with con-stant dividend yield. ANZIAM J.47, 477 – 494 (2006)

106. Zhu, S.P.: An exact and explicit solution for the valuation of American put options. Quant.Financ.6, 229 – 242 (2006)

107. Zou, L.: A Study of Some Nonlinear Water Wave Problems Using Homotopy AnalysisMethod. PhD dissertation, Dalian University of Technology(2008)

108. Zou, L., Zong, Z., Wang, Z., He, L.: Solving the discreteKdV equation with homotopyanalysis method. Phys. Lett. A.370, 287 – 294 (2007)

Chapter 2Basic ideas of the homotopy analysis method

Abstract The basic ideas and all fundamental concepts of the homotopyanalysismethod (HAM) are described in details by means of two simple examples, includ-ing the concept of the homotopy, the flexibility of constructing equations for con-tinuous variations, the way to guarantee convergence of solution series, the essenceof the convergence-control parameterc0, the methods to accelerate convergence,and so on. The corresponding Mathematica codes are given in appendixes and freeavailable online. Beginners of the HAM are strongly suggested to read it first.

2.1 Concept of homotopy

The homotopy analysis method (HAM) [10–23,32] proposed by Shijun Liao [10] in1992 is based on the concept of the homotopy, a fundamental concept in topologyand differential geometry [3, 29]. The concept of the homotopy can be traced backto Jules Henri Poincare (1854 - 1912), a French mathematician. Shortly speaking, ahomotopy describes a kind ofcontinuousvariation or deformation in mathematics.For example, a circle can becontinuouslydeformed into a square or an ellipse,the shape of a coffee cup can deform continuously into the shape of a doughnut.However, the shape of a coffee cup can not be distorted continuously into the shapeof a football. Essentially, a homotopy defines a connection between different thingsin mathematics, which contain same characteristics in someaspects.

For example, the two different real functions sin(πx) and 8x(x−1) in the intervalx∈ [0,1] can be connected by constructing such a family of functions

H (x;q) = (1−q)sin(πx)+q [8x(1− x)] , (2.1)

whereq ∈ [0,1] is called the embedding parameter. Note thatH (x;q) dependson not only the independent variablex ∈ [0,1] but also the embedding parameterq∈ [0,1]. Especially, whenq= 0, we have

13

14 2 Basic ideas of the homotopy analysis method

H (x;0) = sin(πx), x∈ [0,1],

and whenq= 1, it holds

H (x;1) = 8x(x−1), x∈ [0,1],

respectively. So, as the embedding parameterq ∈ [0,1] increases from 0 to 1, thereal functionH (x;q) variescontinuouslyfrom a trigonometric function sin(πx)to a polynomial 8x(x− 1), as shown in Fig. 2.1. In topology,H (x;q) is called ahomotopy, sin(πx) and 8x(x−1) are calledhomotopic, denoted by

H : sin(πx)∼ 8x(x−1).

Fig. 2.1 Continuous defor-mation of the homotopyH (x;q) : sin(πx)∼ 8x(x−1).Dashed line:q = 0; Dot-dashed line:q = 1/4; Solidline: q= 1/2; Dot-dot-dashedline: q = 3/4; Long-dottedline: q= 1.

x0 0.2 0.4 0.6 0.8 1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

q = 0

q = 0.25

q = 0.5

q = 0.75

q = 1

Let C[a,b] denote a set of all continuous real functions in the intervala≤ x≤ b.In general, if a continuous functionf ∈ C[a,b] can be deformed continuously intoanother continuous functiong∈C[a,b], one can construct a homotopy

H : f (x)∼ g(x)

in the wayH (x;q) = (1−q) f (x)+q g(x), x∈ [a,b]. (2.2)

However, a continuous real function cannot be deformed continuously into a dis-continuous function. For example, sin(x) can not be deformed continuously into thestep function

s(x) =

1, whenx< 0,0, whenx= 0,

−1, whenx> 0.

2.1 Concept of homotopy 15

Definition 2.1. A homotopybetween two continuous functionsf (x) andg(x)from a topological spaceX to a topological spaceY is formally defined to be acontinuous functionH : X× [0,1]→Y from the product of the spaceX with theunit interval[0,1] to Y such that, ifx ∈ X thenH (x;0) = f (x) andH (x;1) =g(x).

Fig. 2.2 Continuous deforma-tion of the solutiony(x;q) ofthe homotopy equation (2.3).Solid line:q= 0; Dashed line:q = 1/4; Dash-dotted line:q = 1/2; Dash-dot-dottedline: q= 1.

x

y

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Since curves can be defined by algebraic or differential equations, the conceptof homotopy defined above for functions can be easily expanded to equations. Forexample, let us consider such a family of algebraic equations

E (q) : (1+3q) x2+y2

(1+3q)= 1, q∈ [0,1], (2.3)

whereq∈ [0,1] is the embedding parameter. Whenq= 0, we have the circle equa-tion

E0 : x2+ y2 = 1, (2.4)

whose solution is a circley=±√

1− x2. Whenq= 1, we have the ellipse equation

E1 : 4x2+ y2/4= 1, (2.5)

whose solution is an ellipsey= ±2√

1−4x2. Thus, as the embedding parameterqincreases from 0 to 1, Equation (2.3) varies continuously from the circle equationE0 into the ellipse equationE1, while its solutiony deforms continuously from thecircle y= ±

√1− x2 to the ellipsey= ±2

√1−4x2, as shown in Fig. 2.2. So, more

precisely speaking, the solutiony of (2.3) is dependent not only onx but also onq∈ [0,1], and thus (2.3) should be expressed more precisely in the form

E (q) : (1+3q) x2+y2(x;q)(1+3q)

= 1, q∈ [0,1], (2.6)


which definestwohomotopies: one is thehomotopy of equation

E (q) : E0 ∼ E1,

whereE0 andE1 denote (2.4) and (2.5), respectively, the other is thehomotopy offunction

y(x;q) : ±√

1− x2 ∼±2√

1−4x2.

In other words, the solutiony(x;q) of (2.6) is also a homotopy. Notice that such kindof continuous deformation is completely defined by (2.6). For simplicity, we call(2.6) the zeroth-order deformation equation. The same idea can be easily extendedto other types of equations, such as differential equations, integral equations and soon, as shown later in this book.

Definition 2.2. The embedding parameterq∈ [0,1] in a homotopy of functionsor equations is calledhomotopy-parameter.

Definition 2.3. Given an equation denoted byE1, which has at least one so-lution u. Let E0 denote a proper, simpler equation, calledthe initial equation,whose solutionu0 is known. If one can construct such a homotopy of equationE (q) : E0 ∼ E1 that, as the homotopy-parameterq ∈ [0,1] increases from 0 to1, E (q) deforms (or varies)continuouslyfrom the the initial equationE0 to thethe original equationE1, while its solution variescontinuouslyfrom the knownsolutionu0 of E0 to the unknown solutionu of E1, then this kind of homotopy ofequations is calledthe zeroth-order deformation equation.

Note that we can construct many different homotopies which connect the circleequation (2.4) and the ellipse equation (2.5). For example,the following zeroth-order deformation equation

E (q,µ) : (1+3 qµ) x2+y2(x;q)

(1+3 qµ)= 1, q∈ [0,1], (2.7)

whereµ > 0 is a constant, defines a two-parameter (q,µ) family of homotopy

E (q,µ) : E0 ∼ E1,

whereE0 andE1 denote (2.4) and (2.5), respectively. For different valuesof µ , itdefines a different homotopy. Sinceµ ∈ (0,+∞), there exists an infinite number ofdifferent homotopies of equations which connect the circleequation (2.4) and theellipse equation (2.5), and correspondingly, an infinite number of homotopies offunctions which connect the circley= ±

√1− x2 and the ellipsey= ±2

√1−4x2.

This illustrates the great flexibility of constructing a homotopy for given two homo-

2.2 Example 2.1: generalized Newtonian iteration formula 17

topic functions or equations. All of these belong to the basic concepts in topologyand differential geometry [3,29].

Based on the homotopy mentioned above, some new concepts canbe derived.Note that the homotopy

H (x;q) = (1−q)sin(πx)+q [8x(x−1)]

can be rewritten in the form

H (x;q) = sin(πx)+ [8x(x−1)− sin(πx)] q,

and thus we have

∂H (x;q)∂q

= 8x(x−1)− sin(πx), q∈ [0,1], (2.8)

which describes theratio (or thespeed) of the continuous deformation from sin(πx)to 8x(x−1), called the1st-order homotopy-derivative. In general, the homotopy

H (x;q) = (1−q) f (x)+q g(x), x∈ [a,b]

completely defines the corresponding 1st-order homotopy-derivative

∂H (x;q)∂q

= g(x)− f (x), q∈ [0,1]. (2.9)

Generalizing this concept, we can further define the high-order homotopy-derivatives,as shown later.

Since topology is not a common course for undergraduate students, most readersare unfamiliar with the concept of homotopy. Fortunately, the HAM is based on thesimple fundamental concept of homotopy only, and other knowledge in topology arealmost unnecessary, as shown later in this book. Thus, it is easy to understand andapply the HAM to gain analytic approximations of nonlinear differential equationsin practice. In the following part of this chapter, we will use two simple examples todescribe the basic ideas of the HAM in details. One is a nonlinear algebraic equation,the other is a 2nd-order differential equation for periodicoscillations.

2.2 Example 2.1: generalized Newtonian iteration formula

Let us first consider a nonlinear algebraic equation

f (x) = 0,

wheref (x) ∈C∞[a,b] is a continuous real function. Assume that the above equationhas at least one solution in the regionx∈ [a,b].


Let x0 ∈ [a,b] denote an initial guess of the unknown solutionx. Obviously,f (x)− f (x0) ∈ C∞[a,b] can be deformed continuously tof (x) ∈ C∞[a,b], i.e. theyare homotopic. Thus, we can construct such a homotopy of function

H (x;q) = (1−q) [ f (x)− f (x0)]+q f(x),

whereq∈ [0,1] is the homotopy-parameter. Whenq= 0 andq= 1, we have

H (x;0) = f (x)− f (x0), H (x;1) = f (x),

respectively. Thus, asq increases from 0 to 1,H (x;q) varies continuously fromf (x)− f (x0) to f (x). Such kind of continuous variation is calleddeformationintopology [29].

Now, enforcingH (x;q) = 0, i.e.

(1−q) [ f (x)− f (x0)]+q f(x) = 0, q∈ [0,1],

we have now a parameter-family of algebraic equations. The solution of the aboveparameter-family of algebraic equations is dependent uponthe homotopy-parameterq. Replacingx by x(q), this family of equations can be rewritten more precisely inthe form

(1−q) f [x(q)]− f (x0)+q f [x(q)] = 0. (2.10)

Whenq= 0, we havef [x(0)]− f (x0) = 0,

whose solution isx(0) = x0.

Whenq= 1, we havef [x(1)] = 0,

which is exactly the same as the original algebraic equationf (x) = 0. So, it holds

x(1) = x.

Therefore, as the homotopy-parameterq increases from 0 to 1, ˜x(q) deforms fromthe initial guessx0 to the solutionx of the original equationf (x) = 0. So, Equation(2.10) defines a homotopy of function ˜x(q) : x0 ∼ x. For simplicity, the parameter-family of equations (2.10) is calledthe zeroth-order deformation equation, becauseit defines a continuous deformation from the initial guessx0 to the solutionx of theoriginal equationf (x) = 0.

Note that ˜x(q) is a function of the homotopy-parameterq. Assume that ˜x(q) isanalyticat q= 0 so that it can be expanded into a Maclaurin series with respect tothe homotopy-parameterq, i.e.

x(q)∼ x0++∞

∑k=1

xk qk, (2.11)


wherex(0) = x0 is employed, and

xk =1k!

dkx(q)dqk

∣

∣

∣

∣

q=0= Dk [x(q)] . (2.12)

Here, the series (2.11) is calledhomotopy-Maclaurin series, Dk is calledthe homotopy-derivative operator, andDk [x(q)] is called the kth-order homotopy-derivativeofx(q). We will give more rigorous definitions and some related theorems about theoperatorDk in Chapter 3. Note thatD1 [x(q)] = x1 is exactly the above-mentioneddeformation-ratio of ˜x(q) atq= 0. So,Dk [x(q)], thekth-order homotopy-derivative(k≥ 1) of x(q), can be regarded as the high-order deformation-ratio of ˜x(q) atq= 0.

Assuming that ˜x(q) is analyticin q∈ [0,1], then the homotopy– Maclaurin series(2.11) is convergent atq= 1 to x(1). Thus, using the relationship ˜x(1) = x, we havefrom (2.11) the so-calledhomotopy-series solution

x= x0++∞

∑k=1

xk. (2.13)

It should be emphasized that we had to make such an assumptionhere, because aMaclaurin series of a functionf (x) may not converge tof (x). The assumption maycome true by means of properly constructing the zeroth-order deformation equation,as shown later. In practice, only finite terms can be obtained, and theMth-orderapproximation ofx is given by

x≈ x0+M

∑k=1

xk. (2.14)

So, as long asx1,x2, · · · ,xM become known, we obtain theMth-order approximationof x by means of the above formula.


Definition 2.4. Given a nonlinear equation denoted byE1, which has at leastone solutionu(z, t), wherez andt denote the spatial and temporal independentvariables, respectively. Letq ∈ [0,1] denote a homotopy-parameter andE (q)the zeroth-order deformation equation, which connects theoriginal equationE1

and an initial equationE0 with the known initial approximationu0(z, t). Assum-ing that the zeroth-order deformation equationE (q) is so properly constructedthat its solutionφ(z, t;q) exists and is analytic atq= 0, we have thehomotopy-Maclaurin series:

φ(z, t;q)∼ u0(z, t)++∞

∑n=1

un(z, t) qn, q∈ [0,1]

and thehomotopy-series:

φ(z, t;1)∼ u0(z, t)++∞

∑n=1

un(z, t).

The equations related to the unknownun(z, t) are calledthe nth-order deforma-tion equations.

Definition 2.5. If the solutionφ(z, t;q) of the zeroth-order deformation equationE (q) : E0 ∼ E1 exists and is analytic aboutq in q ∈ [0,1], then we have thehomotopy-series solutionof the original equationE1:

u(z, t) = u0(z, t)++∞

∑n=1

un(z, t),

and theMth-order homotopy-approximation

u(z, t)≈ u0(z, t)+M

∑n=1

un(z, t).

According to the fundamental theorem in calculus about Taylor series, the coeffi-cientxk of the homotopy-Maclaurinseries (2.11) is unique. Therefore, the governingequation ofxk is unique, too, and can be deduced directly from the zeroth-order de-formation equation (2.10). First, differentiating the zeroth-order deformation equa-tion (2.10) with respect to the homotopy-parameterq, we have

f ′[x(q)]dx(q)

dq+ f (x0) = 0. (2.15)

Then, settingq= 0 in the above equation and using the relationship ˜x(0) = x0, wehave


f ′(x0)dx(q)

dq

∣

∣

∣

∣

q=0+ f (x0) = 0.

According to the definition (2.12), we have

dx(q)dq

∣

∣

∣

∣

q=0= x1.

Thus, we obtain the so-called1st-order deformation equation

x1 f ′(x0)+ f (x0) = 0,

whose solution is

x1 =− f (x0)

f ′(x0).

Similarly, differentiating the zeroth-order deformationequation (2.10) twice withrespect to the homotopy-parameterq and dividing it by 2!, we have

12

f ′′(x)

(

dxdq

)2

+ f ′(x)

(

12!

d2xdq2

)

= 0. (2.16)

Then, settingq= 0 and using ˜x(0) = x0, the above equation reads

12

f ′′(x0)

(

dxdq

∣

∣

∣

∣

q=0

)2

+ f ′(x0)

(

12!

d2xdq2

∣

∣

∣

∣

q=0

)

= 0.

Using the definition (2.12), we have the2nd-order deformation equation

12

x21 f ′′(x0)+ x2 f ′(x0) = 0,

whose solution is

x2 =−x21 f ′′(x0)

2 f ′(x0)=− f 2(x0) f ′′(x0)

2[ f ′(x0)]3.

Alternatively, using the linear operatorDk defined by (2.12) and directly taking the2nd-order homotopy-derivative on both sides of the zeroth-order deformation equa-tion (2.10), we obtain exactly the same 2nd-order deformation equation as mentionabove. In this way, we obtainxk one by one in the orderk = 1,2,3, · · ·. In general,given a zeroth-order deformation equation, it is straightforward to give the corre-sponding high-order deformation equations, as described in Chapter 3. Here, wewould like to emphasize that all of above high-order deformation equations arelin-ear, and therefore are easy to solve.

Using (2.14), we have the1st-order homotopy-approximation

x≈ x0+ x1 = x0−f (x0)

f ′(x0), (2.17)


and the2nd-order homotopy-approximation

x≈ x0+ x1+ x2 = x0−f (x0)

f ′(x0)− f 2(x0) f ′′(x0)

2[ f ′(x0)]3. (2.18)

Note that (2.17) is exactly the famous Newton’s iteration formula given by Sir IsaacNewton (1643 -1727), and (2.18) is the Olver’s iteration formula. In fact, one cangive a family of iteration formulas in a similar way. This shows the potential of thehomotopy approach.

The above approach has some interesting characteristics. First of all, it is in-dependentof any physical parameters at all: no matter whether a nonlinear prob-lem contains small/large physical parameters or not, one can always introduce thehomotopy-parameterq∈ [0,1] to construct a zeroth-order deformation equation andthen to obtain the homotopy-series solution or a homotopy approximation. Sec-ondly, as mention above, all high-order deformation equations are linear and thusare easy to solve. In this way, this approach transforms a nonlinear equation intoan infinite number of linear sub-problems, butwithoutany small/large physical pa-rameters. Giving up the dependence on small/large physicalparameters, the HAMcan be applied to solve more complicated nonlinear problems, as shown later in thisbook.

However, the above approach has a limitation: the convergence of homotopy-Maclaurin series (2.11) atq = 1 is not guaranteed, therefore the homotopy-seriessolution (2.13) might be divergent. This is mainly because the above approach isbased on such an assumption that ˜x(q) is analyticin q∈ [0,1] so that the homotopy-Maclaurin series (2.11) is convergent atq = 1, but it does not provide a way toguarantee that such an assumption indeed holds, especiallyfor nonlinear problemswith strong nonlinearity. This explains why the famous Newton’s iteration formula(2.17) and Olver’s iteration formula (2.18) often fail.

To overcome this limitation of the early HAM mentioned above, Liao [11] greatlymodified the approach by introducing a non-zero auxiliary parameterc0, called nowthe convergence-control parameter1, to construct such a more generalized zeroth-order deformation equation

(1−q) f [x(q)]− f (x0)= q c0 f [x(q)]. (2.19)

Sincec0 6= 0, whenq= 1, the above equation becomes

c0 f [x(1)] = 0,

which is equivalent to the original equationf (x) = 0, providedx= x(1). All otherformulas like (2.11) and (2.13) are the same, except the high-order deformationequation, which can be derived as follows.

1 The same non-zero auxiliary parameter was denoted byh when Liao [11] first introduced it intothe frame of the HAM in 1997. However, sinceh has a special meaning in quantum mechanics, wereplaceh by c0 in the whole book, which means the “basic” convergence-control parameter.


Taking the 1st-order homotopy-derivativeon both sides of the zeroth-order defor-mation equation (2.19), i.e. differentiating (2.19) with respect toq and then settingq= 0, we have the corresponding 1st-order deformation equation

x1 f ′(x0)− c0 f (x0) = 0,

whose solution is

x1 = c0f (x0)

f ′(x0).

Taking the 2nd-order homotopy-derivative on both sides of the zeroth-order defor-mation equation (2.19), i.e. differentiating (2.19) twicewith respect toq and thensettingq= 0 and dividing by 2!, we have the 2nd-order deformation equation

x2 f ′(x0)− (1+ c0)x1 f ′(x0)+12

x21 f ′′(x0) = 0,

whose solution is

x2 = (1+ c0)x1−x2

1 f ′′(x0)

2 f ′(x0)= c0(1+ c0)

f (x0)

f ′(x0)− c2

0 f 2(x0) f ′′(x0)

2[ f ′(x0)]3.

Similarly, we have

x3 = c0(1+ c0)2 f (x0)

f ′(x0)− c2

0(1+ c0)f 2(x0) f ′′(x0)

[ f ′(x0)]3

+c3

0 f 3(x0)

3[ f ′′(x0)]2− f ′(x0) f ′′′(x0)

6[ f ′(x0)]5,

and so on.Then, according to (2.14), we have the corresponding first-order homotopy-

approximation

x≈ x0+ x1 = x0+ c0f (x0)

f ′(x0), (2.20)

the 2nd-order homotopy-approximation

x≈ x0+ x1+ x2 = x0+ c0(c0+2)f (x0)

f ′(x0)− c2

0 f 2(x0) f ′′(x0)

2[ f ′(x0)]3, (2.21)

and the 3rd-order homotopy-approximation

x ≈ x0+ x1+ x2+ x3

= x0+ c0(c20+3c0+3)

f (x0)

f ′(x0)− c2

0 (2c0+3)f 2(x0) f ′′(x0)

2[ f ′(x0)]3

+c3

0 f 3(x0)

3[ f ′′(x0)]2− f ′(x0) f ′′′(x0)

6[ f ′(x0)]5. (2.22)


By means of symbolic computations software, it is easy to derive an iteration for-mula at any given order of approximation.

It is interesting that Newton’s iteration formula (2.17) and Olver’s iteration for-mula (2.18) are only special cases of (2.20) and (2.21) in case of c0 = −1, respec-tively. In practice, the convergence-control parameterc0 in (2.17) can be regarded asan iteration factor, which is widely used in numerical computations to modify New-ton’s iteration formula (2.17) and Olver’s iteration formula (2.18). It is well-knownthat a properly chosen iteration factor can greatly modify the convergence of manynumerical iteration approaches. Similarly, the convergence-control parameterc0 cangreatly modify the convergence of the homotopy-series solution. This is indeed true.We will show this point in details later in this book: it is theconvergence-controlparameterc0 that provides us a simple way to ensure the convergence of homotopy-series solution. That is the reason why we callc0 the convergence-controlparameter.In this way, the above mentioned limitation of the early HAM is overcome. Indeed,the convergence-parameterc0 in the zeroth-order deformation equation introducedby Liao [11] greatly modifies the early HAM. We will show this point in details bymeans of Example 2.

The Pade technique can be applied to give a new iteration formula. Regardingq as an independent variable, then using [1,1] Pade approximant aboutq to thehomotopy-series

x(q)≈ x0+ x1 q+ x2 q2+ x3 q3+ · · · ,and finally settingq= 1, we have the iteration formula

x≈ x0+2 f (x0) f ′(x0)

f (x0) f ′′(x0)−2[ f ′(x0)]2. (2.23)

It is very interesting that the above formula isindependentof the convergence-control parameterc0. The so-called homotopy-Pade technique mentioned above pro-vides us an alternative way to modify the convergence of homotopy series solution,as shown later in details in§ 2.3.5.

2.3 Example 2.2: nonlinear oscillation

The approach described above works also for other types of nonlinear equations.For example, let us consider here a body moving on a smooth horizontal plane actedby a horizontal forcef . Let m denote the mass of the body,t the time, andx(t) thehorizontal co-ordinate of the body, as shown in Fig. 2.3. Assume that there is nofriction force between the body and the plane. According to Newtow’s second law,the motion of the body is described by

mx(t) = f ,

2.3 Example 2.2: nonlinear oscillation 25

where the dot denotes the differentiation with respect to the timet. For simplicity,we consider here such a case

x(0) = x∗, x(0) = 0, m= 1, f =−(λx+ εx3)

that the motion of the body is described by

x(t)+λx(t)+ εx3(t) = 0, x(0) = x∗, x(0) = 0, (2.24)

whereλ ∈ (−∞,+∞) andε ∈ (−∞,+∞) are constant physical parameters.

Fig. 2.3 A body moving on asmooth horizontal plane actedby a horizontal forcef

o

f x

m

2.3.1 Analysis of the solution characteristic

Let us first consider the case ofε = 0, i.e.

x(t)+λx(t) = 0, x(0) = x∗, x(0) = 0, (2.25)

whose solution is

x(t) =

x∗ cos(√

λ t), whenλ > 0,x∗, whenλ = 0,x∗ cosh(

√

|λ | t), whenλ < 0.(2.26)

As shown in Fig. 2.4, the solutionx(t) has quite different characteristic for differentvalues ofλ : it is a periodic function whenλ > 0 but quickly tends to infinity whenλ < 0.

We can explain this from the physical points of view. In general, the equilibriumpoint of a dynamic system is defined byf = 0. Enforcing f = −λ x= 0, we havethe equilibrium pointx = 0 for both λ > 0 andλ < 0. Whenλ > 0, the forcef = −λ x = −|λ | x always points tox = 0 so thatx = 0 is a stable equilibriumpoint, thus the body oscillates aroundx= 0 and is expressed by a periodic function.


However, whenλ < 0, the forcef = −λx= |λ | x never points to the equilibriumpointx= 0 so thatx= 0 is an unstable equilibrium point, and thus the body departsfrom the starting pointx = x∗ farther and farther, and will never return, i.e.x(t) isexpressed by an exponential function. Whenλ = 0, the body is still, because thereis no force acted on the body. These are the physical reasons why the solutionx(t)has quite different characteristic forλ > 0, λ = 0 andλ < 0. Note that,

√λ denotes

the frequency of oscillation in case ofλ > 0, and√

|λ | denotes the escaping-speedof the body from the starting-point in case ofλ < 0.

Fig. 2.4 Solutions of equationx+λ x= 0 with x(0) = 1 andx(0) = 0. Solid line: whenλ = 9/4; Dashed line: whenλ = 0; Dash-dotted line: whenλ =−9/4.

t

x(t)

0 2 4 6 8 10

0

2

4

6

8

λ < 0

λ = 0

λ > 0

In summary, the characteristic of solution and the equilibrium point of linearequation ¨x+λx= 0 are listed in the Table 2.1.

Table 2.1 The characteristic of the solution and the equilibrium point of x+λx= 0 with x(0) =x∗, x(0) = 0

λ > 0 λ = 0 λ ≤ 0

Solution expression periodic constant exponential

Equilibrium point x= 0, stable x= x∗, neutral x= 0, unstable

In case ofε 6= 0, the equilibrium point is determined by

f =−λx− ε x3 =−ε x

(

λε+ x2

)

= 0. (2.27)

When λ/ε ≥ 0, Equation (2.27) has only one real solutionx = 0. Whenλ/ε <0, it has three real solutionsx = 0 andx = ±

√

|λ/ε| . Physically, we have oneequilibrium pointx= 0 in case ofλ/ε ≥ 0, but have three equilibrium pointsx= 0andx= ±

√

|λ/ε| in case ofλ/ε < 0. The characteristics of the equilibrium pointand solution for different values ofε andλ are listed below:


• In case ofε > 0 andλ ≥ 0, the forcef = −(λ + ε x2)x always points to theunique equilibrium pointx = 0, so thatx = 0 is a stable equilibrium point andthus the body always oscillates aboutx= 0, i.e.x(t) is a periodic function.

• In case ofε > 0 butλ < 0, the force

f =−λ x− ε x3 = |λ | x− ε x3 =−(

ε x2−|λ |)

x

points to either the equilibrium pointx = +√

|λ |/ε or the equilibrium pointx = −

√

|λ |/ε. Thus, we have two stable equilibrium pointsx = ±√

|λ |/ε butone unstable equilibrium pointx= 0. So, the body always oscillates about one ofthe stable equilibrium pointsx= ±

√

|λ |/ε, and the solution is a periodic func-tion.

• In case ofε < 0 andλ > 0, the force

f =−λx− ε x3 = |ε| x

(

x2−∣

∣

∣

∣

λε

∣

∣

∣

∣

)

points to the equilibrium pointx= 0 when|x∗|<√

|λ/ε|, but never points to anyequilibrium points when|x∗|>

√

|λ/ε|. So,x= 0 is a stable equilibrium point,andx=±

√

|λ/ε| are two unstable equilibrium points. Thus, the body oscillatesabout the stable equilibrium pointx = 0 when|x∗| <

√

|λ/ε|, but departs fromthe starting point to infinity when|x∗|>

√

|λ/ε|.

• In case ofε < 0 andλ ≤ 0, the forcef = −(λ + ε x2)x = (|λ |+ |ε|x2)x neverpoints to the unique equilibrium pointx= 0, so thatx= 0 is a unstable equilib-rium point and thus the body departs from the starting point to infinity.

Table 2.2 Characteristic of equilibrium point of the nonlinear dynamic system (2.24)

ε > 0 ε < 0

λ ≥ 0 one equilibrium point: three equilibrium points:x= 0, stable x= 0, stable;

x=±√

|λ/ε |, unstable

λ < 0 three equilibrium point: one equilibrium point:x= 0, unstable; x= 0, unstablex=±

√

|λ/ε | , stable

The characteristic of the equilibrium point and solution expression of the nonlin-ear dynamic system (2.24) for different values ofε andλ are listed in Tables 2.2and 2.3, respectively. Note that, in case ofε > 0, the body always oscillates around


Table 2.3 Characteristic of the solution expression of the nonlineardynamic system (2.24)

ε > 0 ε < 0

λ ≥ 0 periodic periodic: when|x∗|<√

|λ/ε |exponential: when|x∗|>

√

|λ/ε |

λ < 0 periodic exponential

a stable equilibrium point, forarbitrary value ofλ , so thatx(t) is always a periodicfunction. Besides, in case ofε < 0, there exists a periodic solution whenλ > 0 and|x∗| <

√

|λ/ε|. These periodic solution describe the periodic oscillations arounda stable equilibrium point. In other cases, the solution is an exponential function,describing the quick escaping of the body from the starting point to infinity.

Therefore, dependent on the physical parametersλ ,ε and the starting positionx∗,there are two different types of solutions for the nonlineardynamic system (2.24).One is a periodic function, describing the periodic oscillation around a stable equi-librium point. In this case, the important physical quantity is the frequency of themotion, denoted byω . The other is a non-periodic function, which tends to infin-ity exponentially ast → +∞, describing the quick escaping of the body from thestarting pointx∗ to infinity. In this case, there exists an important physicalquantity,denoted byµ , which describes the escaping speed of the body from the startingpoint x∗ to infinity. Each ofω andµ corresponds to a kind of time scale, so thatwe can define a dimensionless time variavleτ = ω t for the periodic function, orτ = µ t for the non-periodic function, respectively. Although we now do not knowthe details of the solution of the nonlinear dynamic system (2.24), we are quite surethat the periodic solution can be expressed by

x= a0++∞

∑k=1

ak cos(kτ), τ = ω t, (2.28)

and the non-periodic solution can be expressed by

x= b0++∞

∑k=−∞

bk exp(kτ), τ = µ t, (2.29)

respectively, whereak,bk are constant to be determined. Our aim is to find conver-gent series in above forms for all possible physical parameters−∞ < λ < +∞ and−∞< ε <+∞. For simplicity, we call (2.28)the solution expressionfor periodic os-cillations of the nonlinear dynamic system (2.24), and (2.29) the solution expressionfor non-periodic ones, respectively.


Definition 2.6. Given a differential equationE that has at least one solutionu. Ifthere exists such a complete set of base functionsek (0≤ k< ∞) that the series+∞∑

k=0ak ek converges to the solutionu, thenu =

+∞∑

k=0ak ek is called thesolution

expressionof the given equationE .

Based on the above analysis, we define two different sets of functions

Sp :=

x(τ)|x(τ) = a0++∞

∑k=1

ak cos(kτ), ak is constant

(2.30)

and

Se :=

x(τ)|x(τ) = b0++∞

∑k=−∞

bk exp(kτ), bk is constant

(2.31)

Obviously, if f1 ∈ Sp and f2 ∈ Sp are two periodic functions, thenf1 can be con-tinuously deformed intof2. So, f1 ∈ Sp and f2 ∈ Sp are homotopic. Similarly, twoexponential functionsf1 ∈ Se and f2 ∈ Se are homotopic, too. However, it is ratherdifficult (or even impossible) for a periodic functionf1 ∈ Sp to be deformed into anexponential functionf2 ∈ Se.

The above analysis of the characteristic of the solution andthe equilibrium pointof the nonlinear dynamic system (2.24) is rather helpful forthe choice of the initialguess ofx(t) and the auxiliary linear operatorL in the frame of the HAM, asmentioned below.

When the starting positionx(0) = x∗ is at the equilibrium point, the solution isx(t) = x∗. So,x = x∗ is assumed to be not the equilibrium point. In addition, onlythe periodic solutions are considered for the sake of simplicity. As mentioned above,the periodic solutions exist whenε > 0, orε < 0,λ > 0 and|x∗|<

√

|λ/ε|.Let ω andT = 2π/ω denote the frequency and the period of the solutionx(t),

respectively. Using the transformationτ = ω t, Equation (2.24) becomes

γ x′′(τ)+λ x(τ)+ ε x3(τ) = 0, x(0) = x∗,x′(0) = 0, (2.32)

where the prime denotes differentiation with respect toτ, andγ = ω2 is a unknownconstant, which depends onε, λ and x∗. As mentioned above, although the fre-quency squareγ is unknown, we are quite sure thatx(τ) is a function with the knownperiod 2π , expressed by (2.28), i.e.x(τ) ∈ Sp. Our aim is to give such kind of a se-ries solution convergent forall possible physical parametersλ ,ε and the startingpointx∗.


2.3.2 Mathematical formulations

Let x0(τ) denote the initial approximation ofx(τ). According to the solution expres-sion (2.28) and considering the initial conditions in (2.24), we choose

x0(τ) = β +(x∗−β ) cosτ, (2.33)

wherex = β corresponds to the stable equilibrium point near the starting positionx= x∗. According to Tables 2.2 and 2.3, we have

β =

0 , whenε > 0 andλ ≥ 0,+√

|λ/ε| , whenε > 0,λ < 0 andx∗ > 0,−√

|λ/ε| , whenε > 0,λ < 0 andx∗ < 0,0 , whenε < 0,λ ≥ 0 and|x∗|<

√

|λ/ε|.

(2.34)

Note that the above definedx0(τ) satisfies the initial conditionsx(0) = x∗ andx′(0) = 0 of the nonlinear dynamic system (2.32).

Let L denote an auxiliary linear operator with the propertyL [0] = 0. We willshow how to chooseL later. Here, we just emphasize that we have great freedomto choose the so-called auxiliary linear operatorL .

Letc0 denote the convergence-controlparameter,q∈ [0,1] the homotopy-parameter,respectively. For the sake of simplicity, we define such a nonlinear operator

N x= γ x′′(τ)+λ x(τ)+ ε x3(τ). (2.35)

Note thatx0 ∈ Sp is a periodic function, whereSp is defined by (2.30). Obviously,if x(τ) ∈ Sp, thenc0 N [x(τ)] ∈ Sp. Assume thatL is properly chosen so that

L [x(τ)− x0(τ)] ∈ Sp, if x(τ) ∈ Sp. (2.36)

Thus,L [x(τ)− x0(τ)] andc0 N [x(τ)] are periodic functions with the same periodand therefore can be deformed into each other. So, we can construct such a homo-topy of functions

H (x;q) := (1−q)L [x(τ)− x0(τ)]− c0 q N [x(τ)]. (2.37)

Whenq= 0 andq= 1, we have respectively

H (x,0) := L [x(τ)− x0(τ)] , whenq= 0, (2.38)

H (x;1) := −c0 N [x(τ)], whenq= 1. (2.39)

Thus, asq increases from 0 to 1, the homotopyH (x,q) continuouslychanges (ordeforms) from the periodic function

L [x(τ)− x0(τ)] ∈ Sp

to the periodic function


−c0 N [x(τ)] ∈ Se.

Then, enforcingH (x;q) = 0,

we have a two-parameter (q,c0) family of differential equations

(1−q)L [x− x0(τ)] = c0 q N [x],

i.e.(1−q)L [x− x0(τ)] = c0 q

γ x′′+λ x+ ε x3 , (2.40)

subject to the initial conditions

x= x∗, x′ = 0, whenτ = 0, (2.41)

where the dot denotes the differentiation with respect toτ. Obviously, the solutionx of the above dynamic system is dependent not only on the dimensionless timeτ ≥ 0 but also on the homotopy-parameterq∈ [0,1] that has no physical meaning.So,x should be expressed more precisely by ˜x(τ;q). Note thatγ = ω2 is aunknownconstant in the original equation (2.32), soγ in (2.40) can be regarded as a constant,too. However, because we have great freedom to construct thefamily of differentialequation (2.40), we can also regardγ as a function ofq, denoted byγ(q), whichvaries continuously fromγ(0) = γ0 = ω2

0 to γ(1) = γ = ω2, whereω0 is the initialguess of the unknown frequencyω . In other words, we regardγ(q) as a kind ofhomotopy, denoted byγ(q) : γ0 ∼ γ, which connects the initial guessγ0 = ω2

0 andthe unknown quantityγ = ω2. We will explain later why we should regardγ as sucha kind of continuous function ofq. Here, we just point out that it is necessary inorder to avoid the so-called secular terms such asτ cos(τ).

Then, replacingx andγ in (2.40) by x(τ;q) and γ(q), respectively, we have atwo-parameter (q,c0) family of differential equationE (q):

(1−q)L [x(τ;q)− x0(τ)] = c0 q

γ(q) x′′(τ;q)+λ x(τ;q)+ ε x3(τ;q)

, (2.42)


x(0;q) = x∗, x′(0;q) = 0, (2.43)

where the prime denotes the differentiation with respect toτ. Whenq= 0, we havethe equation

E0 : L [x(τ;0)− x0(τ)] = 0, x(0;0) = x∗, x′(0;0) = 0. (2.44)

According to the linear property ofL , i.e.L [0] = 0, and using the fact thatx0(τ)satisfies the initial conditions, it is obvious that

x(τ;0) = x0(τ). (2.45)


Whenq= 1, sincec0 6= 0, Equations (2.42) and (2.43) are equivalent to the equation

E1 : γ(1) x′′(τ;1)+λ x(τ;1)+ ε x3(τ;1) = 0, x(0;1) = x∗, x′(0;1) = 0, (2.46)

which is exactly the same as the original equation (2.32), provided

x(τ;1) = x(τ), γ(1) = γ. (2.47)

So, as the homotopy-parameterq∈ [0,1] increases from 0 to 1, the equation-familyE (q) varies from the equationE0 into the equationE1, while x(τ;q) deformscon-tinuouslyfrom the initial guessx0(τ) to the unknown solutionx(τ) of (2.32), sodoesγ(q) from its initial guessγ0 = ω2

0 to the unknown quantityγ = ω2. In otherwords,E (q) is a homotopy of equations, denoted byE (q) : E0 ∼ E1, andx(τ;q) isa homotopy of functions, denoted by ˜x(τ;q) : x0(τ) ∼ x(τ). Note that,γ(q) : γ0 ∼ γhas been defined as a homotopy, as mentioned above. Note also that both ofγ0 andγ are unknown now, which will be determined later. This kind ofcontinuous vari-ations is called deformation in topology. So, Equations (2.42) and (2.43) are calledthe zeroth-order deformation equations.

Sincex(τ;q) andγ(q) depend on the embedding parameterq∈ [0,1], they can beexpanded in a power series ofq as follows

x(τ;q) ∼ x0(τ)++∞

∑n=1

xn(τ) qn, (2.48)

γ(q) ∼ γ0++∞

∑n=1

γn qn, (2.49)

where

xn(τ) =1n!

∂ nx(τ;q)∂qn

∣

∣

∣

∣

q=0=Dn [x(τ;q)] , γn =

1n!

dnγ(q)dqn

∣

∣

∣

∣

q=0=Dn [γ(q)] . (2.50)

Here, x(τ;0) = x0(τ) and γ(0) = γ0 are used. We call (2.48) and(2.49) thehomotopy-Maclaurin series of ˜x(τ;q) and γ(q), respectively. According to (2.47),we obtain the solution of (2.32), if the homotopy-Maclaurinseries (2.48) and (2.49)are convergent atq= 1 to x(τ;1) andγ(1), respectively. So, it is very important toguarantee the convergence of the homotopy-Maclaurin series (2.48) and (2.49) atq= 1. However, a power series has in general a bounded convergence radius. Fortu-nately, both ˜x(τ;q) andγ(q) are dependent uponnot onlythe homotopy-parameterq∈ [0,1] but alsothe convergence-control parameterc0 and the auxiliary linear op-eratorL , because both ofc0 andL appear in the zeroth-order deformation equa-tion (2.42). More importantly, we have greatfreedomto choose the convergence-control parameterc0 and the auxiliary linear operatorL . Thus, assuming that theconvergence-control parameterc0 and the auxiliary linear operatorL are properlychosen so that the homotopy-Maclaurin series (2.48) and (2.49) are convergent tox(τ;q) andγ(q) atq= 1, we have, according to the expressions (2.47) to (2.49),thehomotopy-series solution


x(τ) = x0(τ)++∞

∑n=1

xn(τ), (2.51)

γ = γ0++∞

∑n=1

γn, (2.52)

andthe mth-order homotopy-approximation

x(τ) ≈ x0(τ)+m

∑n=1

xn(τ), (2.53)

γ ≈ γ0+m

∑n=1

γn. (2.54)

For simplicity, define the sets

Xm = x0(τ),x1(τ),x2(τ), · · · ,xm(τ) , Γm = γ0,γ1,γ2, · · · ,γm .

According to the fundamental theorems in calculus [5],xn(τ) andγn areunique, andare completely determined by ˜x(τ;q) andγ(q), respectively, which are governed bythe zeroth-order deformation equations (2.42) and (2.43).There are two differentways to derive the corresponding equations forxn(τ) andγn. But, each of them givesthe same results, as shown below. First, differentiatingn times (2.42) and (2.43) withrespect toq, then dividing byn!, and finally settingq= 0, we have, according to thedefinition (2.50) ofxn(τ) andγn, the so-callednth-order deformation equation

L [xn(τ)− χn xn−1(τ)] = c0 δn−1(Xn−1,Γn−1), (2.55)


xn(0) = 0, x′n(0) = 0, (2.56)

where

δk(Xk,Γk)

= Dk[

γ(q) x′′(τ;q)+λ x(τ;q)+ ε x3(τ;q)]

=k

∑i=0

γi x′′k−i(τ)+λ xk(τ)+ εk

∑i=0

xk−i(τ)i

∑j=0

xi− j(τ) x j(τ), (2.57)

and

χn =

0, n≤ 1,1, n> 1.

(2.58)

The detailed derivation of (2.55) and (2.56) is given in Appendix 2.1. Alternatively,one can directly substitute the homotopy-Maclaurin series(2.48) and (2.49) into thezeroth-order deformation equations (2.42) and (2.43), then equates the coefficientsof the like power ofq. As proved by Hayat and Sajid [6] in general, the 2nd approach


gives exactly thesameequations as (2.55) and (2.56) by the 1st approach, as shownin Appendix 2.2. This is mainly because, due to the fundamental theorems in cal-culus [5],xn(τ) in the homotopy-Maclaurin series (2.48) isunique, and thus mustbe governed by theuniqueequation. So doesγn. In general, given a zeroth-orderdeformation equation, it is straight-froward to obtain thecorresponding high-orderdeformation equations. We will show this point in Chapter 3 in details.

It should be emphasized here that the high-order deformation equations (2.55)and (2.56) arelinear. Thus, according to (2.51) and (2.52), the original nonlineardifferential equation (2.32) has been transformed into an infinite number of lineardifferential equations (2.55) and (2.56). Note that, different from perturbation tech-niques [4, 7, 8, 25–27], such kind of transformation doesnot need any small/largephysical parameters. Thus, giving up the dependence of perturbation techniqueson small/large physical parameters, the HAM is valid for more equations, espe-cially for those with strong nonlinearity. More importantly, without the restrictionof small/large physical parameters, the HAM provides us great freedom to choosethe auxiliary linear operatorL , as shown later. So, it is an obvious advantage of theHAM to be independent of any small/large physical parameters.

Note that the auxiliary linear operatorL in the zeroth-order deformation equa-tion (2.42) is not determined up to now. As mentioned before,it has the propertyL [0] = 0. Besides, ifx(τ) is a periodic function, thenL x(τ) should be also a peri-odic function, too. Even so,L is still rather general. Since the HAM is independentof small/large physical parameters, we have great freedom to choose the auxiliarylinear operatorL , as shown later in this book.

Note that the original equation (2.32) contains the linear partγ x′′+λ x. Regard-ing ε as a small parameter (i.e. perturbation quantity) and substituting

x= x0++∞

∑n=1

xn εn

into (2.32) and then equating the like-power ofε, one has the perturbed equations

γ x′′0 +λ x0 = 0, x0(0) = x∗, x′0(0) = 0,

γ x′′1 +λ x1 =−ε x30, x1(0) = 0, x′1(0) = 0,

...

The solution ¯x0 of the above perturbed equations is periodic whenλ > 0 but is non-periodic whenλ < 0. However, as listed in Table 2.3, whenε > 0, the solution of(2.32) is periodic even in case ofλ < 0. So, the above perturbation method fails toget good approximations in case ofε > 0 andλ < 0. Therefore,L x= γ x′′+λ x isnot a proper linear operator to get periodic approximationsof (2.32).

Fortunately, the HAM is independent of any small/large physical parameters,as mentioned above. So, we have great freedom to choose a proper auxiliary linearoperator, as shown below. Because the original nonlinear oscillation problem is gov-erned by the 2nd-order differential equation (2.32), it seems natural for us to choose


such a 2nd-order differential operator

L x(τ) = x′′(τ)+A1(τ) x′(τ)+A2(τ) x(τ), (2.59)

where the prime denotes the differentiation with respect toτ, andA1(τ),A2(τ) areperiodic real functions so thatL x(τ) ∈ Sp if x(τ) ∈ Sp. Let y1(τ),y2(τ) denote thenon-zero solutions ofL x(τ) = 0, andxs

n(τ) a special solution of (2.55). Then, thegeneral solution of (2.55) reads

xn(τ) = χn xn−1(τ)+ xsn(τ)+C1 y1(τ)+C2 y2(τ), (2.60)

whereC1,C2 are integral coefficients. Note that our aim is to obtain convergent se-ries solutionx(τ) with the known period 2π , i.e.x(τ)∈Sp, which implies, accordingto (2.51), thatxn(τ) must be a periodic function with the period 2π , i.e.xn(τ) ∈ Sp.So, we choose the simplest periodic function with the period2π :

y1(τ) = cosτ, y2(τ) = sinτ.

Becausey1(τ) andy2(τ) are non-zero solutions (i.e. kernel) ofL x= 0, it holds forany non-zero constant coefficientsC1 andC2 that

L [C1 cosτ +C2 sinτ] = 0.

Then, using the above equation and the definition (2.59) ofL , we have

[A2(τ)−1]cosτ −A1(τ)sinτC1

+ [A2(τ)−1]sinτ +A1(τ)cosτC2 = 0. (2.61)

The above equation holds for arbitrary coefficientsC1 andC2 in the intervalτ ∈[0,+∞), if and only if

A1(τ) = 0, A2(τ) = 1. (2.62)

Substituting them into the general definition (2.59) ofL , we obtain the auxiliarylinear operator

L x= x′′+ x, (2.63)

which has the propertyL [C1 cosτ +C2 sinτ] = 0 (2.64)

for any constantsC1 andC2. In other words, cosτ and sinτ belong to the kernel ofthe auxiliary linear operatorL . Then, the general solution of (2.55) reads

xn(τ) = χn xn−1(τ)+ xsn(τ)+C1 cosτ +C2 sinτ, (2.65)

where the integral coefficients

C1 =−χn xn−1(0)− xsn(0), C2 = 0


are determined by the initial condition (2.56). Thus, the solution of thenth-orderdeformation equations (2.55) and (2.56) reads

xn(τ) = χn xn−1(τ)+ xsn(τ)− [χn xn−1(0)+ xs

n(0)] cosτ. (2.66)

Sincexn ∈ Sp, the above expression implies thatxsn ∈ Sp, i.e. the special solution

xsn(τ) must be a function with the period 2π . Note that both ofxs

n(τ) andγn−1 areunknown, but we have only one governing equation forxs

n(τ). So, one additionalalgebraic equation is needed to determineγn−1. To show how to getxs

n(τ) andγn−1,let us consider the first-order equation

L [x1(τ)] = c0 δ0(x0,γ0), x1(0) = 0,x′1(0) = 0,

where we have, according to (2.33) and (2.57), that

δ0(x0,γ0) = γ0 x′′0(τ)+λ x0(τ)+ ε x30(τ)

= A1,0+A1,1cosτ +A1,2cos2τ +A1,3cos3τ, (2.67)

in whichA1,0,A1,1,A1,2 andA1,3 are constant coefficients, especially

A1,1 = (β − x∗)

[

γ0−λ − 34

ε(

5β 2−2β x∗+(x∗)2)]

.

According to the property (2.64), ifA1,1 6= 0, thenx1(τ) contains the so-called sec-ular termτ cosτ, which however isnot periodic, i.e.xs

n /∈ Sp. Such kind of non-periodic solution must be avoided, since our aim is to obtainperiodic solution of(2.32). For this reason, we had to enforceA1,1 = 0, which provides us with oneadditional algebraic equation ofγ0, i.e.

(β − x∗)

γ0−λ − 34

ε[

4β 2+(β − x∗)2]

= 0.

Sinceβ 6= x∗, we have the initial guess

γ0 = λ +34

ε[

4β 2+(β − x∗)2] , (2.68)

corresponding to the initial guess of the frequency

ω0 =

√

λ +34

ε [4β 2+(β − x∗)2] . (2.69)

Note thatγ0 = ω20 defined above is always positive for all possible values ofλ ,ε

andx∗ in case of periodic solutionx(τ), thereforeω0 is always a real positive num-ber. The above expression indicates that, physically, the frequency of the periodicoscillation of the nonlinear dynamic system (2.32) dependsnot only on the physical


parametersλ andε, but also onx∗, the starting position, andβ , the position (i.e. thecoordinate) of the stable equilibrium point.

Thereafter, it is easy to get the special solution

xs1(τ) = A1,0−

A1,2

3cos2τ − A1,3

8cos3τ.

Then, substituting it into (2.66), we have

x1(τ) = A1,0−(

A1,0−A1,2

3− A1,3

8

)

cosτ − A1,2

3cos2τ − A1,3

8cos3τ.

Similarly, we can solveγ1, x2(τ), γ2, x3(τ), and so on. In this way, it is easy tosuccessively solve the linear deformation equations (2.55) and (2.56) by means ofsymbolic computation software such as Mathematica, Maple and so on, and thenobtain a high-order approximation ofx(τ) and γ = ω2 in a few seconds using alaptop.

It should be emphasized that the HAM mentioned above provides us great free-dom to construct the zeroth-order deformation equation, mainly because it is inde-pendent of any small/large physical parameters. It is due tothis kind of freedom thatwe can choose the auxiliary linear operatorL x= x′′+x to obtainall periodic solu-tions forall possible values ofλ ,ε andx∗, even includingλ < 0. Besides, it is dueto this freedom that we can regardγ as a function ofq, i.e. a homotopyγ : γ0 ∼ γ,in the zeroth-order deformation equation (2.42), so that the secular termτ cosτ isavoided and the periodic solutions are obtained, as shown above. Furthermore, it isdue to this kind of freedom that we can introduce the convergence-controlparameterc0 into the zeroth-order deformation equation, which provides us a simple way toguarantee the convergence of homotopy-series solution, asshown below. By meansof this freedom, we can obtain much better approximations ofmany problems withstrong nonlinearity. This is the 2nd advantage of the HAM.

2.3.3 Convergence of homotopy-series solution

First of all, let us prove the following two theorems about the homotopy-series+∞∑

n=0xn(τ) and

+∞∑

n=0γn.

Theorem 2.1.If the homotopy-series+∞∑

k=0xk(τ) and

+∞∑

k=0x′′k(τ) are convergent, then

+∞∑

k=0δk = 0, whereδk is defined by (2.57).

Proof. According to (2.55) and (2.58), we have

L x1 = c0 δ0,


L (x2− x1) = c0 δ1,

L (x3− x2) = c0 δ2,

...

L (xm− xm−1) = c0 δm−1,

whereL u= u′′+u is the auxiliary linear operator. SinceL is a linear operator, thesum of all above equations gives

L xm = c0

m−1

∑n=0

δn.

Since the homotopy-series+∞∑

n=0xn and

+∞∑

n=0x′′n converge, it holds

limn→+∞

xn = 0, limn→+∞

x′′n = 0.

Then, we have

c0

+∞

∑n=0

δn = limm→+∞

(L xm) = limm→+∞

(

x′′m+ xm)

= limm→+∞

x′′m+ limm→+∞

xm = 0,

which gives, sincec0 6= 0, that+∞∑

n=0δn = 0. ⊓⊔

Theorem 2.2.If the convergence-control parameter c0 is so properly chosen that

the homotopy-series+∞∑

n=0xn(τ) and

+∞∑

n=0γn are absolutely convergent to x(τ) and γ,

respectively, and besides+∞∑

n=0x′′n(τ) is convergent to x′′(τ), then the homotopy-series

+∞∑

n=0xn(τ) and

+∞∑

n=0γn satisfy the original nonlinear differential equation (2.32).

Proof. Since the homotopy-series+∞∑

n=0xn and

+∞∑

n=0x′′n are converge, we have according

to Theorem 2.1 that+∞∑

n=0δn = 0, i.e.

+∞

∑n=0

[

n

∑k=0

γk x′′n−k(τ)+λ xn(τ)+ εn

∑k=0

xn−k(τ)k

∑j=0

xk− j(τ) x j(τ)

]

= 0.

Since+∞∑

n=0xn and

+∞∑

n=0γn are absolutely convergent tox(τ) andγ, respectively, and

besides+∞∑

n=0x′′n is convergent tox′′(τ), we have due to the theorems of Cauchy product


that

+∞

∑n=0

(

n

∑k=0

γk x′′n−k

)

=+∞

∑k=0

+∞

∑n=k

γk x′′n−k =+∞

∑k=0

+∞

∑m=0

γk x′′m =

(

+∞

∑j=0

γ j

)(

+∞

∑m=0

x′′m

)

and+∞

∑n=0

(

n

∑k=0

xn−k

k

∑j=0

xk− j x j

)

=

(

+∞

∑n=0

xn

)3

.

Besides, it holds obviously

+∞

∑n=0

λ xn = λ

(

+∞

∑n=0

xn

)

.

Substituting the above three expressions into the equationfrom+∞∑

n=0δn = 0 gives

(

+∞

∑j=0

γ j

)(

+∞

∑n=0

xn

)′′

+λ

(

+∞

∑n=0

xn

)

+ ε

(

+∞

∑n=0

xn

)3

= 0.

According to (2.33), it holdsx0(0) = x∗ andx′0(0) = 0. Then, using (2.56), we have

+∞

∑n=0

xn(0) = x0(0) = x∗,+∞

∑n=0

x′n(0) = x′0(0) = 0.

Thus, the absolutely convergent homotopy-seriesx =+∞∑

n=0xn andγ =

+∞∑

n=0γn satisfy

the original equation (2.32). This ends the proof. ⊓⊔

According to Theorem 2.2, it is important to guarantee the convergence ofhomotopy-series. Theorem 2.1 provides us a convenient way to check the conver-gence of homotopy-series, as shown below. The above two theorems are tenable ingeneral, as proved in Chapter 3.

Obviously, it is very important to guarantee the convergence of an approximationseries. Unfortunately, near all previous analytic approximation methods such as per-turbation perturbation, Adomian decomposition method, the δ -expansion methodand so on, can not guarantee the convergence of analytic approximation series, asmentioned in Chapter 1. This is the essential reason why these previous analytictechniques are valid basically for weakly nonlinear problems.

Fortunately, independent of small/large physical parameters, the HAM providesus great freedom. Using this kind of freedom, we introduce a non-zero auxil-iary parameterc0, namely the convergence-control parameter, into the so-calledzeroth-order deformation equation (2.42). Note that the high-order deformationequation (2.55) also contains the convergence-control parameterc0. Therefore, the


homotopy-series+∞∑

n=0xn(τ) and

+∞∑

n=0γn contain the convergence-control parameterc0,

too. More importantly, as mentioned above, we have great freedom to choose thevalue ofc0 so that we can find some proper values ofc0 to guarantee the conver-gence of the homotopy-series. Thus, the convergence-control parameterc0 providesus a convenient way to guarantee the convergence of homotopy-series, as shownbelow.

According to Theorem 2.2, the residual2 of the original governing equation (2.32)

tends to zero inτ ∈ [0,2π ] if the homotopy-series+∞∑

n=0xn(τ) and

+∞∑

n=0γn are absolutely

convergent and besides+∞∑

n=0x′′n(τ) is convergent. The averaged value of the squared

residual of the governing equation clearly indicates the accuracy of an analytic ap-proximation. So, in order to choose a proper value ofc0, we use the squared residual

Em(c0) =1

2π

∫ 2π

0[∆m(τ;c0)]

2 d τ, (2.70)

where∆m(τ;c0) = γ x ′′(τ)+λ x(τ)+ ε x3(τ) (2.71)

is the residual of the governing equation (2.32), and

x=m

∑n=0

xn(τ), γ =m

∑n=0

γn

are themth-order approximation ofx(τ) andγ, respectively. For the sake of com-putational efficiency, we calculate the discrete squared residualEm(c0) numerically,i.e.

Em(c0)≈1

(N+1)

N

∑k=0

[∆m(τk;c0)]2 , τk =

2kπN

, (2.72)

whereN is an integer. Obviously, the above expression is a good approximation ofEm(c0) for large enoughN. In this chapter, we useN= 50. Note thatEm(c0) dependson the convergence-controlparameterc0. Obviously, the smaller the value ofEm(c0)for givenm, the better the approximation. At the given order of approximationm,the “best” or “optimal” approximation is defined by the minimum of Em(c∗0) withthe corresponding optimal convergence-control parameterc∗0.

Note that, it is convenient to solve the linear high-order deformation equations(2.55) and (2.56) by means of the computer algebra systems like Mathematica,Maple and so on. The corresponding Mathematica code (without iteration approach)is free available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

Without loss the generality, let us first consider the caseε = 1, λ = −9/4 andx∗ = 1. As shown in Fig. 2.5, as the order of approximation increases, the discrete

2 For precise definition about residual of equations, please refer to the website athttp://en.wikipedia.org/wiki/Residual


Fig. 2.5 Discrete squaredresidualEm(c0) in case ofε = 1,λ =−9/4 andx∗ = 1.Solid line: 1st-order approx.;Dashed line: 3rd-order ap-prox.; Dash-dotted line: 5th-order approx.; Dot-dot-dashedline: 7th-order approx.

c0

Em

-0.25 -0.2 -0.15 -0.1 -0.05 010-6

10-5

10-4

10-3

10-2

10-1

100

λ = -9/4, ε = 1, x* = 1

Table 2.4 Discrete squared residualEm(c0) in case ofε = 1,λ = −9/4,x∗ = 1 by means of dif-ferent values ofc0

order of approx. c0 =−1/5 c0 =−3/20 c0 =−1/10 c0 =−1/20

5 8.6×10−4 6.5×10−5 1.1×10−3 2.0×10−2

10 1.1×10−4 2.8×10−7 2.3×10−5 2.0×10−3

20 5.2×10−6 9.4×10−12 3.0×10−8 4.7×10−5

30 4.4×10−7 4.4×10−16 5.7×10−11 1.8×10−6

40 4.5×10−8 2.3×10−20 1.2×10−13 8.7×10−8

50 5.4×10−9 1.3×10−24 2.8×10−16 4.5×10−9

60 6.9×10−10 8.3×10−29 6.8×10−19 2.5×10−10

70 9.3×10−11 5.3×10−33 1.7×10−21 1.4×10−11

80 1.3×10−11 3.5×10−37 4.4×10−24 8.0×10−13

Table 2.5 Approximations ofγ = ω2 in case ofε = 1,λ = −9/4,x∗ = 1 by means of differentvalues ofc0

order of approx. c0 =−1/5 c0 =−3/20 c0 =−1/10 c0 =−1/20

5 3.8944 3.9322 3.9550 4.080110 3.9381 3.9280 3.9307 3.968720 3.9297 3.9278 3.9279 3.932530 3.9283 3.9278 3.9278 3.928540 3.9280 3.9278 3.9278 3.927950 3.9279 3.9278 3.9278 3.927860 3.9278 3.9278 3.9278 3.927870 3.9278 3.9278 3.9278 3.927880 3.9278 3.9278 3.9278 3.9278


Table 2.6 Approximations ofγ = ω2 in case ofε = 1,λ = −9/4 andx∗ = 1 by means of theoptimal convergence-control parameterc∗0 =−17/100

m, order of approx. γ = ω2 Em(c∗0)

0 4.6875 0.491 3.9223 2.4×10−3

5 3.9263 1.6×10−5

10 3.9281 1.0×10−7

15 3.9278 2.2×10−10

20 3.9278 1.3×10−12

25 3.9278 7.0×10−15

30 3.9278 4.8×10−17

40 3.9278 2.6×10−21

50 3.9278 1.7×10−25

60 3.9278 1.2×10−29

70 3.9278 8.5×10−34

80 3.9278 6.5×10−38

squared residualEm(c0) decreases in the interval

Rc = c0|−0.2≤ c0 ≤−0.05.

Therefore, ifc0 ∈ Rc, then the corresponding homotopy-series solution converges.To confirm this, we investigate the convergence of the homotopy-series solutions bydifferent values ofc0, such asc0 = −1/5,−3/20,−1/10 and−1/20, respectively.It is found that, as the order of approximation increases, the discrete squared resid-ual of all these homotopy-series decreases monotonously, as shown inTable 2.4,and besidesall these homotopy-series give thesamevalueγ = 3.9278, as shown inTable 2.5. Our calculations illustrate that there indeed exists such a setRc that, ifc0 ∈ Rc, then the corresponding homotopy-series converges to thesameresult, al-though with different convergence-rate. Mathematically,according to Theorem 2.2,all convergent homotopy-series ofx(τ) andγ satisfy the original equation (2.32).Since (2.32) has unique solution, then all of these convergent homotopy-series ofx(τ) andγ must be the same. Physically, the convergence-control parameterc0 isan artificial parameter, which has no physical meanings at all. Thus, from phys-ical points of view, all convergent homotopy-series must beindependent of theconvergence-control parameterc0 (otherwise, it is physically wrong). In this way,we can explain all of our above results from both mathematical and physical view-points.


Definition 2.7. Each unknown auxiliary parameter in the zeroth-order deforma-tion equation, except the homotopy-parameterq∈ [0,1], is called aconvergence-control parameter, if it can influence the convergence of the homotopy-series. All of these convergence-control parameters construct the so-calledconvergence-control vector, denoted byc = (c0,c1,c2, · · ·), wherec0,c1,c2, · · ·are convergence-control parameters.

Definition 2.8. A setRc of all possible values of a convergence-control param-eterc0 is calledthe effective-regionof the convergence-control parameterc0, ifthe corresponding homotopy-series converges for eachc0 ∈ Rc.

Definition 2.9. A setRc of all possible values of the convergence-control vectorc is calledthe effective-regionof the convergence-control vectorc, if the corre-sponding homotopy-series converges for eachc∈ Rc.

According to Fig. 2.5 and Table 2.4, the homotopy-series given by differentvalues ofc0 ∈ Rc converges in a quite different rate. For example, the homotopy-series given byc0 = −3/20 converges much faster than the homotopy-series givenby c0 = −1/5 and c0 = −1/20, as shown in Table 2.4. Besides, according toFig. 2.5, the minimum ofEm(c0) exists nearc0 = −0.17. So, we have the opti-mal convergence-control parameterc∗0 =−0.17 in the case ofε = 1, λ =−9/4 andx∗ = 1. This is indeed true. By means of the optimal convergence-control parameterc∗0 = −17/100, the discrete squared residualEm(c0) quickly decreases andγ tendsto a fixed value 3.9278 rather faster, as shown in Table 2.6. Thus, the convergence-control parameterc0 indeed provides us a convenient way to guarantee the quickconvergence of homotopy-series solution. Note that, although the initial guess ofγhas 19.3% relative error, the 3rd and 5th-order approximation of γ have only 0.1%and 0.04% relative error, respectively. Thus, in this case,only a few terms can givea rather accurate approximation ofx(τ) by means of the optimal value ofc0, asshown in Fig. 2.6. Note also that, the homotopy-series givenby c0 =−3/20 quicklyconverges, too. Thus, in practice, it is unnecessary to use the “exact” value of theoptimal convergence-control parameterc0.

The above approach has general meaning. For example, we further consider thefollowing three cases:

• ε =+1, λ = 9/4, x∗ =+1;

• ε =+1, λ = 0, x∗ =+1;

• ε =−1, λ = 4, x∗ =−1.


Fig. 2.6 Comparison of nu-merical result with analyticapproximations ofx(τ) incase ofε = 1,λ = −9/4and x∗ = 1 by means ofc0 = −0.17. Solid line: 5th-order approx.; Dashed line:initial approx.; Symbols: nu-merical result.

t

x(t)

0 1 2 3 4 5 60.5

1

1.5

2

For each case, we can investigate the curves of the discrete squared residualEm(c0)versusc0 in a similar way so as to find out a region ofc0 for the convergence ofhomotopy-series and besides the optimal value ofc0. According to Figs. 2.7 to 2.9,we have the optimal valuec0 = −1/3 in case ofε = +1, λ = 9/4, x∗ = +1, theoptimal valuec0 =−4/3 in case ofε =+1, λ = 0, x∗ =+1, and the optimal valuec0 = −3/10 in case ofε = −1, λ = 4, x∗ = −1, respectively. Using the corre-sponding optimal value ofc0, the discrete squared residualEm decreases quicklyfor each case, as shown in Table 2.7, and besidesγ = ω2 quickly tends to a fixedvalue, as shown in Table 2.8. Furthermore, even the corresponding 1st or 2nd-orderapproximations ofx(t) are rather accurate, as shown in Fig. 2.10. All of these indi-cate the convergence of the corresponding homotopy-seriessolution. Therefore, theconvergence-control parameterc0 indeed provides us a convenient way to guaranteethe convergence of the homotopy-series solution. It shouldbe emphasized that otheranalytic techniques have no ways to guarantee the convergence of series. So, this isan obvious advantage of the HAM.

Table 2.7 Discrete squared residualEm(c0) of mth-order approximation in different cases

m, order of approx. ε = 1,λ = 9/4 ε = 1,λ = 0 ε =−1,λ = 4x∗ = 1,c0 =−1/3 x∗ = 1,c0 =−4/3 x∗ =−1,c0 =−3/10

0 0.032 0.032 0.0321 3.1×10−5 5.1×10−4 4.6×10−5

5 1.3×10−14 1.5×10−8 6.7×10−14

10 5.8×10−26 7.3×10−14 2.4×10−24

15 5.9×10−37 7.7×10−19 1.1×10−34

Considering the theoretical rigorousness of the HAM, we often mention thehomotopy-series and investigate the convergence of this kind of infinite series. How-ever, this does not mean that we must use many terms to get an accurate enoughapproximation! In many cases, only a few terms of the homotopy-approximations


Table 2.8 Themth-order approximation ofγ = ω2 for different cases

m, order of approx. ε = 1,λ = 9/4 ε = 1,λ = 0 ε =−1,λ = 4x∗ = 1,c0 =−1/3 x∗ = 1,c0 =−4/3 x∗ =−1,c0 =−3/10

0 3 0.75 3.251 2.9921875 0.71875 3.242968755 2.9921730367 0.7177741910 3.242777097810 2.9921730364 0.7177700399 3.242777091715 2.9921730364 0.7177700110 3.242777091720 2.9921730364 0.7177700110 3.242777091725 2.9921730364 0.7177700110 3.242777091730 2.9921730364 0.7177700110 3.2427770917

Fig. 2.7 Discrete squaredresidualEm(c0) in case ofε = 1,λ = 9/4 andx∗ = 1.Solid line: 1st-order approx.;Dashed line: 3rd-order ap-prox.; Dash-dotted line: 5th-order approx.

c0

Em

-0.5 -0.4 -0.3 -0.2 -0.1 010-14

10-12

10-10

10-8

10-6

10-4

10-2

ε = 1, λ = 9/4, x* = 1

Fig. 2.8 Discrete squaredresidualEm(c0) in case ofε =1,λ = 0 andx∗ = 1. Solid line:1st-order approx.; Dashedline: 3rd-order approx.; Dash-dotted line: 5th-order approx.

c0

Em

-2 -1.5 -1 -0.5 010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

ε = 1, λ = 0, x* = 1


Fig. 2.9 Discrete squaredresidualEm(c0) in case ofε = −1,λ = 4 andx∗ =−1. Solid line: 1st-orderapprox.; Dashed line: 3rd-order approx.; Dash-dottedline: 5th-order approx.

c0

Em

-0.5 -0.4 -0.3 -0.2 -0.1 010-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1 ε = -1, λ = 4, x* = -1

Fig. 2.10 Comparison of nu-merical results with analyticapproximations ofx(τ). Solidline: 1st-order of approx. incase ofε = 1,λ = 9/4,x∗ = 1by c0 = −1/3; Dashed line:2nd-order approx. in caseof ε = 1,λ = 0,x∗ = 1 byc0 =−4/3; Dot-dashed line:2nd-order approx. in case ofε = −1,λ = 4,x∗ = −1 byc0 = −3/10; Symbols: nu-merical results.

t

x(t)

0 2 4 6 8-1.5

-1

-0.5

0

0.5

1

1.5

can give very accurate result. For example, we obtain ratheraccurate 1st-orderhomotopy-approximation

x≈ 196

[95cos(τ)+ cos(3τ)] , τ =√

3 t, (2.73)

in the case ofε =+1, λ = 9/4, x∗ =+1 by means of the optimal valuec0 =−1/3,the rather accurate 2nd-order homotopy-approximation

x≈ 1576

[551cosτ +24cos(3τ)+ cos(5τ)] , τ =

√

2332

t, (2.74)

in the case ofε = +1, λ = 0, x∗ = +1 by means of the optimal valuec0 = −4/3,and the rather accurate 2nd-order homotopy-approximation

x ≈ −1.00952 cosτ +9.60938×10−3cos(3τ)


− 8.78906×10−5cos(5τ), τ =

√

41511280

t, (2.75)

in the case ofε = −1, λ = 4, x∗ =−1 by means of the optimal valuec0 = −3/10,respectively, as shown in Fig. 2.10. In practice, using the optimal convergence-control parameter, we often obtain accurate enough approximations in a few termsby means of the HAM. However, as the nonlinearity of equations becomes stronger,more and more terms are needed, due to the complexity of strong nonlinear prob-lems.

Without computer, it is hard to find the optimal valuec0, corresponding to theminimum of the squared residualEm(c0). However, in the times of computers, itis easy to do so by means of symbolic computation software such as Mathemat-ica, Maple, MathLab and so on. For example, using Mathematica, we obtain up-to 30th-order approximations just in a few seconds even by a laptop, although, asmentioned above, such high-order approximations arenot necessary at all for theabove cases. However, when necessary, we can obtain rather high order homotopy-approximations by symbolic computation software, and moreimportantly, find outan optimal convergence-control parameterc0 to guarantee the convergence of thecorresponding homotopy-series so as to have an accurate enough result by manyenough terms. Note also that, a laptop can now save/read an enormous amount ofdata to/from a diskette in a few seconds. Besides, rather lengthy expressions can becalculated in a few seconds by a computer. Using a modern laptop, we generallycan not feel obvious difference between calculating a shortformula and evaluat-ing a rather complicated expression which might be one hundred A4 pages long ifprinted. So, the HAM is indeed for the times of computer: it combines the great flex-ibility of constructing a homotopy in theory and the increasing computing-power ofan electronic computer in practice.

2.3.4 Essence of the convergence-control parameterc0

To reveal the essence of the convergence-control parameterc0, let us further con-sider a little more general caseλ = 0,x∗ = 1 andε > 0, which has the exact solution

ω =Γ (3/4)Γ (5/4)

√

π ε8

,

whereΓ denotes Gamma function. Thus, we have the exact solution

γ = ω2 ≈ 0.7177700110ε. (2.76)

According to (2.34) and (2.68), we have the initial guessx0(τ) = cosτ and

γ0 =34

ε.


Regardingc0 as unknown and using the same approach mentioned above, we obtainthe 1st-order homotopy-approximation

γ ≈ 34

ε +3

128c0 ε2,

the 2nd-order homotopy-approximation

γ ≈ 34

ε +364

c0 ε2+9

512c2

0 ε3,

the 3rd-order homotopy-approximation

γ ≈ 34

ε +9

128c0 ε2+

27512

c20 ε3+

1779131072

c30 ε4,

and so on. It is found thatγ is a kind of power series ofε with coefficients dependentupon the convergence-control parameterc0. So, mathematically speaking, we havedifferent approximations ofγ for different values ofc0. The comparison of the exactformula (2.76) with the 20th-order approximations ofγ by means of different valuesof c0 is given in Fig. 2.11, which clearly indicates that the convergence radius ofthe homotopy-series ofγ is larger whenc0 < 0 closer to zero is used. Therefore,different values ofc0 correspond to different convergence radius of the homotopyseries ofγ = ω2. In other words, we can adjust and control the convergence regionof the homotopy-series by choosing different values ofc0: this is exactly the reasonwhy we callc0 the convergence-control parameter.

Fig. 2.11 Comparison ofexact formula (2.76) withthe 20th-order homotopy-approximations ofγ in case ofλ = 0,x∗ = 1 andε > 0. Solidline: the exact formula (2.76);Dashed line:c0 = −1; Dot-dashed line:c0 =−1/2; Dot-dot-dashed line:c0 =−1/4.

ε

γ

0 2 4 6 8 100

2

4

6

8

10

c0 = -1

c0 = -0.5

c0 = -0.25

According to Fig. 2.11, the smaller the absolute value ofc0 < 0, the larger theconvergence radius of the power series ofγ aboutε. This suggests us to choosec0 =−1/(1+ ε), which gives the first-order approximation


γ ≈ 3ε(32+31ε)128(1+ ε)

(2.77)

and the second-order approximation

γ ≈ 3ε(128+248ε +123ε2)

512(1+ ε)2 , (2.78)

respectively. Compared to the exact formula (2.76), the above expressions have1.2% and 0.4% relative error, respectively, forall possible values ofε, i.e. 0≤ε < +∞. Using the 2nd-order approximation (2.78) ofγ = ω2, we have a simpleapproximate formula of the period

T ≈ 32π (1+ ε)

√

23 ε(128+248ε +123ε2)

, (2.79)

which agrees quite well with the exact periodT = 7.4163/√

ε in thewholeregion0< ε <+∞, as shown in Fig. 2.12.

Fig. 2.12 Comparison of theexact periodT = 7.4163/

√ε

with the 2nd-order homotopy-approximation (2.79) in caseof λ = 0, x∗ = 1 by means ofc0 = −(1+ ε)−1. Solid line:formula (2.79); Symbols:exact result.

ε

T

10-1 100 101 102 103

5

10

15

20

λ = 0, x* = 1, c0 = -1/(1+ε)

This is mainly because we have great freedom to choose different values ofthe convergence-control parameterc0. Especially, when we choosec0 = −1/γ0 =−4/(3ε), the homotopy-approximations ofγ quickly converges even to the exactformula (2.76), as shown in Table 2.9. Note that the maximum relative errors of the1st and 3rd-order approximation ofγ given byc0 = −4/(3ε) are only 0.14% and0.008%, respectively! It is very interesting that the corresponding 17th and 19th-order approximations ofγ are thesameas the exact formula (2.76) ! So, it seemsthatc0 = −1/γ0 is the optimal convergence-control parameter in this special case.It illustrates the great potential of the HAM for strongly nonlinear problems.

Note that, the larger the value ofε, the stronger the nonlinearity of (2.32). How-ever, in case ofλ = 0 andx∗ = 1, for all possible values ofε > 0, even includingε →+∞, we obtain simple but rather accurate approximations (2.77) and (2.78), and


Table 2.9 Approximation ofγ = ω2 in case ofλ = 0 andx∗ = 1 by means ofc0 =−4/(3ε)

Order of approx. γ

1 0.71875ε3 0.7178276910ε5 0.7177741910ε7 0.7177703474ε9 0.7177700399ε11 0.7177700136ε13 0.7177700113ε15 0.7177700111ε17 0.7177700110ε19 0.7177700110ε

even the exact formula (2.76), by choosing a proper convergence-control parameterc0. All of these indicate that the convergence-control parameter c0 indeed providesus a convenient way to guarantee the convergence of homotopy-series so that theHAM is valid for strongly nonlinear problems.

Fig. 2.13 Curve of the realfunction (1+ z)−1 defined inthe intervalz∈ (−∞,+∞).Solid line: the part whichcan be expressed by the

series+∞∑

n=0(−z)n; Dashed line:

the part which can not beexpressed by the same series.

z

1/(

1+

z)

-6 -4 -2 0 2 4 6-3

-2

-1

0

1

2

3

Finally, we use a simple example to explain why an auxiliary parameter cangreatly influence the convergence of a series. The real function (1+ z)−1 is welldefined in an infinite interval−∞ < z< +∞, except the singular pointz= −1.However, according to the so-called Newtonian binomial theorem, the series

+∞

∑n=0

(−z)n = 1− z+ z2− z3+ · · · (2.80)

converges to(1+ z)−1 only in a rather small interval|z|< 1, as shown in Fig. 2.13.This is becausez= −1 is a singular point of the function(1+ z)−1, and according


to the traditional theorems, this kind of singular point determines the convergence

radius of the power series+∞∑

n=0(−z)n, as shown in Fig. 2.13.

It is interesting that, using the famous Newtonian binomialtheorem, we can rig-orously prove the following theorem:

Theorem 2.3.For a real number z and an auxiliary parameter c0 6= 0, it holds

11+ z

= limm→+∞

m

∑n=0

µm+1,n+10 (c0) (−z)n, (2.81)

in the interval

−1< z<2|c0|

−1, when c0 < 0,

or

− 2c0

−1< z<−1, when c0 > 0,

where

µm,n0 (c0) = (−c0)

nm−n

∑i=0

(

n−1+ ii

)

(1+ c0)i . (2.82)

The detailed proof is given in Appendix 2.3.

Fig. 2.14 Comparison of(1+ z)−1 with the series(2.81) by different values ofc0. Dashed line:c0 =−1; Dot-dashed line:c0 = −1/2; Dot-dot-Dashed line:c0 = −1/3;Solid line: the exact function(1+z)−1 whenz>−1.

z-1 0 1 2 3 4 5 6

0

1

2

3

c0 = -1/2c0 = -1/3

c0 = -1

According to the above theorem, the convergence region of the power series

11+ z

= limm→+∞

m

∑n=0

µm+1,n+10 (c0)(−z)n

is determined by the auxiliary parameterc0. As shown in Fig. 2.14, whenc0 =−1,it is exactly the traditional Newtonian binomial and thus converges to(1+ z)−1 inthe interval−1< z< 1; whenc0 = −1/2, it converges to(1+ z)−1 in the interval


Fig. 2.15 Comparison of(1+ z)−1 with the series(2.81) by different values ofc0. Dashed line:c0 = 1; Dot-dashed line:c0 = 1/2; Dot-dot-Dashed line:c0 = 1/3;Solid line: the exact function(1+z)−1 whenz<−1.

z-8 -7 -6 -5 -4 -3 -2 -1

-3

-2

-1

0

c0 = 1c0 = 1/2c0 = 1/3

−1< z< 3; whenc0 = −1/3, it converges to(1+ z)−1 in the interval−1< z< 5,respectively. Especially, according to Theorem 2.3, the above series converges to(1+z)−1 in the infinite interval−1< z<+∞ whenc0 < 0 tends to zero ! Similarly,as shown in Fig. 2.15, whenc0 = 1, it converges to(1+ z)−1 in the interval−3<z< −1; whenc0 = 1/2, it converges to(1+ z)−1 in the interval−5< z< −1; andwhenc0 = 1/3, it converges to(1+ z)−1 in the interval−7 < z< −1. It is veryinteresting that, according to Theorem 2.3, the above series converges to(1+ z)−1

in the infinite interval−∞ < z< −1 whenc0 > 0 tends to zero ! Thus, by simplyintroducing such a kind of auxiliary parameterc0, the convergence of the powerseries of(1+z)−1 is modified fantastically: different from the traditional Newtonian

binomial(1+ z)−1 =+∞∑

n=0(−z)n which is valid only in the small interval|z| < 1, the

power series (2.81) can converge to(1+ z)−1 in the infinite interval−∞ < z<+∞,except the singular pointz=−1 only!

The above simple example illustrates that such an auxiliaryparameterc0 can in-deed fantastically modify the convergence of a series. Notethat Figs. 2.14 and 2.15are in essence similar to Fig. 2.11. Therefore, in essence, the auxiliary parameterc0

in the frame of the HAM provides us a convenient way to adjust and control the con-vergence of the homotopy-series. This is the reason why the auxiliary parameterc0

is called the convergence-control parameter. There are some different ways to intro-duce such a kind of non-zero auxiliary parameter in approximation series. The waymentioned above is one of them. Another example is the famousEuler transform,which also uses an non-zero auxiliary parameter to enlarge the convergence regionof a series. Unfortunately, these two approaches can not be directly applied to non-linear differential equations in general. It is the HAM which introduces such a kindof non-zero auxiliary parameterc0 for nonlinear differential equations in general,which provides us a convenient way to guarantee the convergence of homotopy-series solution and to get optimal approximation by means ofan optimal value ofc0. In fact, the HAM even logically contains the Euler transform: we can derive the


famous Euler transform in the frame of the HAM, and give a similar but more gen-eral transform, called the generalized Euler transform, asshown later in Chapter 5.

From the mathematical points of view, We can explain why the convergence ofthe power series (2.81) of(1+z)−1 can be fantastically modified by the convergence-control parameterc0. RegardS0 = (1,−1,1,−1, · · ·) as a point of an infinite-dimension spaceR∞ (for example, a Hilbert space). Then, the traditional binomial+∞∑

n=0(−z)n corresponds to anuniquelimit which tends to this point along such a tra-

ditional path:(1,0,0,0,0, · · ·) ∈ R∞ ,(1,−1,0,0,0, · · ·) ∈ R∞ ,(1,−1,1,0,0, · · ·) ∈ R∞ ,(1,−1,1,−1,0, · · ·) ∈ R∞ ,

...

However, Liao (2003) proved in the book “Beyond Perturbation” that the functionµm,n

0 (c0) has the property

limm→+∞

µm,n0 (c0) =

1, when−2< c0 < 0,∞, otherwise,

for a given integern≥ 0. Thus, when−2< c0 < 0, the series (2.81) corresponds toa family of limits which tend to thesamepoint S0 = (1,−1,1,−1, · · ·) ∈ R∞ alongdifferent approach paths dependent of the value ofc0:

(µ0,00 (c0),0,0,0,0, · · ·) ∈ R∞ ,

(µ1,00 (c0),−µ1,1

0 (c0),0,0,0, · · ·) ∈ R∞ ,

(µ2,00 (c0),−µ2,1

0 (c0),µ2,20 (c0),0,0, · · ·) ∈ R∞ ,

(µ3,00 (c0),−µ3,1

0 (c0),µ3,20 (c0),−µ3,3

0 (c0),0, · · ·) ∈ R∞ ,...

It is well-known that a limit of a function with multiple variables might be quitedifferent for different approach paths. For example, the limit

lim(x,y)→(0,0)

√

x2+ y2

|x| =√

1+α2

is dependent on an arbitrary real numberα, if we gain the limit along the differentapproach pathsy=αx. This is the mathematical reason why the convergence regionof the series (2.81) of(1+ z)−1 (whenz> 0) can be greatly enlarged when−2<c0 < 0.


It is a pity that, this can not explain why the series (2.81) converges to(1+ z)−1

(whenz<−1) by means ofc0 > 0. Note that, limm→+∞

µm,n0 (c0) = ∞ in case ofc0 > 0,

as proved by Liao (2003) in his book. For example, whenc0 = 1, the polynomialm∑

n=0µm+1,n+1

0 (c0) (−z)n reads

−3− z, whenm= 1,−15−17z−7z2− z3, whenm= 3,−63−129z−111z2−49z3−11z4− z5, whenm= 5,−255−769z−1023z2−769z3−351z4−97z5−15z6− z7, whenm= 7,

...

Note that, the constant term tends to infinity, so do the coefficients ofall terms,conformed to the property lim

m→+∞µm,n

0 (c0) = ∞ for c0 > 0. However, it is very inter-

esting that, asm→ +∞, the corresponding series (2.81) converges to(1+ z)−1 in−3< z< −1, according to Theorem 2.3. So, in case ofc0 > 0, even if each termseems to be divergent, the series of (2.81) as a whole is convergent. Although thisphenomena can not be clearly explained by the traditional method of determiningthe convergence radius of a power series, it shows the great power of such kind ofauxiliary parameterc0.

Fig. 2.16 The feedback loopto control dynamic behaviorof a system

System output

Sensor

ReferenceController

Systeminput

System

Measured output

Measurederror

Finally, from the view points of control theory, we explain why the convergence-control parameterc0 can ensure the convergence of homotopy series. The feedbackloop, as shown in Fig. 2.16, is a basic concept in control theory to control dynamicbehaviors of a system. Let us regard the governing equation and the related bound-ary/initial conditions as a system. Then, the initial guess, the auxiliary linear op-erator and the convergence-control parameterc0 can be regarded as “the systeminput”, and themth-order homotopy-approximation as “the system output”, respec-tively. The squared residual of the governing equations of themth-order homotopy-approximation can be regarded as the “measured error”. For different inputs, espe-cially the different convergence-control parameterc0, the “measured error” is differ-ent. Our strategy is: keepc0 as a unknown parameter in the “system input” and thenchoose its value in such a way that the “measured error”, i.e.the squared residual ofthe governing equations, is the minimum. In essence, this constructs a negative feed-back loop to control the residual of governing equations! Note that, only by means ofthe computer algebra systems like Mathematica and Maple, the convergence-controlparameterc0 can be used as a unknown variable and can appear in solution expres-


sions. This is an essential difference between the symboliccomputation and numer-ical techniques: all input of numerical methods must be assigned numerical valuesat the beginning of computation so that the input completelydetermine the conver-gence of iteration. For example, the so-called under-relaxation factor is widely usedin most numerical techniques of strongly nonlinear equations to make an iterationapproach convergent. However, at the beginning of iteration, one had to assign avalue to the under-relaxation factor, and as long as its value is chosen, one can not“control” the convergence of the iteration approach any more: in principle, all nu-merical iterations can not construct a negative feedback loop from the viewpointsof control theory. However, using computer algebra systemslike Mathematica andMaple, we neednot assign a value to the convergence-control parameterc0 at thebeginning of computation, whose value is determinedafter the finish of the com-putation by means of the minimum of the squared residual of governing equations.So, from the viewpoint of the control theory, the convergence-control parameterc0 provides us in principlea negative feedback loopso that we can guarantee theconvergence of the homotopy-series.

2.3.5 Convergence acceleration by Homotopy-Pade technique

Let

x0(τ)++∞

∑n=1

xn(τ) qn

denote a homotopy-series gained by the HAM in general. Sincethe homotopy-seriessolution is given by

x(τ) = x0(τ)++∞

∑n=1

xn(τ),

it is very important to guarantee the convergence of the homotopy-series atq= 1.As shown above, the optimal convergence-control parameterc0 of the HAM pro-vides us a convenient way to guarantee the quick convergenceof homotopy-series.In addition, there exist some other techniques to accelerate the convergence of agiven series. Among them, the so-called Pade approximant developed by the Frenchmathematician Henri Eugene Pade (1863 -1953) is widely applied, which gives the“best” approximation of a given function by a rational function of given order.

For a power series+∞

∑n=0

αn zn,

the corresponding[m,n] Pade approximant is expressed by


m∑

k=0am,k zk

n∑

k=0bm,k zk

,

wheream,k,bm,k are determined by the coefficientsα j ( j = 0,1,2,3, · · · ,m+n). Inmany cases the traditional Pade technique can greatly increase the convergence re-gion and rate of a given series. Note that such a traditional Pade approximant isa fraction, whose numerator and denominator are polynomials of z that is often aphysical parameter.

The so-called homotopy-Pade technique [16] is a combination of the traditionalPade technique with the homotopy analysis method. Regarding a homotopy-series

x(τ;q)∼ x0(τ)++∞

∑n=1

xn(τ) qn

as a power series ofq, we first employ the traditional[m,n] Pade technique aboutthe homotopy-parameterq to obtain the[m,n] Pade approximant

m∑

k=0Am,k(τ) qk

n∑

k=0Bm,k(τ) qk

, (2.83)

where the coefficientsAm,k(τ) andBm,k(τ) are determined by the firstm+n terms

x0(τ),x1(τ),x2(τ), · · · ,xm+n(τ)

of the homotopy-series. Then, settingq= 1 in (2.83), we have the so-called[m,n]homotopy-Pade approximation

x(τ)≈

m∑

k=0Am,k(τ)

n∑

k=0Bm,k(τ)

. (2.84)

Note that the Pade approximant (2.83) is a fraction, whose numerator and denom-inator are polynomials ofq, the homotopy-parameter without physical meanings.Therefore, both of the numerator and denominator of the[m,n] Homotopy-Pade ap-proximation (2.84) areunnecessaryto be polynomials: they can be any proper basefunctions. This provides us great freedom to choose different base functions. Thus,the so-called homotopy-Pade technique mentioned above ismore general than thetraditional ones.

For example, let us reconsider the nonlinear differential equation (2.32) in caseof λ = 0,x∗ = 1 andε > 0. By means of the HAM mentioned before, we have

x0 = cosτ,


x1 =132

c0 ε (cosτ − cos3τ) ,

x2 =c0ε

1024[(32+23c0ε)cosτ − (32+24c0ε)cos3τ + c0ε cos5τ] ,

...

wherec0 is the convergence-control parameter. By means of the homotopy-Padetechnique mentioned above, we have the [1,1] homotopy-Pad´e approximation

x(τ) ≈ 21 cosτ(23−2cos2τ)

, (2.85)

and the [2,2] homotopy-Pade approximation

x(τ)≈ 7723cosτ −513cos3τ − cos5τ(9099−1940cos2τ +50cos4τ)

, (2.86)

respectively, whereτ = ωt. Note that the numerator and denominator of these twoHomotopy-Pade approximations are not polynomials but trigonometric functions!Similarly, based on the homotopy-series

γ(q) =34

ε +(

3128

c0ε2)

q+

[

3512

c0ε2 (4+3c0ε)]

q2+ · · · ,

we obtain the corresponding[m,m] homotopy-Pade approximations, as shown in Ta-ble 2.10. Note that the homotopy-Pade approximation ofγ converges rather quicklyto the exact resultγ = 0.7177700110ε, even more quickly than the homotopy-series ofγ given in§2.3.4 byc0 = −4/(3ε). Besides, it should be emphasized thatall of these homotopy-Pade approximations ofx(τ) andγ are independentof theunknownconvergence-control parameterc0. Therefore, even if a bad value of theconvergence-control parameterc0 is chosen so that the corresponding homotopy-series is convergent slow or even divergent, we can obtain quickly convergenthomotopy-series by means of the homotopy-Pade technique!

Besides, the homotopy-Pade approximations ofx(τ) reveals thatx(τ) is inde-pendent of the physical parameterε. This is indeed true. As shown in§2.3.4, ratheraccurate approximations are obtained by means of the convergence-control param-eterc0 = −1/γ0 = −4/(3ε). Substitutingc0ε = −4/3 into x1,x2 and so on, it isfound that the corresponding homotopy-series ofx(τ) is indeed independent ofε.Note that the [1,1] homotopy-Pade approximation (2.85) isvery simple, but ratheraccurate, as shown in Fig. 2.17. Therefore,

x≈ 21 cosτ(23−2cos2τ)

, τ = 0.7177700110ε t (2.87)

is a simple but accurate approximation ofx(t) for all possible values ofε ∈(−∞,+∞). All of these illustrate the great potential of the so-called homotopy-Padetechnique.


Table 2.10 [m,m] homotopy-Pade approximations ofγ = ω2 in case ofλ = 0,x∗ = 1 for arbitraryε > 0 and the arbitrary unknown convergence-control parameterc0

m γ = ω2 given by the[m,m] homotopy-Pade approx.

1 0.71875ε2 0.7177996422ε3 0.7177708977ε4 0.7177700374ε5 0.7177700118ε6 0.7177700111ε7 0.7177700110ε8 0.7177700110ε9 0.7177700110ε10 0.7177700110ε

Fig. 2.17 Comparison ofthe [1,1] homotopy-Padeapproximation (2.85) ofx(τ)with the numerical ones incase ofλ = 0, x∗ = 1 andε >0. Solid line: approximationformula (2.85); Symbols:numerical result.

τ

x(τ)

0 1 2 3 4 5 6

-1

-0.5

0

0.5

1 λ = 0, x* = 1, ε >0

Furthermore, for arbitraryλ , ε andx∗, replacingx(τ) = x∗ y(τ), we can rewrite

γ x′′(τ)+λ x(τ)+ ε x3(τ) = 0, x(0) = x∗, x′(0) = 0

byγ y′′(τ)+λ y(τ)+ ε y3(τ) = 0, y(0) = 1, y′(0) = 0,

whereε = ε (x∗)2. In fact, this is the reason why we mainly considerx∗ = 1 in §2.3.So, in case ofλ = 0, using (2.87 ), we have a simple but accurate homotopy-Pad´eapproximation

y≈ 21 cosτ(23−2cos2τ)

, τ = 0.7177700110ε t,

which gives a rather simple but accurate approximation of the periodic oscillation

x(t)≈ 21x∗ cosτ(23−2cos2τ)

, τ = 0.7177700110ε (x∗)2 t (2.88)


with the corresponding period

T ≈ 7.4163√ε |x∗| (2.89)

for arbitrary values ofx∗ ∈ (−∞,+∞) andε ∈ (0,+∞). Note that the above accu-rate approximation ofT clearly reveals the relationship between the period and thephysical parameters and initial conditions. This illustrates that, using the HAM, wecan indeed obtain rather simple but accurate approximations of some problems withvery strong nonlinearity!

2.3.6 Convergence acceleration by optimal initial approximation

In the frame of the HAM, we have great freedom to choose the initial approximationx0(x) in the so-called zeroth-order deformation equation (2.42): any a periodic realfunction satisfyingx0(0) = x∗,x′(0) = 0 can be used. In§ 2.3.2, we choose the initialapproximation

x0(τ) = β +(x∗−β ) cosτ,

whereβ is defined by (2.34), although it can be an arbitrary real number in the-ory. According to the analysis of solution characteristic in § 2.3.1, whenx = 0 isthe stable equilibrium point, the body oscillates aboutx= 0 with the amplitudex∗,so that the centre of the motionx(τ) is exactly the stable equilibrium pointx = 0.In this case, we haveβ = 0 by the definition (2.34), which is physically correctand thus can give accurate homotopy-approximations, as mentioned before. How-ever, whenx= ±

√

|λ/ε| is the stable equilibrium point, the body oscillates aboutthe stable equilibrium point, but due to the non-asymmetry of the force f near thestable equilibrium point, the centre of the motionx(τ) is different from the stableequilibrium pointx= ±

√

|λ/ε|. In this case,β given by (2.34) is only an approxi-mation of the center of motionx(τ): the more asymmetric the forcef near the stableequilibrium pointx= ±

√

|λ/ε|, the worse the approximation. Therefore, in somecases ofβ 6= 0, the homotopy-series obtained by the “normal” approach ofthe HAMmentioned in§ 2.3.2 converge slowly.

For example, let us consider the case

λ =−32, ε = 2, x∗ = 1.

According to the analysis of solution characteristic in§ 2.3.1,x= 4 is a stable equi-librium point, and the body oscillates nearx= 4 but not exactly around it. Accordingto (2.34), we haveβ = 4, which gives the initial approximation

x0 = 4−3cosτ.

As mentioned in§ 2.3.2 and§ 2.3.3, we can obtain the curves of the discretesquared residualEm(c0) versusc0, as shown in Fig. 2.18, which suggests an op-timal convergence-control parameterc∗0 =−0.008. However, even by means of this


Fig. 2.18 Discrete squaredresidualE5(c0) versusc0 incase ofλ = 32, ε = 2 andx∗ = 1 by means of differentinitial approximationx0(τ).Solid line: x0 = 4− 3cosτ ;Dashed line:x0 = 2.9116−1.9116cosτ .

c0

E5(

c 0)

-0.03 -0.02 -0.01 0100

101

102

103

104

λ = -32, ε = 2, x* = 1

optimal convergence-control parameter, the corresponding homotopy-series con-verges rather slowly, as shown in Table 2.11. Note that the discrete squared resid-ual E0(c∗0) is rather large, which indicates that the initial approximation x0(τ) =β +(x∗−β )cosτ with the definition (2.34), i.e.

x0(τ) = 4−3cosτ,

is a bad initial approximation.

Table 2.11 The discrete squared residualEm(c0) and homotopy-approximations ofγ = ω2 in caseof λ = −32, ε = 2 andx∗ = 1 by β = 4 and the optimal convergence-control parameterc∗0 =−1/125.

m, order of approx. Em(c∗0) γ = ω2

0 18046 77.55 127.2 32.84910 30.1 33.22715 10.1 31.48920 2.5 31.94525 1.4 31.34830 0.3 31.64340 5.8×10−2 31.53750 2.2×10−2 31.49160 1.2×10−2 31.468

A simple way to modify the homotopy-approximations is to usea better initialapproximationx0(τ). Note that we have great freedom to construct the homotopy ofequationsE(q) : E0 ∼ E1, which governs the homotopy of functions ˜x(τ;q) : x0(τ)∼x(τ). Note also that, in theory,β in the initial approximation can be anarbitraryreal number! So, using the same initial approximationx0(τ) = β +(x∗−β )cosτ butregardingβ as a unknown constant, we can getγ0 by (2.68) that is now dependent on


Table 2.12 The discrete squared residualEm(c0) and homotopy-approximations ofγ = ω2 in caseof λ = −32, ε = 2 andx∗ = 1 by β = 2.9116 and the optimal convergence-control parameterc∗0 =−2/125.


0 649.3 24.3465 6.0 30.88910 0.29 31.35415 3.3×10−2 31.35920 5.8×10−3 31.39625 1.2×10−3 31.41030 2.7×10−4 31.41535 6.5×10−5 31.41840 1.6×10−5 31.41945 4.3×10−6 31.42050 1.2×10−6 31.42060 8.8×10−8 31.420

Table 2.13 The [m,m] homotopy-Pade approximations ofγ = ω2 in case ofλ =−32, ε = 2 andx∗ = 1 by different initial approximationsx0 and different optimal convergence-control parameter.

m x0 = 4−3cosτ , x∗0 = 2.9116−1.9116cosτ ,c∗0 =−1/125 c∗0 =−2/125

2 35.576 31.6324 32.222 31.5646 31.585 31.4498 31.451 31.42310 31.426 31.42012 31.421 31.42014 31.420 31.42016 31.420 31.42018 31.420 31.42020 31.420 31.420

the unknown constantβ . Then, the discrete squared residualE0 given by the aboveinitial approximationx0(τ) is a function ofβ . Obviously, the “best” or “optimal”initial approximationx0(τ) is given by the minimum ofE0 at β = β ∗, whereβ ∗ iscalled the “best” or “optimal” value ofβ .

In case ofλ =−32,ε = 2 andx∗ = 1, by means of the above-mentioned approachand using the Mathematica commandMinimize under the conditionβ > 1, weobtain the minimum of the squared residual of governing equation for the initialapproximation atβ ∗ = 2.9116, which gives us the optimal initial approximation

x0(τ) = 2.9116−1.9116cosτ.

Similarly, using this optimal initial approximation, we can obtain the correspond-ing discrete squared residualE5(c0) versusc0, as shown in Fig. 2.18, which sug-


Fig. 2.19 Comparison ofthe numerical result with thehomotopy-approximations incase ofλ = 32, ε = 2 andx∗ = 1 by means ofc0 =−2/125 and the optimalβ ∗ =2.9116. Symbols: numericalresult; Solid line: 15th-orderhomotopy-approximationDot-Dashed line: initial guessx0 given by β ∗ = 2.9116;Dashed line: initial guessx0given byβ = 4.

t

x(t)

0 0.5 1 1.5 2 2.50

2

4

6

8λ = -32, ε = 2, x* = 1

gests us an optimal convergence-control parameterc∗0 = −2/125. By means of theabove optimal initial approximation and the optimal convergence-control parame-ter c∗0 = −2/125, the corresponding homotopy-series ofγ = ω2 converges muchfaster than that given by the normal initial approximationx0 = 4−3cosτ, as shownin Table 2.12. Note that, the corresponding[m,m] homotopy-Pade approximationsconverge faster, too, as shown in Table 2.13. Besides, the corresponding homotopy-approximations ofx(τ) agree well with the numerical ones, as shown in Fig.2.19.Note that the optimal initial approximation

x0(τ) = 2.9116−1.9116cosτ

is more close to the exact ones than the normal initial approximation

x0(τ) = 4−3cosτ.

According to Tables 2.11 and 2.12, the discrete squared residual given by the normalinitial approximationx0(τ) = 4−3cosτ is 27.5 times larger than that given by theoptimal onesx0(τ) = 2.9116− 1.9116cosτ. This is the essential reason why thehomotopy-series given by the optimal initial approximation converges much faster.

The optimization of initial approximations has general meanings in the frame ofthe HAM. In general, better homotopy-approximations are obtained by better initialapproximations, if others are the same. Besides, as shown later in Chapter 8, usingthe great freedom on the choice of initial approximations, one can obtain multiplesolutions of some nonlinear differential equations by means of the HAM.

2.3.7 Convergence acceleration by iteration

As mentioned before, the HAM provides us great freedom to choose the initial ap-proximation. It is due to this freedom that we can choose an optimal initial approx-


imation so as to greatly accelerate the convergence of homotopy-series, as show in§ 2.3.6. Here, we further illustrate that, by means of the freedom on the choice ofinitial approximation, an iteration approach can be introduced in the frame of theHAM, which can greatly accelerate the convergence of the homotopy-series, too.

Note that any a real function, which satisfies the initial condition x(0) = x∗ andx′(0) = 0, can be used as an initial approximationx0(τ) in the zeroth-order deforma-tion equation (2.40). In§ 2.3.6, we illustrate that the convergence of the homotopy-series can be greatly accelerated by means of an optimal initial approximation. Ob-viously, the better the initial approximations, the fasterthe convergence of the cor-responding homotopy-series. Clearly, theMth-order homotopy-approximation

x(τ)≈ x0(τ)+M

∑k=0

xk(τ) (2.90)

satisfies the initial conditionx(0) = x∗ andx′(0) = 0, and is often better than theinitial approximationx0(τ), if the convergence-control parameterc0 is properlychosen. So, it is natural to use the aboveMth-order homotopy-approximation asa new initial approximationx0, i.e. x0(τ) = x(τ). In this way, a better homotopy-approximation can be obtained in general. This provides us an iteration approach inthe frame of the HAM. For simplicity, we call the above iteration approach as theMth-order homotopy-iteration.

Table 2.14 The discrete squared residual and homotopy-approximations of γ = ω2 in case ofλ = −32, ε = 2 andx∗ = 1 by the 3rd-order homotopy-iteration approach with the normal initialapproximationx0 = 4−3cosτ and the corrsponding optimal convergence-control parameter c∗0 =−1/125.

m, iteration times Em(c∗0) γ = ω2

1 286.5 43.8502 34.3 32.8183 9.8 31.7894 2.9 31.5276 0.2 31.4278 1.7×10−2 31.41910 1.2×10−3 31.41911 3.1×10−4 31.42012 8.3×10−5 31.42014 5.8×10−6 31.42016 4.0×10−7 31.42018 2.7×10−8 31.42020 1.9×10−9 31.420

Theoretically speaking, the periodic oscillation governed by (2.32) should beexpressed by an infinite series


x(τ) =+∞

∑k=0

ak cos(kτ).

However, the first few terms are much more important than the high frequencyterms, because the high frequency terms contribute very little to the accuracy ofx(τ). So, in practice, a truncated expression

x(τ)≈N

∑k=0

ak cos(kτ) (2.91)

is accurate enough for a large enough value ofN. In other words, our homotopy-approximations contain the first(N + 1) terms, and all other terms with higherfrequency, such as cos(N+ 1)τ,cos(N+ 2)τ and so on, are neglected. To do so,we simply delete all higher-frequency terms ofδn−1 in the high-order deformationequation (2.55) before we solve it. Mathematically speaking, we approximateδn−1

by

δn−1 ≈N

∑k=0

An,k cos(k τ),

where

An,0 =1

2π

∫ 2π

0δn−1(τ)dτ, An,k =

1π

∫ 2π

0δn−1(τ)cos(kτ)dτ, 1≤ k≤ N.

In this way, the homotopy-approximation at each iteration can be expressed by(2.91) with a finite number of terms. This strategy of truncation is necessary for theiteration approach of the HAM. Otherwise, the length of homotopy-approximationsincreases exponentially in a few iterations so that the computational efficiencyquickly becomes unacceptable. The corresponding Mathematica code (with itera-tion approach) is free available at

http://numericaltank.sjtu.edu.cn/HAM.htm.For example, let us consider again the caseλ =−32,ε = 2 andx∗ = 1. To com-

pare the homotopy-approximations by the iteration approach with those given in§ 2.3.6, we use here the same optimal convergence-control parameterc0 =−1/125and the same initial approximationx0 = 4−3cosτ. Without loss of generality, letus first useM = 3, i.e. a 3rd-order homotopy-iteration approach. Besides,we useN = 21, say,x(τ) is approximated by the first 21 low-frequency terms (from cosτ tocos21τ) only. The homotopy-approximations at each iteration and the correspond-ing discrete squared residual are listed in Table 2.14. Comparing these results withthose in Table 2.11, the approximations given by the homotopy-iteration approachconverge much faster than the normal homotopy-analysis method: the 11st iterationgives the exact valueγ = 31.420 in only 4.8 seconds by a laptop (MacBook Pro,2.8GHz Inter Core 2 CPU, 4 GB EMS memory). Note that, the normal approachtakes about 46.3 seconds, i.e. more than 9 times longer, to get the 60th-order ap-proximationγ = 31.468, which still has a small difference from the exact resultγ = 31.420.


Fig. 2.20 CPU times versusm in case ofλ = 32, ε = 2and x∗ = 1 by means ofx0 = 4− 3cosτ and c0 =−1/125. Solid line: 3rd-order homotopy-iterationapproach (m denotes theiteration times); Dashed line:normal approach withoutiteration (m denotes the orderof homotopy-approximation).

m

CP

Utim

e

10 20 30 40 50 600

10

20

30

40

50

λ = -32, ε =2, x* = 1, c0=-0.008

Fig. 2.21 Squared residualversusm in case ofλ =32, ε = 2 andx∗ = 1 bymeans ofx0 = 4− 3cosτandc0 =−1/125. Solid line:3rd-order homotopy-iterationapproach (m denotes theiteration times); Dashed line:normal approach withoutiteration (m denotes the orderof homotopy-approximation).

m

Squ

ared

resi

dual

10 20 30 40 50 6010-21

10-16

10-11

10-6

10-1

λ = -32, ε =2, x* = 1, c0=-0.008

Fig. 2.22 Squared residualversus CPU times in case ofλ = 32, ε = 2 andx∗ = 1by means ofx0 = 4−3cosτandc0 = −1/125. Dashedline: normal approach withoutiteration. Solid line: 3rd-orderhomotopy-iteration approach;Dot-dashed line: 2nd-orderhomotopy-iteration approach;Dot-dot-dashed line: 4th-order homotopy-iterationapproach.

CPU times

Squ

ared

resi

dual

10 2010-21

10-16

10-11

10-6

10-1

λ = -32, ε =2, x* = 1, c0=-0.008


Let us compare the accuracy and used CPU time of the iterationapproach andthe normal HAM without iteration in details. It is found thatthe CPU time in-creaseslinearly with the iteration times for the iteration approach, but increasesexponentiallywith the order of approximation for the non-iteration approach, asshown in Fig. 2.20. Thus, the iteration approach is computationally more efficient.More importantly, for the iteration approach, the squared residual of the homotopy-approximations decreasesexponentiallyas the iteration timesm increases, as shownin Fig. 2.21, until the minimum of the squared residualEmin(N) = 5.2× 10−21

is arrived. However, for normal approach without iteration, the squared residualdecreasesalgebraically as the order of approximationm increases, as shown inFig. 2.21. Thus, the approximations given by iteration approach converges muchfaster.

The discrete squared residual versus the CPU time of two approaches are givenin Fig. 2.22, which indicates that the squared residual given by the iteration ap-proach decreasesexponentiallywith the CPU time until its minimum valueEmin(N),whereas the squared residual given by the non-iteration approach decreases muchmore slowly. That means, using the same CPU time, one can obtain much moreaccurate homotopy-approximations by means of the iteration approach. All of theabove conclusions have general meanings: they are correct for different-order ofhomotopy-iteration approaches, as shown in Fig. 2.22. Notethat the approxima-tions given by the 3rd-order homotopy-iteration approach converges a little fasterthan those given by the 2nd and 4th-order homotopy-iteration approaches, whereasthe approximations given by the 4th-order homotopy-iteration approach convergea little faster than those given by the 2nd-order. In practice, the 2nd, 3rd and 4th-order homotopy-iteration approaches are good enough, and higher-order iterationformulas are often unnecessary.

As shown in Figs. 2.21 and 2.22, when the above-mentioned minimum of thesquared residualEmin = 5.2× 10−21 is arrived, one can not get better homotopy-approximations by more iterations. This is mainly because we use finite number(denoted byN) of terms in (2.91) to approximatex(τ). Theoretical speaking, theminimum squared residualEmin of the iteration approach is dependent uponN, andit should decrease if more terms in (2.91) are employed. Thisis indeed true in prac-tice, as shown in Table 2.15, which illustrates thatEmin is dependent uponN, thenumber of terms in (2.91), but has nothing to do withM, the order of homotopy-approximation formula in (2.90) at each iteration.

Letan denote the coefficient of the term cos(nτ) in (2.91). Obviously, each coeffi-cientan (0≤ n≤ 21) is modified at each homotopy-iteration, as listed in Table 2.16.Note that, as the iteration times increases, each coefficient an (0≤ n≤ 21) convergesquickly to a fixed value. For example, the coefficientsa0,a1,a5,a10 anda21 convergeto 2.8027, -2.1860,−3.87×10−3, 1.33×10−6 and−3.22×10−14, respectively. Ob-viously, the coefficients for other higher-frequency termsare rather small, and thuscan be neglected without loss of the accuracy of the homotopy-approximationx(τ).

To conform the generality of the homotopy-iteration approach, we further ap-ply it to all cases mentioned above in§ 2.3. The 3rd-order homotopy-iteration for-mula is used with the normal initial approximationsx0 = β +(x∗−β )cosτ. At the


Table 2.15 The minimum of the squared residualEmin in case ofλ = −32,ε = 2,x∗ = 1 withc0 =−1/125 andx0 = 4−3cosτ by means of the different order homotopy-iteration approachesand the different number of termsN in (2.91).

N 2nd-order iteration 3rd-order iteration 4th-order iterationformula (M = 2 ) formula (M = 3 ) formula (M = 4 )

5 0.389 0.389 0.38910 5.51×10−7 5.51×10−7 5.51×10−7

15 2.96×10−13 2.96×10−13 2.96×10−13

21 5.2×10−21 5.2×10−21 5.2×10−21

23 1.4×10−23 1.4×10−23 1.4×10−23

Table 2.16 The modification of the coefficientan of (2.91) at them times iteration in case ofλ = −32, ε = 2 andx∗ = 1 by the 3rd-order homotopy-iteration approach with the normal initialapproximationx0 = 4−3cosτ and the corresponding optimal convergence-control parameterc∗0 =−1/125.

m a0 a1 a5 a10 a21

1 2.9863 -2.2963 -3.60×10−4 0 03 2.8236 -2.1977 -2.02×10−3 1.52×10−7 -4.65×10−17

5 2.8063 -2.1863 -3.17×10−3 6.10×10−7 -2.57×10−15

10 2.8027 -2.1861 -3.83×10−3 1.26×10−7 -2.44×10−14

15 2.8027 -2.1860 -3.87×10−3 1.33×10−6 -3.15×10−14

20 2.8027 -2.1860 -3.87×10−3 1.33×10−6 -3.22×10−14

25 2.8027 -2.1860 -3.87×10−3 1.33×10−6 -3.22×10−14

30 2.8027 -2.1860 -3.87×10−3 1.33×10−6 -3.22×10−14

Table 2.17 The discrete squared residual and homotopy-approximations of γ = ω2 in case ofλ =−9/4, ε = 1 andx∗ = 1 by the 3rd-order homotopy-iteration approach with the normal initialapproximationx0 = (3− cosτ)/2 and the corresponding optimal convergence-control parameterc∗0 =−0.1632.

m, iteration times Em(c∗0) γ = ω2

1 5.0×10−4 3.99322 1.1×10−6 3.92783 5.9×10−9 3.92784 3.3×10−11 3.92785 1.9×10−13 3.9278

beginning of iteration, the convergence-control parameter c0 is unknown. An opti-mal value ofc0 is determined at the first iteration by the minimum of the squaredresidual of the 3rd-order homotopy-approximation, and this optimal value is usedfor all other iterations. It is found that, for all of these cases, the approximationsgiven by the homotopy-approach converge rather quickly, asshown in Tables 2.17to 2.19. Note that, we obtain exact values ofγ = ω2 mostly by two or three it-erations. And the squared residual decreases rather quickly. Note that, in case of


Table 2.18 The homotopy-approximations ofγ = ω2 in three different cases by the 3rd-orderhomotopy-iteration approach with the normal initial approximation and the corresponding optimalconvergence-control parameterc∗0

iteration times λ = 9/4,ε = 1, λ = 0, ε = 1, λ = 4,ε =−1,x∗ = 1, x∗ = 1, x∗ =−1,c∗0 =−0.3333 c∗0 =−1.3314 c∗0 =−0.3075

1 2.9921875001 0.7187500657 3.24278846442 2.9921730367 0.7177745854 3.24277709173 2.9921730364 0.7177700220 3.24277709174 2.9921730364 0.7177700111 3.24277709175 2.9921730364 0.7177700110 3.2427770917

Table 2.19 The discrete squared residualEm(c∗0) in three different cases by the 3rd-orderhomotopy-iteration approach with the normal initial approximation and the corresponding opti-mal convergence-control parameterc∗0

iteration times λ = 9/4,ε = 1 λ = 0, ε = 1 λ = 4,ε =−1x∗ = 1, x∗ = 1, x∗ =−1,c∗0 =−0.3333 c∗0 =−1.3314 c∗0 =−0.3075

1 6.0×10−10 2.2×10−6 4.0×10−10

2 5.7×10−20 1.7×10−11 5.8×10−22

3 4.0×10−31 9.5×10−17 6.9×10−33

4 3.3×10−38 4.9×10−22 6.6×10−39

5 3.3×10−38 7.4×10−26 6.6×10−39

λ = 9/4,ε = 1,x∗ = 1 andλ = 4,ε = −1,x∗ = −1, the discrete squared resid-ual stops decreasing at the 5th iteration, mainly because weapproximatex(τ) bythe first 21 low-frequency terms. All of these illustrate that, like the homotopy ap-proach using the optimal initial approximation, the homotopy-iteration approachgives quickly convergent results with rather high computational efficiency in gen-eral.

The results given by the homotopy-iteration approach is still analytic approxima-tions, because, different from numerical solutions, our homotopy-approximationsare expressed by the continuous base functions cos(nτ). Therefore, it is easy for usto givearbitrary order derivatives ofx(τ) at anytime 0≤ τ <+∞. This is impossi-ble for any numerical methods. So, in essence, the homotopy-approximations givenby the iteration approach are still analytic results.

The homotopy-iteration approach has general meanings and can be widely ap-plied to greatly accelerate the convergence of homotopy-approximations, as shownlater in this book (see Chapter 9, Chapter 8 and so on).


2.3.8 Flexibility on the choice of auxiliary linear operator

In addition, we have great freedom to choose the auxiliary linear operatorL in theso-called zeroth-order deformation equation (2.42). As pointed out in§ 2.3.2, thelinear operatorL0(x) = x′′ + λ x, which is exactly the linear part of the originalgoverning equation, can not give periodic approximations whenλ < 0 or λ = 0.However, due to the freedom on the choice of the auxiliary linear operatorL , wecan simply choose the auxiliary linear operatorL x = x′′ + x in the zeroth-orderdeformation equation (2.42) forall possible physical parameters related to periodicsolutions, even includingλ ≤ 0. So, different from perturbation techniques, whoseauxiliary linear operators are closely connected to the small/large physical param-eters and types of considered equations, we have rather large freedom to choosea proper auxiliary linear operator to get accurate approximations by means of theHAM, as shown in§ 2.3.

Besides, due to the great freedom on constructing homotopy of equations andfunctions, we introduce the convergence-control parameter c0 in the zeroth-orderdeformation equation (2.42). As shown in§ 2.3.3 and§ 2.3.4, the convergence-control parameterc0 provides us a convenient way to guarantee the convergenceof homotopy-series solution. Thus, the convergence-control parameterc0 has anessential role in the frame of the HAM. It is interesting that, if we define such anauxiliary linear operatorL = L /c0, then the zeroth-order deformation equation(2.42) can be rewritten as

(1−q)L [x(τ;q)− x0(τ)] = q

γ(q) x′′(τ;q)+λ x(τ;q)+ ε x3(τ;q)

.

So, mathematically, the convergence-control parameterc0 can be regarded as a partof the auxiliary linear operatorL . In essence, different values of the convergence-control parameterc0 corresponds to different auxiliary linear operatorsL =L /c0.Especially, the optimal convergence-control parameterc∗0 gives the optimal auxil-iary linear operatorL . So, in essence, it is due to the freedom on the choice ofthe auxiliary linear operator that we introduce the so-called convergence-control pa-rameterc0 in the zeroth-order deformation equation (2.42). Note thatthe concept ofconvergence-control is a key of the HAM.

Liao and Tan (2007) illustrated that, in the frame of the HAM,we have muchlarger freedom on the choice of the auxiliary linear operator L than we had thoughttraditionally. For example, let us consider the zeroth-order deformation equation(2.42). Using the freedom on the choice of the auxiliary linear operatorL in (2.42),we can choose such an auxiliary linear operator

L x=∂ 2κx∂τ2κ +(−1)κ+1x, κ = 1,2,3, · · · , (2.92)

whereκ ≥ 1 is anarbitrary positive integer. Whenκ = 1, we haveL x= x′′+x, sothatL is exactly the same as the 2nd-order auxiliary linear operator L defined by(2.63). Thus, the above auxiliary linear operator is more general than (2.63). Note


that all mathematical formulas in§ 2.3.2 and§ 2.3.3, such as the high-order defor-mation equations (2.55) and so on, keep the same in form, except thatL defined by(2.63) is replaced byL defined by (2.92), a more general auxiliary linear operator.Note that we search for periodic solution expressed by (2.28). Thus, as mentionedin § 2.3.7,δn−1 in (2.55) can be expressed as a sum of some cosine functions. Thus,by means of the general auxiliary linear operator defined by (2.92), the high-orderdeformation equation (2.55) has a special solution

xsn(τ) = χn xn−1(τ)+ c0 L

−1 [δn−1(Xn−1,Γn−1)] , (2.93)

where the inverse operatorL −1 is defined by

L−1[cos(mτ)] =

(−1)κ+1 cos(mτ)(1−m2κ)

, m 6= 1, (2.94)

andL

−1(C) = (−1)κ+1C (2.95)

for any constantC.Whenκ = 2, the auxiliary linear operator (2.92) becomesL x= x(4)− x, where

u(4) denotes the 4th-order differentiation with respect toτ. This 4th-order linearoperator has the property

L(

C1cosτ +C2sinτ +C3 eτ +C4 e−τ)= 0 (2.96)

for any constant coefficientsC1,C2,C3 andC4. So, whenκ = 2, the general solutionof the high-order deformation equation (2.55) reads

xn(τ) = χn xn−1(τ)+ xsn(τ)+C1cosτ +C2sinτ +C3 eτ +C4 e−τ , (2.97)

where the special solutionxsn(τ) is given by (2.93). Note that there arefour integral

coefficientsC1,C2,C3 andC4, but we have onlytwo initial conditions defined by(2.56). Note that we search for periodic solutions of (2.32), expressed by (2.28).However, the termseτ and e−τ are not periodic functions, and thusdisobeythesolution expression (2.28). Therefore, they can not appearin the expression ofxn(τ).So, in order to obtain periodic solution ofxn(τ), we must enforceC3 = C4 = 0. Inother words, the solution expression (2.28) implies one additional period condition

xn(τ) = xn(τ +2π),

which enforcesC3 = C4 = 0. This condition comes from the periodic property ofx(τ), i.e.

x(τ) = x(τ +2π).

Thereafter,C1 andC2 are uniquely determined by the two initial conditions definedby (2.56).

Similarly, whenκ = 3, the auxiliary linear operator (2.92) becomes


L x=d6xdτ6 + x,

and the general solution of the corresponding high-order deformation equation(2.55) reads

xn(τ) = χn xn−1(τ)+ xsn(τ)+C1cosτ +C2sinτ

+ e√

3τ/2(C3cosτ +C4sinτ)+e−√

3τ/2(C5cosτ +C6sinτ)], (2.98)

where the special solutionxsn(τ) is given by (2.93), andCi is a constant coefficient.

Similarly, to obtain periodic solution ofxn(τ), we must enforce

C3 =C4 =C5 =C6 = 0.

The left two constantsC1 andC2 are uniquely determined by the two initial condi-tions defined by (2.56).

Similarly, for any positive integersκ ≥ 1, we can always obtain aperiodicsolu-tion xn(τ) of the high-order deformation equations (2.55) and (2.56) by means of the(2κ)th-order auxiliary linear operatorL defined by (2.92). Here, it should be em-phasized that, althougheτ , eτ cosτ andeτ sinτ, which tend to infinity asτ → +∞,are the so-called secular terms, the termse−τ , e−τ cosτ, e−τ sinτ tend to zero asτ → +∞ and thus do not belong to the traditional definition of secular term. So,using the traditional ideas of “avoiding the secular terms”, we cannot avoid the ap-pearance of the non-periodic termse−τ , e−τ cosτ, e−τ sinτ in xn(τ). Therefore, theconcept ofsolution expressionis more general than the traditional ideas of “avoid-ing secular terms”. Indeed, the concept of solution expression has a very importantrole in the frame of the HAM.

Fig. 2.23 Squared residualversusc0 in case ofλ = 0,ε = 0 andx∗ = 1 by meansof the 4th-order auxiliarylinear operatorL x = x′′′′ −x (corresponding toκ =2). Dashed line: 1st-orderapprox.; Dot-dashed line: 3rd-order approx.; Dot-dot-dashedline: 5th-order approx.; Solidline: 7th-order approx.

c0

Em(c

0)

0 5 10 15 20 2510-6

10-5

10-4

10-3

10-2

10-1

λ = 0, ε =1, x* = 1, κ = 2

Without loss of generality, let us consider here the caseλ = 0,ε = 1 andx∗ = 1.In this case, the corresponding initial approximation readsx0 = cosτ. First of all, letus consider the case ofκ = 2, i.e. the 4th-order differential operatorL x= x′′′′− x


is used as the auxiliary parameter. In a similar way as mentioned in§ 2.3.3, we canobtain the curves of the discrete squared residualEm(c0) versus the convergence-control parameterc0, as shown in Fig. 2.23. The discrete squared residual at the7th-order of approximation arrives the minimum atc0 = 15.97. So, we choose theoptimal convergence-control parameterc∗0 = 16. It is found that the discrete squaredresidual decreases monotonously asm, the order of approximation, increases, andγ =ω2 converges to the exact value 0.7177700110, as shown in Table2.20. Besides,the corresponding[m,m] homotopy-Pade approximations ofγ = ω2 also convergeto the exact value 0.7177700110, as shown in Table 2.21. Furthermore, even the 5th-order approximation is rather accurate: the 5th-order approximation ofγ has only0.006% relatively error, and the 5th-order approximation of x(τ) agrees well withthe numerical ones, as shown in Fig. 2.24. All of these indicate that, the homotopy-approximations given by means of the 4th-order auxiliary linear operator

L x= x′′′′− x,

corresponding toκ = 2 in (2.92), are indeed convergent. Therefore, we can applythe HAM to obtain convergent homotopy-series solution of the original2nd-ordernonlinear differential equation (2.32) even by means of the4th-orderdifferentialoperator as the auxiliary linear operatorL . This indicates that the HAM indeedprovides us extremely large freedom to choose the auxiliarylinear operator.

Table 2.20 The discrete squared residual and homotopy-approximations of γ = ω2 in case ofλ = 0, ε = 1 andx∗ = 1 by the 4th-order (κ = 2) auxiliary linear operatorL x = x′′′′ − x withthe normal initial approximationx0 = cosτ and the corresponding optimal convergence-controlparameterc∗0 = 16.


3 8.3×10−5 0.71725861695 9.1×10−6 0.717724536410 1.5×10−7 0.717841918020 1.1×10−9 0.717772953630 2.4×10−11 0.717770350440 1.2×10−12 0.717770072550 9.0×10−14 0.717770025760 8.7×10−15 0.717770015270 1.1×10−15 0.717770012480 1.6×10−16 0.717770011590 2.8×10−17 0.7177700112100 5.4×10−18 0.7177700111

Similarly, we can further investigate the 6th-order auxiliary linear operator, corre-sponding toκ = 3 in (2.92). It is found that a convergent homotopy-series solutioncan be found by means of this 6th-order auxiliary linear operator, too. Therefore,one can obtain convergent series solution by theκ th-order auxiliary linear oper-ator (2.92) for anarbitrary positive integerκ ≥ 1. However, it is found that the


Table 2.21 [m,m] homotopy-Pade approximations ofγ = ω2 in case ofλ = 0, ε = 1 andx∗ = 1by means of the 4th-order auxiliary linear operatorL x= x′′′′ −x (corresponding toκ = 2)

m γ = ω2 given by the[m,m] homotopy-Pade approx.

5 0.717758579010 0.717770070415 0.717770011720 0.717770011125 0.717770011030 0.717770011035 0.717770011040 0.717770011045 0.717770011050 0.7177700110

Fig. 2.24 Comparison ofthe homotopy-approximationwith the numerical result incase ofλ = 0,ε = 0,x∗ = 1 bymeans ofc∗0 = 16 and the 4th-order auxiliary linear operatorL x= x′′′′ −x (correspondingto κ = 2). Solid line: 5th-orderapproximation; Symbols:numerical result.

t

x(t)

0 2 4 6 8 10-1.5

-1

-0.5

0

0.5

1

1.5

λ = 0, ε = 1, x* = 1, c0=16, κ=2

Fig. 2.25 Discrete squaredresidualEm versus the ordermof homotopy-approximationin case ofλ = 0, ε = 0,x∗ = 1by means of different auxil-iary linear operators definedby (2.92). Solid line:κ = 1with c∗0 = −4/3; Dashedlined: κ = 2 with c∗0 = 16;Dot-dashed line:κ = 3 withc∗0 =−160.

m

Em(c

0* )

20 40 60 80 10010-35

10-30

10-25

10-20

10-15

10-10

10-5

κ = 1

κ = 2

κ = 3

λ = 0, ε = 1, x* = 1


homotopy-series given by the 6th-order auxiliary linear operator converges moreslowly than that given by the 4th-order ones, as shown in Fig.2.25. Besides, thehomotopy-series given by the 2nd-order auxiliary linear operator converges fasterthan that given by the 4th and 6th-order ones, as shown in Fig.2.25. In general,the larger the value ofκ , the more slowly the corresponding homotopy-series con-verges. Therefore, the 2nd-order auxiliary linear operator L x = x′′+ x may be thebest among the infinite number of auxiliary linear operatorsdefined by (2.92).

It is found that the general auxiliary linear operator defined by (2.92) only worksfor the cases withx = 0 being a stable equilibrium point. This fact warns us thatsuch kind of freedom is valuable only when it can be properly used. In practice,if such kind of freedom is used properly, one can solve some nonlinear problemsin a very simple way. For example, as shown in Chapter 14, a 2nd-order nonlineardifferential equation (Gelfand equation) can be successfully solved by means of a4th-order auxiliary linear operator in a rather simple way.

In summary, in the frame of the HAM, the 2nd-order nonlinear differential equa-tion (2.32) can be even transferred into an infinite number of4th or 6th-order lineardifferential equations. To the best of my knowledge, this isagainst the mainstream ofthe traditional thought: it is often believed that onemth-order nonlinear differentialequation should be transferred into an infinite number of linear differential equa-tions with the same or lower order. So, although this simple example has no practi-cal meaning (because the 2nd-order auxiliary linear operator L x= x′′+x seems tobe the best for this nonlinear oscillation problem), it shows in theory that we indeedhave much larger freedom to solve nonlinear problems than wethought and believedbefore. I personally believe, if this kind of extremely large freedom is properly used,more accurate and better solutions of some difficult nonlinear problems should befound. Indeed, “the essence of mathematics lies entirely inits freedom”, as pointedout by Georg Cantor (1845-1918).

2.4 Concluding remarks and discussions

In this chapter, the basic ideas of the HAM are described by means of two simple ex-amples: one is the nonlinear algebraic equationf (x) = 0, the other is the 2nd-ordernonlinear differential equation (2.32) for periodic oscillations of a body. Althoughthese two examples are rather simple, all methods mentionedin this chapter havegeneral meanings, and nearly all conclusions are qualitatively correct in general.

The HAM has a solid base, i.e. the homotopy in topology, whichconnects twodifferent things with a few same mathematical aspects by a continuous mapping.Given an equation that has at least one solution, denoted byE1, one first finds aproper, much simpler equation with a known solution, denoted byE0, and then con-structs such a homotopy of equationE (q) : E0∼ E1 that it deforms (or varies)contin-uouslyfrom the initial equationE0 to the original equationE1 when the homotopy-parameterq ∈ [0,1] increases from 0 to 1. The so-called zeroth-order deformationequation defined by such kind of continuous deformation (or variation) of equa-

2.4 Concluding remarks and discussions 75

tions is a base of the HAM. In theory, given a nonlinear equationE1 having at leastone unknown solution, one has extremely large freedom to construct such a kindof zeroth-order deformation equation in many different ways. Besides, it has noth-ing to do with the existence of any small/large physical parameters: one can alwaysconstruct such a zeroth-order deformation equation, even if the given equationE1

does not contain any small/large physical parameters. In essence, it is the extremelylarge freedom on constructing a homotopy of equations that makes the HAM ratherpowerful for strongly nonlinear problems and quite different from other traditionalanalytic techniques.

Firstly of all, it has nothing to do with the existence of any small/large physicalparameters to construct a homotopy of equations, i.e. the zeroth-order deformationequation. So, different from perturbation techniques, theHAM works even if a givennonlinear problem does not contain any small/large physical parameters, called tra-ditionally perturbation quantities. So, in theory, the HAMcan be applied much morewidely than perturbation techniques, especially for thosewith strong nonlinearity.

Secondly, it is due to the freedom on constructing the zeroth-order deformationequation that the so-called convergence-control parameter c0 can be introduced,which provides a convenient way to guarantee the convergence of homotopy-series.In other words, by means of properly choosing the value of such a non-zero aux-iliary parameterc0, one cancontrol the convergence of homotopy-series and givean optimal approximation with high accuracy. Thus, the convergence-control is afundamental concept of the HAM. In fact, it is the so-called convergence-controlparameterc0 that makes the HAM differs from all other analytic techniques.

Thirdly, it is due to the freedom on constructing the zeroth-order deformationequation that we have extremely large freedom to choose the so-called auxiliarylinear operatorL . In general, using such kind of freedom, one can obtain better ap-proximations by choosing a better auxiliary linear operator. For example, it is basedon such kind of freedom that we choose the auxiliary linear operatorL x= x′′+x toget all periodic solutions for all possible physical parameters of the nonlinear oscil-lation equation (2.32), even includingλ = 0 andλ < 0. Note that, whenλ ≤ 0, theauxiliary linear operatorL x= x′′+ x is qualitatively quite different from the linearpartx′′ +λx of the original nonlinear equation (2.32). Such kind of freedom is solarge that, in§ 2.3.8, the convergent homotopy-series solution of the 2nd-order non-linear differential equation (2.32) can be obtained even bymeans of the 4th-orderauxiliary linear operator

L x= x′′′′− x

and the 6th-order auxiliary linear operator

L x=d6xdτ6 + x,

respectively! This illustrates that we indeed have extremely large freedom on thechoice of the auxiliary linear operatorL .

Besides, it is due to the freedom on constructing the zeroth-order deformationequation that we can regard some constants, such asγ = ω2 in (2.32), as a func-


tion of the homotopy-parameterq∈ [0,1], as illustrated in§ 2.3.2. In this way, theso-called secular terms are easily avoided and the accurateapproximations of theperiodic solutions are obtained.

Finally, it is due to the freedom on constructing the zeroth-order deformationequation that we have freedom to choose different initial approximations. Based onsuch kind of freedom, the optimal initial approximation approach in§ 2.3.6 andthe iteration approach in§ 2.3.7 are developed, which can greatly accelerate theconvergence of homotopy-series. In practice, the optimal initial approximation, andespecially the iteration approach, are strongly suggestedfor problems with strongnonlinearity.

Note that, the convergence-control is a fundamental concepts of the HAM. Asrevealed in Theorem 2.3, a non-zero auxiliary parameter canfantastically modifythe convergence of an infinite series. In the frame of the HAM,such kind of auxil-iary parameter is introduced in a natural way to approximations of general nonlinearproblems. Without such kind of guarantee of the convergenceof the homotopy se-ries, the freedom on the choice of the auxiliary linear operator and initial approxima-tions has no meanings at all: a divergent series is useless. In essence, the extremelylarge freedom of constructing the zeroth-order deformation equation is based onthe concept of the convergence-control by means of the convergence- parameterc0. As shown later in this book, one can introduce more such kindof unknownconvergence-control parameters so as to enhance the ability of convergence-control.Since the introduction of the convergence-control parameter (it is previously calledh-parameter), it has been widely applied to obtain accurate approximations of manynonlinear problems [1,1,6,9,28,30,31,33].

The solution expression is another fundamental concept of the HAM, especiallyfor nonlinear problems with periodic solutions. For example, it is due to the peri-odic solution expression (2.28) that an additional algebraic equation forγn is givenand the secular terms are avoided. In addition, it is also dueto the periodic solutionexpression (2.28) that the additional integral coefficients of the 4th and 6th-orderauxiliary linear operators are enforced to be zero so that the 2nd-order differentialequation (2.32) can be solved by the rather general auxiliary linear operator (2.92).As mentioned in§ 2.3.7, the concept of solution expression has more general mean-ings than the idea of avoiding the secular terms, because it provides more restric-tions. Note that the periodic solution expression (2.28) isbased on the physicalanalysis on the nonlinear oscillation system (2.32). In other words, before solvingthis equation, we know from physical viewpoints that the solutions are periodic nearsome stable equilibrium point, but otherwise are non-periodic. So, like the idea of“avoiding secular terms”3, the solution expression (2.28) contains some empiricalknowledge and/or even a priori knowledge. This is easy for anapplied mathemati-cians who have solid physical background, but is difficult for a pure mathematician.However, for any given equations, one can find some properties, especially someasymptotic properties, by analyzing it in details. Such kind of properties are helpfulto determine a set of proper base functions for the unknown solution. So, the more

3 When one talks about avoidance of secular terms, it implies that she or he must know that theunknown solution is periodic before she or he solves it successfully.


such kind of properties, the better. Fortunately, many solutions can be expressed bydifferent base functions.

In this chapter, we attempt to give more general and more accurate definitionsof some important concepts in the frame of the HAM. This is necessary for the de-velopment of the HAM, because these concepts are the base of it. Note that, thesymbolh was used to denote the same non-zero auxiliary parameter in the so-calledzeroth-order deformation equation when Liao [11] first introduced it into the HAMin 1997. Besides, the so-calledh-curves were often used to determine a interval ofthe auxiliary parameterh corresponding to convergent homotopy-series solutions.Sinceh has a very special meaning in quantum mechanics, we suggest to replacehby c0 so as to avoid the possible confusion. Besides, since the essence of this auxil-iary parameterc0 is to control the convergence of homotopy-series, we renameit theconvergence-control parameter. Due to the extremely largefreedom on constructinga homotopy of equations, more convergence-control parameters can be introducedinto zeroth-order deformation equations, as shown later inthis book. However,c0

is the most important, and thus can be regarded as a basic convergence-control pa-rameter. In 2007, Yabushita, Yamashita and Tsuboi [33] firstused an optimal valueof the auxiliary parameter in the frame of the HAM, which was determined by theminimum of the squared residual of two governing equations.This idea has gen-eral meanings, because we can always calculate the squared residual of governingequations no matter whether there exist closed-form analytic solutions or not. Moreimportantly, the curves of squared residual versus the convergence-control param-eterc0 not only give the effective-region of the convergence-control parameterc0,but also its optimal value. Therefore, the optimal convergence-control parameterc∗0determined by the minimum of squared residual of governing equations is stronglysuggested to use in practice. For many nonlinear problems, it is enough to obtainan accurate enough homotopy-approximation by means of an optimal convergence-control parameterc0. However, for some problems with strong nonlinearity, oneshould accelerate convergence of the homotopy-series by means of the homotopy-Pade technique, or the optimal initial approximation, andespecially the homotopy-iteration approach.

Note that, like perturbation techniques [4, 7, 8, 25–27], the HAM can give accu-rate but very simple approximations for some nonlinear problems. For example,the 1st-order homotopy-approximation (2.73) ofx(τ), the 2nd-order homotopy-approximations (2.74) and (2.75) ofx(τ) are rather simple but very accurate, asshown in Fig. 2.10. Besides, the 2nd-order homotopy-approximation (2.79) ofthe periodT is very simple, but agrees quite well with the exact periodT =7.4163/

√ε in thewholeregion 0< ε < +∞, as shown in Fig. 2.12. Furthermore,

the [1,1] homotopy-Pade approximation (2.85) ofx(τ) is very simple but accurate,as shown in Fig. 2.17. Especially, the homotopy-approximation (2.88) ofx(t) andthe homotopy-approximation (2.89) of its periodT are very simple but accurate forall physical parameter−∞ < ε <+∞ andall initial position−∞ < x∗ <+∞ in caseof λ = 0. In general, by means of the optimal convergence-control parameterc0,one often obtains accurate enough approximations in a few terms by means of theHAM. However, as the nonlinearity of equations becomes stronger and the complex-


ity of problems increases, more and more terms are needed. Indeed, by means of theHAM together with computer algebra system, one often gets high-order homotopy-approximations in a few seconds. However, this completely does not means that theHAM alwayshad touse many terms: this is a big misunderstanding to the HAM.On the other hand, for complicated problems with strong nonlinearity, it is oftenimpossible to get an accurate enough approximation by a few terms. For these diffi-cult problems, the HAM is more powerful, because it is essentially a method for thetimes of computer and internet.

In summary, the HAM is in essence based on the extremely largefreedom ofconstructing a homotopy of equations. It is such kind of fantastic freedom, combinedwith more and more powerful computer algebra systems, that makes the HAM rathergeneral and valid for problems with strong nonlinearity. Inessence, it is due tosuch kind of fantastic freedom that the HAM greatly differs from other traditionalanalytic methods.

Acknowledgements In a private discussion, Dr. Pradeep Siddheshwar (Bangalore University, In-dia) suggested me to renamec0 the convergence-control parameter, which was called the auxiliaryparameter in my previous articles [11–15, 17–20] and in my book “Beyond Perturbation” [16].Thanks to Dr. Zhiliang Lin (Shanghai Jiao Tong University, China) for his help on plotting Fig. 2.3.

Appendix 2.1 Derivation ofδn in (2.57) 79

Appendix 2.1 Derivation of δn in (2.57)

The definition ofδn in (2.57) reads

δn = Dn[

γ(q) x′′(τ;q)+λ x(τ;q)+ ε x3(τ;q)]

.

It holds according to the definition (2.50) that

Dn[

γ(q) x′′(τ;q)]

=1n!

∂ n

∂qn

[

γ(q) x′′(τ;q)]

∣

∣

∣

∣

q=0

=1n!

n

∑k=0

(

nk

)

dk γ(q)dqk

∂ n−k x′′(τ;q)∂qn−k

∣

∣

∣

∣

∣

q=0

=

n

∑k=0

[

1k!

dkγ(q)dqk

][

1(n− k)!

∂ n−k x′′(τ;q)∂qn−k

]

∣

∣

∣

∣

∣

q=0

=n

∑k=0

[

1k!

dkγ(q)dqk

]

∣

∣

∣

∣

∣

q=0

∂ 2

∂τ2

[

1(n− k)!

∂ n−k x(τ;q)∂qn−k

]∣

∣

∣

∣

q=0

=n

∑k=0

γk x′′n−k(τ).

Similarly, we have according to the definition (2.50) that

Dn [λ x(τ;q)] =1n!

∂ n

∂qn [λ x(τ;q)]

∣

∣

∣

∣

q=0= λ xn(τ)

and

Dn[

ε x3(τ;q)]

=1n!

∂ n

∂qn

[

ε x3(τ;q)]

∣

∣

∣

∣

q=0

=εn!

n

∑k=0

(

nk

)


∂ k[x2(τ;q)]∂qk

∣

∣

∣

∣

∣

q=0

= ε

n

∑k=0

1k!(n− k)!


k

∑j=0

(

kj

)

∂ k− j x(τ;q)∂qk− j

∂ j x(τ;q)∂q j

∣

∣

∣

∣

∣

q=0

= εn

∑k=0

[

1(n− k)!


∣

∣

∣

∣

q=0

]

×k

∑j=0

[

1(k− j)!

∂ k− j x(τ;q)∂qk− j

∣

∣

∣

∣

q=0

] [

1j!

∂ j x(τ;q)∂q j

∣

∣

∣

∣

q=0

]


= εn

∑k=0

xn−k(τ)k

∑j=0

xk− j(τ) x j(τ).

Substituting the above expressions into the definition ofδn, we have

δn =n

∑k=0


∑k=0

xn−k(τ)k

∑j=0

xk− j(τ) x j(τ). (2.99)

Note that, directly using the related Theorems proved in Chapter 4, one can ob-tain the same result.

Appendix 2.2 Derivation of (2.55) by the 2nd approach 81

Appendix 2.2Derivation of (2.55) by the 2nd approach

As pointed out by Hayat and Sajid [6], directly substitutingthe homotopy-Maclaurin(2.48) and (2.49) into the zeroth-order deformation equations (2.42) and (2.43), andequating the coefficients of the like power ofq, one can get exactly thesameequa-tions as (2.55) and (2.56).

Using the homotopy-Maclaurin series (2.48), we have

(1−q)L [x(τ;q)− x0(τ)]

= (1−q)L

[

+∞

∑k=0

xk(τ) qk− x0(τ)

]

= (1−q)L

[

+∞

∑k=1

xk(τ) qk

]

=+∞

∑k=1

L [xk(τ)]qk−+∞

∑k=1

L [xk(τ)]qk+1

=+∞

∑k=1

L [xk(τ)]qk−+∞

∑k=2

L [xk−1(τ)]qk

= L [x1(τ)] q++∞

∑k=2

L [xk(τ)− xk−1(τ)]qk

=+∞

∑n=1

L [xn(τ)− χn xn−1(τ)]qn, (2.100)

whereχn is defined by (2.58).Substituting the homotopy-Maclaurin series (2.48) and (2.49) into the definition

(2.35), we have

γ(q) x′′(τ;q)+λ x(τ;q)+ ε x3(τ;q)

=

(

+∞

∑i=0

γi qi

)(

+∞

∑j=0

x′′j (τ) q j

)

+λ+∞

∑n=0

xn(τ) qn

+ε

(

+∞

∑i=0

xi(τ) qi

)(

+∞

∑j=0

x j(τ) q j

)(

+∞

∑l=0

xl (τ) ql

)

=+∞

∑n=0

[

n

∑k=0

γk x′′n−k(τ)

]

qn+λ+∞

∑n=0

xn(τ) qn

+ε+∞

∑n=0

[

n

∑k=0

xn−k(τ)k

∑j=0

x j(τ) xk− j(τ)

]

qn


=+∞

∑n=0

[

n

∑k=0


∑k=0

xn−k(τ)k

∑j=0

xk− j(τ) x j(τ)

]

qn.

According to (2.57) or (2.99), the above expression reads

γ(q) x′′(τ;q)+λ x(τ;q)+ ε x3(τ;q) =+∞

∑n=0

δn(Xn,Γn) qn. (2.101)

Substituting (2.100) and (2.101) into the zeroth-order deformation equations(2.42), we have

+∞

∑n=1

L [xn(τ)− χn xn−1(τ)]qn = c0

+∞

∑n=0

δn(Xn,Γn) qn+1.

Equating the coefficient of like-power ofq in above equation, we have

xn(τ)− χn xn−1(τ) = c0 δn−1(Xn−1,Γn−1),

which is exactly thesameas the high-order deformation equation (2.55).Besides, substituting the homotopy-Maclaurin series (2.48) into the initial con-

ditions x(0;q) = 1 andx′(0;q) = 0, equating the coefficient of the like power ofq,we have

xk(0) = 0, x′k(0) = 0, k= 1,2,3, · · · ,which is exactly thesameas the initial conditions (2.56).

Therefore, no matter whether one regards the homotopy-parameterq as a smallparameter or not, one should always obtain thesamehigh-order deformation equa-tions from thesamezeroth-order deformation equations, as proved in general byHayat and Sajid [6]. This is easy to understand, because, according to the funda-mental theorem in calculus [5], the Taylor series of a real function is unique.

Appendix 2.3 Proof of Theorem 2.3 83

Appendix 2.3 Proof of Theorem 2.3

Proof. Defineξ = 1+ c0+ c0 z,

which gives sincec0 6= 0 that

11+ z

=− c0

(1− ξ ).

Enforcing|ξ |= |1+ c0+ c0 z|< 1, we have by Newtonian binomial theorem that

11+ z

= − c0

1− ξ=−c0

(

1+ ξ + ξ 2+ ξ 3+ · · ·)

=−c0

+∞

∑n=0

(1+ c0+ c0 z)n.

In other words, it holds

11+ z

= limm→+∞

[

−c0

m

∑n=0

(1+ c0+ c0 z)n

]

, when|1+ c0+ c0 z|< 1.

For real numbersz∈ (−∞,+∞) andc0 6= 0, we have

−c0

m

∑n=0

(1+ c0+ c0 z)n

= −c0

m

∑n=0

n

∑k=0

(

nk

)

(1+ c0)n−k (c0 z)k

= −c0

m

∑k=0

m

∑n=k

(

nk

)

(1+ c0)n−k ck

0 zk

=m

∑k=0

(−1)k zk(−c0)k+1

m−k

∑i=0

(

k+ ik

)

(1+ c0)i

=m

∑k=0

(−z)k

[

(−c0)k+1

m−k

∑i=0

(

k+ ii

)

(1+ c0)i

]

=m

∑n=0

(−z)n

[

(−c0)n+1

m−n

∑i=0

(

n+ ii

)

(1+ c0)i

]

=m

∑n=0

[

µm+1,n+10 (c0)

]

(−z)n,

where

µm,n0 (c0) = (−c0)

nm−n

∑i=0

(

n−1+ ii

)

(1+ c0)i .


Besides,|1+ c0+ c0 z|< 1 gives

−1< 1+ c0+ c0 z< 1,

i.e.−2− c0 < c0 z<−c0,

which leads to either

−1< z<2|c0|

−1, whenc0 < 0,

or

− 2c0

−1< z<−1, whenc0 > 0.

Thus, it holds1

1+ z= lim

m→+∞

m

∑n=0

µm+1,n+10 (c0)(−z)n,

in the region

−1< z<2|c0|

−1, whenc0 < 0,

or

− 2c0

−1< z<−1, whenc0 > 0.

⊓⊔

Appendix 2.4 Mathematica code (without iteration) for Example 2.2 85

Appendix 2.4Mathematica code (without iteration) for Example 2.2

Periodic solutions of nonlinear oscillation equation

x′′+λx+ εx3 = 0, x(0) = x∗,x′(0) = 0

is solved by means of the HAM without iteration approach, whereλ andε are phys-ical parameters,x∗ is the starting position of oscillation, respectively. This Mathe-matica code is free available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

A Simple Users GuideInput data:lambda : physical parameterλ ;epsilon : physical parameterε;xstart : start-positionx∗ = x(0);c0 : convergence-control parameterc0;KAPPA: value ofκ of the auxiliary linear operator (2.92);

Control parameter:xOptimal : Optimal initial approximation is used whenxOptimal = 1 ;

Calculated results:X[k] : kth-order homotopy-approximation ofx(τ) ;u[k] : kth-order homotopy-approximation ofx(t) ;GAMMA[k] : kth-order homotopy-approximation ofγ ;Err[k] : squared residualEk defined by (2.72) ;

Main codes:ham[1,11] : First, gain 1st to 10th-order homotopy-approximation;ham[12,35] : Then, gain 12th to 35th-order homotopy-approximation;GetErr[k] : Gain squared residualEk defined by (2.72)


Mathematica code (without iteration) for Example 2.2by Shijun LIAO

Shanghai Jiao Tong UniversityJune 2010

<<Calculus‘Pade‘;<<Graphics‘Graphics‘;

( *************************************************** *********** )( * Define physical and control parameters * )( *************************************************** *********** )KAPPA = 1;lambda = -9/4;epsilon = 1;c0 = -17/100;xstart = 1;

( *************************************************** *********** )( * Using optimal initial approximation when xOptimal = 1 * )( *************************************************** *********** )xOptimal = 0;If[epsilon > 0 && lambda >= 0, beta = 0];If[epsilon > 0 && lambda < 0 && xstart > 0,

beta = N[Sqrt[Abs[lambda/epsilon]],100]];If[epsilon > 0 && lambda < 0 && xstart < 0,

beta =-N[Sqrt[Abs[lambda/epsilon]],100]];If[epsilon < 0 && lambda >= 0 &&

Abs[xstart] < Sqrt[Abs[lambda/epsilon]], beta = 0];If[!NumberQ[beta//N],Print[" There is no periodic soluti on

for these input data !"]];If[xOptimal == 1, beta = . ];

( *************************************************** *********** )( * Define initial guess * )( *************************************************** *********** )x[0] = beta + (xstart - beta) * Cos[t];X[0] = x[0];u[0] = X[0] /. t->Sqrt[GAMMA[0]] * t ;

( *************************************************** *********** )( * Define the function chi[k] * )( *************************************************** *********** )chi[k_] := If[ k <= 1, 0, 1 ];

( *************************************************** *********** )( * Define inverse operator of auxiliary linear operator L * )( *************************************************** *********** )Linv[Cos[m_ * t]] := (-1)ˆ(KAPPA+1) * Cos[m* t]/(1-mˆ(2 * KAPPA));


( *************************************************** *********** )( * The linear property of the inverse operator Linv * )( * Linv[f_+g_] := Linv[f]+Linv[g]; * )( *************************************************** *********** )Linv[p_Plus] := Map[Linv,p];Linv[c_ * f_] := c * Linv[f] /; FreeQ[c,t];Linv[c_] := (-1)ˆ(KAPPA+1) * c /; FreeQ[c,t];

( *************************************************** *********** )( * Define GetDelta[k] * )( * Calculate the right-hand side term delta[k] * )( *************************************************** *********** )GetDelta[k_]:=Module[temp,temp[1] = xttgamma[k]//TrigReduce;temp[2] = xxx[k]//TrigReduce;temp[3] = temp[1] + lambda * x[k] + epsilon * temp[2];Delta[k]= Expand[temp[3]];];

( *************************************************** *********** )( * Define GetxAll * )( *************************************************** *********** )GetxAll[k_]:=Module[temp,xtt[k] = Expand[D[x[k],t,2]];temp = Sum[x[j] * x[k-j],j,0,k]//Expand;xx[k] = TrigReduce[temp];temp = Sum[x[j] * xx[k-j],j,0,k]//Expand;xxx[k] = TrigReduce[temp];xttgamma[k] = Sum[gamma[j] * xtt[k-j],j,0,k]//Expand;];

( *************************************************** *********** )( * Define Getgamma * )( *************************************************** *********** )Getgamma[k_]:=Module[temp,eq,temp[1] = Delta[k]//TrigReduce;temp[2] = Expand[temp[1]];eq = Coefficient[temp[2],Cos[t]];temp[3] = Solve[eq == 0, gamma[k]];gamma[k] = temp[3][[1,1,2]];];

( *************************************************** *********** )( * Define Getx * )( * Get solution of high-order deformation equation * )( *************************************************** *********** )Getx[k_] := Module[temp,temp[1] = xSpecial + chi[k] * x[k-1];temp[2] = temp[1] /. t->0;x[k] = temp[1] - temp[2] * Cos[t];];


( *************************************************** *********** )( * Define GetErr * )( * Gain squared residual of governing equation * )( *************************************************** *********** )GetErr[k_]:=Module[temp,Xtt,error,delt,tt,Npoint,s um,i,Xtt = D[X[k],t,2];error = GAMMA[k] * Xtt + lambda * X[k] + epsilon * X[k]ˆ3;Npoint = 50;sum = 0;delt = N[2 * Pi/Npoint,24];For[ i = 0, i <= Npoint, i=i+1,

tt = i * delt;temp = error/.t -> tt //Expand;sum = sum + tempˆ2 //Expand;

];Err[k] = sum/(Npoint+1)];

( *************************************************** *********** )( * Main Code * )( *************************************************** *********** )ham[m0_,m1_]:=Module[temp,k,j,zz,For[k=Max[1,m0], k<=m1, k=k+1,

Print[" k = ",k];GetxAll[k-1];GetDelta[k-1];RHS[k] = c0 * Delta[k-1]//Expand;Getgamma[k-1];GAMMA[k-1] = Expand[Sum[gamma[j],j,0,k-1]];T[k-1] = 2 * Pi/Sqrt[GAMMA[k-1]];If[NumberQ[GAMMA[k-1]],

Print[k-1,"th approx. of gamma = ",N[GAMMA[k-1],10]," variation = ",gamma[k-1]//N ];

Print[k-1,"th approx. of T = ",N[T[k-1],10] ];];

If[k == 1 && xOptimal == 1,GetErr[0];If[ xstart > 0,

temp = Minimize[Err[0], beta > xstart,beta] ];If[ xstart < 0,

temp = Minimize[Err[0], beta < xstart,beta] ];zz = beta/.temp[[2]];

beta = N[IntegerPart[zz * 10000]/10000,100];Print[" Optimal initial approx. is used with beta = ",

beta//N];];

temp = RHS[k]/.Cos[t]->0//Expand;xSpecial = Linv[temp];Getx[k];X[k] = X[k-1] + x[k]//Expand;u[k] = X[k] /. t-> Sqrt[GAMMA[k-1]] * t;

];Print["Successful !"];];


( *************************************************** *********** )( * Print physical and contyrol parameters * )( *************************************************** *********** )Print[" lambda = ",lambda];Print[" epsilon = ",epsilon];Print[" x * = ",xstart];Print[" beta = ",beta];Print[" c0 = ",c0];Print[" KAPPA = ",KAPPA];Print[" xOptimal = ",xOptimal];

( * Gain 10th-order homotopy-approximation * )ham[1,10];


Appendix 2.5Mathematica code (with iteration) for Example 2.2

Periodic solutions of nonlinear oscillation equation

x′′+λx+ εx3 = 0, x(0) = x∗,x′(0) = 0

is solved by means of the HAM with iteration approach, whereλ andε are physicalparameters,x∗ is the starting position of oscillation, respectively. This Mathematicacode is free available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

A Simple Users GuideInput data:lambda : physical parameterλ ;epsilon : physical parameterε;xstart : start-positionx∗ = x(0);

Control parameter:IterOrder : value ofM, the order of iteration formula (2.90);Nterms : value ofN in the truncated expression (2.91 );

Calculated results:X[k] : x(τ) given by thekth-iteration ;u[k] : x(t) given by thekth-iteration;GAMMA[k] : γ given by thekth-iteration;Err[k] : squared residual at thekth-iteration ;

Main codes:ham[1,11] : First, gain 1st to 10th-iteration approximations;ham[12,21] : Then, gain 12nd to 21st-iteration approximations;GetErr[k] : Gain squared residual of thekth-iteration approximation;

Appendix 2.5 Mathematica code (with iteration) for Example2.2 91

Mathematica code (with iteration) for Example 2.2by Shijun LIAO



( *************************************************** *********** )( * Physical and control parameters * )( *************************************************** *********** )lambda = 0;epsilon = 1;xstart = 1;If[NumberQ[c0],c0 = N[c0,100]];

( *************************************************** *********** )( * Control parameters for iteration approach * )( *************************************************** *********** )IterOrder = 3;Nterms = 21;

( *************************************************** *********** )( * Define initial approximation * )( *************************************************** *********** )If[epsilon > 0 && lambda >= 0, beta = 0];If[epsilon > 0 && lambda < 0 && xstart > 0,

beta = N[Sqrt[Abs[lambda/epsilon]],100]];If[epsilon > 0 && lambda < 0 && xstart < 0,

beta =-N[Sqrt[Abs[lambda/epsilon]],100]];If[epsilon < 0 && lambda >= 0 &&

Abs[xstart] < Sqrt[Abs[lambda/epsilon]], beta = 0];If[!NumberQ[beta//N],

Print[" No periodic solution for the input data !"]];x[0] = beta + (xstart - beta) * Cos[t];X[0] = x[0];u[0] = X[0] /. t->Sqrt[GAMMA[0]] * t ;

( *************************************************** *********** )( * Define the function chi[k] * )( *************************************************** *********** )chi[k_] := If[ k <= 1, 0, 1 ];

( *************************************************** *********** )( * Define inverse operator of auxiliary linear operator * )( *************************************************** *********** )Linv[Cos[m_ * t]] := Cos[m * t]/(1-mˆ2);


( *************************************************** *********** )( * The linear property of the inverse operator Linv * )( * Linv[f_+g_] := Linv[f]+Linv[g]; * )( *************************************************** *********** )Linv[p_Plus] := Map[Linv,p];Linv[c_ * f_] := c * Linv[f] /; FreeQ[c,t];Linv[c_] := c /; FreeQ[c,t];

( *************************************************** *********** )( * Define GetDelta[k] * )( *************************************************** *********** )GetDelta[k_]:=Module[temp,n,temp[1] = xttgamma[k]//TrigReduce;temp[2] = xxx[k]//TrigReduce;temp[3] = temp[1] + lambda * x[k] + epsilon * temp[2];Delta[k] = temp[3]//Expand;];

( *************************************************** *********** )( * Define GetxAll * )( *************************************************** *********** )GetxAll[k_]:=Module[temp,xtt[k] = Expand[D[x[k],t,2]];temp = Sum[x[j] * x[k-j],j,0,k]//Expand;xx[k] = TrigReduce[temp];temp = Sum[x[j] * xx[k-j],j,0,k]//Expand;xxx[k] = TrigReduce[temp];xttgamma[k] = Sum[gamma[j] * xtt[k-j],j,0,k]//Expand;];

( *************************************************** *********** )( * Define Getgamma * )( *************************************************** *********** )Getgamma[k_]:=Module[temp,eq,temp[1] = Delta[k]//TrigReduce;temp[2] = Expand[temp[1]];eq = Coefficient[temp[2],Cos[t]];temp[3] = Solve[eq == 0, gamma[k]];gamma[k] = temp[3][[1,1,2]];];

( *************************************************** *********** )( * Define Getx * )( * Get solution of high-order deformation equation * )( *************************************************** *********** )Getx[k_]:=Module[temp,temp[1] = xSpecial + chi[k] * x[k-1];temp[2] = temp[1] /. t->0;x[k] = temp[1] - temp[2] * Cos[t];];

Appendix 2.5 Mathematica code (with iteration) for Example2.2 93

( *************************************************** *********** )( * Define GetErr * )( * Get squared residual of governing equation * )( *************************************************** *********** )GetErr[iter_] := Module[temp,Xtt,error,delt, t,Npoint ,sum,i,Xtt = D[X[iter],t,2];error = GAMMA[iter] * Xtt+lambda * X[iter]+epsilon * X[iter]ˆ3;Npoint = 50;sum = 0;delt = N[2 * Pi/Npoint,100];For[ i = 0, i <= Npoint, i=i+1,

tt = i * delt;temp = error/.t -> tt //Expand;sum = sum + tempˆ2 //Expand;

];Err[iter] = sum/(Npoint+1);];

( *************************************************** *********** )( * Define Truncation * )( * This module neglects high-order terms * )( *************************************************** *********** )Truncation[k_,Nterms_] := Module[temp,coef,i,j,temp[1] = RHS[k] //TrigReduce;temp[2] = temp[1] //Expand;coef[0] = temp[2] /. Cos[_] -> 0;For[i = 2, i <= Nterms, i = i + 1,

coef[i] = Coefficient[temp[2],Cos[i * t]];];

temp[3] = coef[0] + Sum[coef[j] * Cos[j * t],j,2,Nterms];RHS[k] = temp[3]//Expand;];

( *************************************************** *********** )( * Main Code * )( *************************************************** *********** )ham[m0_,m1_]:=Module[temp,k,j,zz,For[iter = Max[1,m0], iter <= m1, iter = iter + 1,

For[k = 1, k <= IterOrder, k = k+1,If[k == 1, Clear[gamma]];GetxAll[k-1];GetDelta[k-1];RHS[k] = c0 * Delta[k-1]//Expand;Getgamma[k-1];Truncation[k, Nterms];xSpecial = Linv[RHS[k]];Getx[k];];

X[iter] = Sum[x[i],i,0,IterOrder];GAMMA[iter] = Expand[Sum[gamma[j],j,0,IterOrder-1]] ;variation = GAMMA[iter]-chi[iter] * GAMMA[iter-1]//N;T[iter] = 2 * Pi/Sqrt[GAMMA[iter]];u[iter] = X[IterOrder] /. t-> Sqrt[GAMMA[iter]] * t;GetErr[iter];


If[ iter == 1 ,c0 = .;temp = Minimize[Err[1],c0 < 0, c0];zz = c0/.temp[[2]];c0 = N[IntegerPart[zz * 10000]/10000,100];Print["The optimal value of c0 = ",c0//N];

];Print[ iter, "th iteration: error = ", Err[iter]//N];Print[ " gamma = ",N[GAMMA[iter],24],

" variation = ",variation];If[ Err[iter] < 10ˆ(-30) || Abs[variation] < 10ˆ(-30),

Print[" Desired accuracy is arrived: iteration stops!"];Goto[end];];

x[0] = X[iter];Clear[gamma];];

Label[end];Print[" Successful ! "];];

( *************************************************** *********** )( * Print physical and control parameters * )( *************************************************** *********** )Print[" lambda = ",lambda];Print[" epsilon = ",epsilon];Print[" x * = ",xstart];Print["-------------------------------------------- ---------"];Print[" Oder of iteration formula = ",IterOrder];Print[" N, the number of terms in delta = ",Nterms];

( * Gain 10th homotopy-iteration approximation * )ham[1,10];

Problems 95

Problems

2.1. Non-periodic solution of a nonlinear dynamic systemUse the HAM to obtain the non-periodic solutions of a nonlinear differential equa-tion

x(t)+λ x(t)+ ε x3(t) = 0, x(0) = x∗, x(0) = 0,

whereλ andε are physical constants,x∗ denotes the starting position.

2.2. Periodic solution of a nonlinear dynamic systemUse the HAM to obtain the periodic solution of a nonlinear differential equation

x(t)+λ x(t)+ ε x3(t) = 0, x(0) = 0, x(0) = v0,

whereλ ∈ (−∞,+∞) andε ∈ (−∞,+∞) are physical constants,v0 denotes the start-ing velocity.

2.3. Periodic and non-periodic solution of a nonlinear dynamic systemUse the HAM to obtain the periodic and non-periodic solutions of a nonlinear dif-ferential equation

x(t)+λ x(t)+ ε x2(t) = 0, x(0) = x∗, x(0) = 0,

whereλ ∈ (−∞,+∞) andε ∈ (−∞,+∞) are physical constants,x∗ denotes the start-ing position.


References


2. Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota– Satsuma coupled KdV equation. Phys. Lett. A.361, 478 – 483 (2007)

3. Armstrong, M.A., Basic Topology (Undergraduate Texts inMathematics). Springer, NewYork (1983)

4. Cole, J.D.: Perturbation Methods in Applied Mathematics. Blaisdell Publishing Company,Waltham (1992)

5. Fitzpatrick, P.M.: Advanced Calculus. PWS Publishing Company, New York (1996)6. Hayat, T., Sajid, M.: On analytic solution for thin film flowof a fourth grade fluid down a

vertical cylinder. Phys. Lett. A.361, 316–322 (2007)7. Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge

(1953)8. Hinch, E.J.: Perturbation Methods. In seres of CambridgeTexts in Applied Mathematics ,

Cambridge University Press, Cambridge (1991)9. Liang, S.X., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy pertur-

bation method through an evalution equation. Commun. Nonlinear Sci. Numer. Simulat.14,4057 – 4064 (2009)
















25. Murdock, J.A.: Perturbations: - Theory and Methods. John Wiley & Sons, New York (1991)26. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (1973)27. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (2000)

References 97

28. Niu, Z., Wang, C.: A one-step optimal homotopy analysis method for nonlinear differentialequations. Commun. Nonlinear Sci. Numer. Simulat.15, 2026 – 2036 (2010).

29. Sen, S.: Topology and Geometry for Physicists. AcademicPress, Florida (1983)30. Van Gorder, R.A., Vajravelu, K.: Analytic and numericalsolutions to the Lane-Emden equa-

tion. Phys. Lett. A.372, 6060 – 6065 (2008)31. Wu, Y.Y., Cheung, K.F.: Homotopy solution for nonlineardifferential equations in wave prop-

agation problems. Wave Motion.46, 1 – 14 (2009)32. Xu, H., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy-based solutions of the Navier-

Stokes equations for a porous channel with orthogonally moving walls. Physics of Fluids.22,053601 (2010). doi:10.1063/1.3392770


Chapter 3Optimal homotopy analysis method

Abstract In this chapter, we describe and compare the different optimal approachesof the homotopy analysis method (HAM). A generalized optimal HAM is proposed,which logically contains the basic optimal HAM with only oneconvergence-controlparameter and also the optimal HAM with an infinite number of parameters. It isfound that approximations given by the optimal HAMs converge fast in general. Es-pecially, the basic optimal HAM mostly gives good enough approximations. Thus,the optimal HAMs with a couple of convergence-control parameters are stronglysuggested in practice.

3.1 Introduction

Nonlinear equations are much more difficult to solve than linear ones, especiallyby means of analytic methods. In general, there are two standards for a satisfactoryanalytic method of nonlinear equations:

(a) It canalwaysprovide analytic approximationsefficiently.(b) It can ensure that analytic approximations areaccurateenough forall physical

parameters.

Using above two standards as criterion, let us compare different analytic techniquesfor nonlinear problems.

Perturbation techniques [9, 13, 27, 32, 33, 38] are widely applied in science andengineering. Perturbation techniques are based on small/large physical parameters(called perturbation quantities) in governing equations or initial/boundary condi-tions. In general, perturbation approximations are expressed in series of perturba-tion quantities, and a nonlinear equation is replaced by an infinite number of linear(sometimes even nonlinear) sub-problems, which are completely determined by theoriginal governing equation and especially by the place where perturbation quan-tities appear. Perturbation methods are simple, and easy tounderstand. Especially,based on small physical parameters, perturbation approximations often have clear

99

100 3 Optimal homotopy analysis method

physical meanings. Unfortunately,not every nonlinear problem has such kind ofperturbation quantity. Besides, even if there exists such asmall physical parameter,the sub-problem might have no solutions, or might be so complicated that only a fewsub-problems can be solved. Thus, it isnotguaranteed that one canalwaysgain per-turbation approximations for any a given nonlinear problem. More importantly, it iswell-known that most perturbation approximations are valid only for small physicalparameters. So, it isnot guaranteed that a perturbation result is valid in the wholeregion ofall physical parameters. Thus, perturbation techniques do notsatisfy notonly the standard (a) but also the standard (b) mentioned above.

To overcome the restrictions of perturbation techniques, some traditional non-perturbation methods are developed, such as Lyapunov’s artificial small parame-ter method [28], theδ -expansion method [7, 12], Adomian decomposition method[1–4, 8, 36], and so on. In principle, all of these methods arebased on a so-calledartificial parameter, and approximation solutions are expanded into series of suchkind of artificial parameter. The artificial small parameteris often used in such away that one can always gain approximation solutions for anya given nonlinearequation. Compared with perturbation techniques, this is indeed a great progress.However, in theory, one can put the artificial small parameter in many differentways, but unfortunately there are no theories to guide us howto put it in a betterplace so as to gain a better approximation. For example, Adomian decompositionmethod simply uses the linear operatordk/dxk in most cases, wherek is the highestorder of derivative of governing equations, and therefore it is rather easy to gainsolutions of the corresponding sub-problems by means of integratingk times withrespect tox. However, such simple linear operator gives approximations in power-series, which unfortunately has often a finite radius of convergence. Thus, Adomiandecomposition method can not ensure the convergence of its approximation series.Generally speaking, all traditional non-perturbation methods, such as Lyapunov’sartificial small parameter method [28], theδ -expansion method [7, 12] and Ado-mian decomposition method [1–4, 8, 36], cannot guarantee the convergence of ap-proximation series. So, these traditional non-perturbation methods satisfy only thestandard (a) butnot the standard (b) mentioned before.

In 1992 Liao [15] took the lead to apply the homotopy [11], a basic concept intopology [37], to gain analytic approximations of nonlinear differential equations.The early homotopy analysis method (HAM) was first describedby Liao [15] in hisPhD dissertation in 1992. For a given nonlinear differential equation

N [u(x)] = 0, x∈ Ω ,

whereN is a nonlinear operator andu(x) is a unknown function, Liao [15] con-structed aone-parameterfamily of equations in the embedding parameterq∈ [0,1],called the zeroth-order deformation equation

(1−q)L [φ(x;q)−u0(x)]+q N [φ(x;q)] = 0, x∈ Ω , q∈ [0,1], (3.1)

whereL is an auxiliary linear operator andu0(x) is an initial guess. In theory,the concept of homotopy in topology provides us much larger freedom to choose

3.1 Introduction 101

both of the auxiliary linear operatorL and the initial guess than the traditionalnon-perturbation methods mentioned above, as pointed out by Liao [17, 18, 21]and shown later in this book. Atq = 0 andq = 1, we haveφ(x;0) = u0(x) andφ(x;1) = u(x), respectively. So, as the embedding parameterq ∈ [0,1] increasesfrom 0 to 1, the solutionφ(x;q) of the zeroth-order deformation equations varies(or deforms) from the initial guessu0(x) to the exact solutionu(x) of the originalnonlinear differential equationN [u(x)] = 0. Such kind of continuous variation iscalled deformation in topology, and this is the reason why wecall (3.1) the zeroth-order deformation equation. Sinceφ(x;q) is also dependent upon the embeddingparameterq∈ [0,1], we can expand it into the Maclaurin series with respect toq:

φ(x;q) = u0(x)++∞

∑n=1

un(x) qn, (3.2)

called the homotopy-Maclaurin series. Note that we have extremely large freedomto choose the auxiliary linear operatorL and the initial guessu0(x). Assuming that,the auxiliary linear operatorL and the initial guessu0(x) are so properly chosenthat the above homotopy-Maclaurin series converges atq= 1, we have the so-calledhomotopy-series solution

u(x) = u0(x)++∞

∑n=1

un(x), (3.3)

which satisfies the original equationN [u(x)] = 0, as proved by Liao [17, 18] ingeneral. Here,un(x) is governed by the so-called high-order deformation equation

L [un(x)− χn un−1(x)] =−Dn−1N [φ(x;q)] , (3.4)

whereχk equals to 1 whenk≥ 2 but zero otherwise,Dk is thekth-order homotopy-derivative operator defined by

Dk =1k!

∂ k

∂qk

∣

∣

∣

∣

q=0.

The high-order deformation equation (3.4) is always linearwith the known term onthe right-hand side, therefore is easy to solve, as long as wechoose the auxiliarylinear operatorL properly.

Unfortunately, Liao [16,18] found that the early HAM mentioned above can notalways guarantee the convergence of approximation series of nonlinear equations ingeneral. To overcome this restriction, Liao [16] in 1997 introduced such a non-zeroauxiliary parameterc0 to construct atwo-parameterfamily of equations1, i.e. thezeroth-order deformation equation

1 Liao [16] originally used the symbolh to denote the non-zero auxiliary parameter. But,h is well-known as Planck’s constant in quantum mechanics. To avoid misunderstanding, we replaceh bythe symbolc0 in this book, which denotes the “basic” convergence-control parameter.


(1−q)L [φ(x;q)−u0(x)] = c0 q N [φ(x;q)], x∈ Ω , q∈ [0,1]. (3.5)

In this way, the homotopy-series solution (3.3) is not only dependent upon the phys-ical variablex but also the auxiliary parameterc0. Mathematically, it was found[16–18] that the auxiliary parameterc0 can adjust and control the convergence re-gion and rate of homotopy-series solutions, althoughc0 has no physical meaningsat all. For detailed mathematical proofs, please refer to Chapter 5. In essence, theuse of the auxiliary parameterc0 introduces us one more “artificial” degree of free-dom, which greatly improves the early HAM: it is the auxiliary parameterc0 whichprovides us a convenient way to guarantee the convergence ofhomotopy-series so-lution. For example, Liang & Jeffrey [14] illustrated that,when analytic approxima-tions given by the other analytic method is divergent in the whole domain, one cangain convergent series solution simply by choosing a properauxiliary parameterc0.This is the reason why we callc0 theconvergence-control parameter.

The use of the convergence-control parameterc0 is indeed a great progress inthe frame of the HAM. It seems that more “artificial” degrees of freedom implylarger possibility to gain better approximations by means of the homotopy analysismethod. Thus, Liao [17] in 1999 further introduced more “artificial” degrees offreedom by using the zeroth-order deformation equation in amore general form:

[1−α(q)]L [φ(x;q)−u0(x)] = c0 β (q) N [φ(x;q)], x∈ Ω , q∈ [0,1], (3.6)

whereα(q) andβ (q) are the so-calleddeformation functions2 satisfying

α(0) = β (0) = 0,α(1) = β (1) = 1, (3.7)

whose Taylor series

α(q) =+∞

∑m=1

αm qm, β (q) =+∞

∑m=1

βm qm, (3.8)

are convergent for|q| ≤ 1. In fact, the zeroth-order deformation equation (3.6) canbe further generalized, as shown by Liao [18, 19, 22]. Obviously, there are an infi-nite number of the deformation functions as defined above. Thus, the approxima-tion series given by the HAM can contain so many “artificial” degrees of freedomthat they provide us great possibility to guarantee the convergence of homotopy-series solution. Note thatun(x) is always governed by the same auxiliary linearoperatorL , and we have great freedom to chooseL in such a way thatun(x) iseasy to obtain. More importantly, for given auxiliary linear operatorL and initialguess, we can always gain convergent homotopy-series solution by choosing properconvergence-control parameterc0 and proper deformation functionsα(q) andβ (q).Inversely, the guarantee of the convergence of homotopy-series solutions also pro-vides us freedom to choose the auxiliary linear operatorL and initial guess. It is

2 α(q) andβ (q) were called “approaching function” in some early articles about the homotopy-analysis method. In this book, we name them “deformation function”, which better reveals itsrelationship with the zeroth-order deformation equations


due to such kind of guarantee in the frame of the HAM that a nonlinear ODE withvariable coefficients can be transferred into a sequence of linear ODEs with constantcoefficients [24], that a nonlinear PDE can be transferred into an infinite number oflinear ODEs [20, 23], that several coupled nonlinear ODEs can be transferred intoan infinite number of linear decoupled ODEs [40], and that even a 2nd-order non-linear PDE can be replaced by an infinite number of 4th-order linear PDEs [21].In fact, it is such kind of guarantee for convergence of series solutions, togetherwith the extremely large freedom in choice of the auxiliary linear operators, thatgreatly simplifies finding convergent series of nonlinear equations in the frame ofthe HAM, as illustrated in above-mentioned articles [20,21,23,24,40]. On the otherhand, without such kind of guarantee of convergence, we havein practice notruefreedom to choose the auxiliary linear operatorL , because the freedom to get adivergent series solution has no meanings at all! For example, Liang & Jeffrey [14]pointed out that the series solution given by means of the so-called “homotopy per-turbation method” [10] is divergent at all points except theinitial guess, and thushas completely no scientific meanings. So, unlike perturbation techniques and thetraditional non-perturbation methods mentioned above, the HAM satisfies both thestandard (a) and (b).

How to find a proper convergence-control parameterc0 so as to gain a con-vergent series solution? A straight-forward way to check the convergence of ahomotopy-series solution is to substitute it into originalgoverning equations andboundary/initial conditions and then to check the corresponding squared residual in-tegrated in the whole region. However, when the approximations contain unknownconvergence-control parameters and/or other unknown physical parameters, it istime-consuming to calculate the squared residual at high-order of approximations.To avoid the time-consuming computation, Liao [16–18] suggested to investigatethe convergence of some special quantities which often haveimportant physicalmeanings. For example, one can consider the convergence ofu′(0) andu′′(0) of anonlinear differential equationN [u(x)] = 0. It is found by Liao [16–18] that thereoften exists such an effective-regionRc that anyc0 ∈ Rc gives a convergent se-ries solution of such kind of quantities. Besides, such kindof effective-region canbe found, although approximately, by plotting the curves ofthese unknown quanti-ties versusc0. For example, for a nonlinear differential equationN [u(x)] = 0, onemay approximately determineRc by plotting curvesu′(0) ∼ c0, u′′(0) ∼ c0 and soon. These curves are called “c0-curves” or “curves for convergence-control parame-ter”3, which have been successfully applied to solve many nonlinear problems [18].

However, such kind ofc0-curves can not tell us the best convergence-controlparameterc0, which corresponds to the fastest convergent series. In 2007, Yabushita,Yamashita and Tsuboi [39] applied the HAM to solve two coupled nonlinear ODEs.They suggested the so-called “optimization method” to find out the two optimalconvergence-control parameters by means of the minimum of the squared residualof governing equations. Let

3 Thec0-curve was originally called theh-curve, andRc was originally denoted byRh.


Em =

∫

Ω

N

[

m

∑n=1

un(x)

]2

dΩ

denote the squared residual of themth-order approximation of the governing equa-tion N (u) = 0, integrated in the whole domainΩ . In theory, if the squared resid-

ual Em tends to zero, then+∞∑

n=0un(x) is a series solution of the original equation

N (u) = 0. So, if there exists only one convergence-control parameter c0, the so-called effective-regionRc of the convergence-control parameterc0 is defined by

Rc =

c0∣

∣ limm→+∞

Em(c0) = 0

.

Besides, at the given order of approximation, the minimum ofthe squared residualEm corresponds to the optimal approximation. So, the curves ofthe squared residualEm versusc0 indicate not only the effective-regionRc of the convergence-controlparameterc0, but also the optimal value ofc0 that corresponds to the minimumof Em. Note that one can gain the squared residual of an equation atany order ofapproximations, even if the exact solutions are unknown. Therefore, it is a verygood idea of Yabushita, Yamashita and Tsuboi [39] to use the squared residual tofind out the effective-regionRc and the optimal convergence-control parameters.

In 2008, Akyildiz and Vajravelu [5] gained optimal convergence-control param-eter by the minimum of squared residual of governing equation, and found that thecorresponding homotopy-series solution converges very quickly.

In 2008, Marinca and Herisanu [29] combinedc0 andβ (q) in the zeroth-orderdeformation equation (3.6) as one functionβ (q) with β (0) = 0 but β (1) 6= 1, andconsidered such a family of equations

(1−q)L [φ(x;q)−u0(x)] = β (q) N [φ(x;q)], q∈ [0,1], (3.9)

where the Taylor series

β (q) =+∞

∑n=1

cn qn

converges atq= 1. The above equation is a special case of (3.6), if we choose

α(q) = q, β (q) =1c0

+∞

∑n=1

cn qn =β (q)c0

, c0 =+∞

∑n=1

cn 6= 0. (3.10)

So, the so-called “optimal homotopy asymptotic method” [29,30] is still in the frameof the HAM. Even so, Marinca and Herisanu’s approach [29] isinteresting, whichhas the advantage thatβ (1) = 1 is unnecessaryso that we havelarger freedom tochoose the auxiliary parameterscn: all of them become the so-called convergence-control parameters. Marinca and Herisanu [29] developed the so-called “optimalhomotopy asymptotic method” by minimizing the squared residualEm: at themth-

3.2 An illustrative description 105

order of approximation, a set of nonlinear algebraic equations aboutc1,c2, · · · ,cm

are solved so as to find their optimal values. In theory, the more the convergence-control parameters are used, the better approximation we should obtain by this op-timal HAM. However, with too many unknown parameters, it is time-consumingto find out the optimal convergence-control parameters, especially at high-orderof approximations for a complicated nonlinear problem. Forexample, Niu andWang [34] illustrated that the optimal approach given by Marinca et al. [29, 30]is time-consuming [31, 35], although their optimal HAM [29]is more rigorous intheory than Nou and Wang’s ones. It seems that one had to balance the rigorousnessin theory against the computational efficiency in practice.

To increase the computational efficiency, Liao [26] developed in 2010 an optimalHAM with only three convergence-controlparameters. Like Marinca and Herisanu’sapproach [29,30], this optimal HAM is also based on the zeroth-order deformationequation (3.6). However, two types of special deformation-functionsare used, whichare determined completely by only one characteristic parameter|c1|< 1 and|c2|<1, respectively. In this way, there exist at most only three unknown convergence-control parametersc0,c1 and c2 at any order of approximations. In addition, thediscrete squared residual is first introduced by Liao [26] toefficiently find out theoptimal convergence-control parameters.

In § 3.2.1, the Balsius boundary-layer flow is used as an example to illustratethe basic ideas of the different approaches of the optimal HAM. The comparisonsof the different optimal HAMs are given in§ 3.2.2. A systematic description of theoptimal HAM is given in§ 3.3, followed by some concluding remarks in§ 3.4.

3.2 An illustrative description

3.2.1 Basic ideas

For the sake of simplicity, let us first describe the basic ideas of the optimal HAMvia the so-called Blasius boundary-layer flows in fluid mechanics, governed by anonlinear differential equation

f ′′′(η)+12

f (η) f ′′(η) = 0, f (0) = f ′(0) = 0, f ′(+∞) = 1, (3.11)

whereη is a similarity variable,f (η) is related to the stream-function, and theprime denotes the derivative with respect toη , respectively. Letλ > 0 denote a kindof spatial scale-parameter. By means of the transformation

f (η) = λ−1 u(ξ ), ξ = λ η . (3.12)

the original equation (3.11) becomes


u′′′(ξ )+(

12λ 2

)

u(ξ ) u′′(ξ ) = 0, u(0) = u′(0) = 0,u′(+∞) = 1, (3.13)

where the prime denotes the derivative with respect toξ . For the sake of comparisonwith the results given by the normal HAM, we choose the same valueλ = 4 as thatused by Liao [17].

Mathematically, due to the boundary conditionu′(+∞) = 1, one has the asymp-totic propertyu∼ ξ asξ →+∞. Physically, it is a common knowledge that velocityof boundary-layer flows mostly tends to mainstream flow exponentially. Accordingto these mathematical and physical knowledge,u(ξ ) could be expressed in the form

u(ξ ) = A0,0+ ξ ++∞

∑m=1

+∞

∑n=0

Am,n ξ n exp(−mξ ), (3.14)

whereAm,n is a constant to be determined. It provides us the so-called solution-expression ofu(ξ ), which plays a key role in the frame of the HAM.

Note that there exist three boundary conditions. Accordingto the solution-expression (3.14), the simplest three terms ofu(ξ ) areA0,0, ξ andA1,0exp(−ξ ).So, we choose the initial approximation in the form

u0(ξ ) = A0,0+ ξ + A1,0 exp(−ξ ),

whereA0,0 and A1,0 are unknown constants. Enforcingu0(ξ ) to satisfy the threeboundary conditions, we haveA0,0 =−1 andA1,0 = 1, i.e.

u0(ξ ) = ξ −1+e−ξ . (3.15)

In addition, according to the solution-expression (3.14),we should choose such anauxiliary linear operatorL that the simplest three terms ofu(ξ ), i.e. A0,0, ξ andA1,0exp(−ξ ), should be the common solutions ofL u= 0, say,

L

(

C0+C1ξ +C2 e−ξ)

= 0, (3.16)

which uniquely determines the auxiliary operator

L u= u′′′+u′′, (3.17)

where the prime denotes the derivative with respect toξ , C0,C1 andC2 are integra-tion coefficients.

Based on the governing equation (3.13), we define such a nonlinear operator

N (u) = u′′′(ξ )+(

12λ 2

)

u(ξ ) u′′(ξ ). (3.18)

Let α(q) andβ (q) denote two deformation-functions, i.e.

α(0) = β (0) = 0, α(1) = β (1) = 1,


whose Maclaurin series

α(q)∼+∞

∑k=1

αk qk, β (q)∼+∞

∑k=1

βk qk

converge atq= 1, say,+∞

∑k=1

αk = 1,+∞

∑k=1

βk = 1.

Let q∈ [0,1] denote the embedding parameter,c0 6= 0 the convergence-control pa-rameter, andφ(ξ ;q) a kind of continuous mapping ofu(ξ ), respectively. We con-struct the so-called zeroth-order deformation equation

[1−α(q)]L [φ(ξ ;q)−u0(ξ )] = c0 β (q) N [φ(ξ ;q)], (3.19)

subject to the boundary conditions

φ = 0,∂φ∂ξ

= 0, at ξ = 0 (3.20)

and∂φ∂ξ

= 1, asξ →+∞. (3.21)

Obviously, asq increases from 0 to 1,φ(ξ ;q) varies (or deforms) from the initialguessu0(ξ ) to the exact solutionu(ξ ).

The homotopy-series solution reads

u(ξ ) = u0(ξ )++∞

∑k=1

uk(ξ ), (3.22)

where, according to Theorem 4.18,um(ξ ) is governed by themth-order deformationequation

L

[

um(ξ )−m−1

∑k=1

αm−k uk(ξ )

]

= c0

m

∑k=1

βk δm−k(ξ ), (3.23)


um(0) = u′m(0) = 0,u′m(+∞) = 0, (3.24)

with the definitionδk(ξ ) = DkN [φ(ξ ;q)]

According to Theorem 4.2, we have

DkN [φ(ξ ;q)]= Dk[

φ ′′′(ξ ;q)]

+1

2λ 2 Dk[

φ ′′(ξ ;q) φ(ξ ;q)]

,

which gives according to Theorem 4.3 and Theorem 4.6 that


δk(ξ ) = DkN [φ(ξ ;q)]= u′′′k (ξ )+(

12λ 2

) k

∑j=0

u′′j (ξ ) uk− j(ξ ). (3.25)

Let u∗m(ξ ) denote a special solution of (3.23) andL −1 the inverse operator ofL ,respectively. We have

u∗m(ξ ) =m−1

∑k=1

αm−k uk(ξ )+ c0

m

∑k=1

βk Sm−k(ξ ), (3.26)

whereSk(ξ ) = L

−1 [δk(ξ )] . (3.27)

The common solution reads

um(ξ ) = u∗m(ξ )+B0+B1 ξ +B2 e−ξ ,

where the integral coefficients

B1 = 0, B2 =du∗mdξ

∣

∣

∣

∣

ξ=0, B0 =−u∗m(0)−B2,

are determined by the boundary conditions (3.24).The corresponding Mathematica code is given in the appendixand free available

at http://numericaltank.sjtu.edu.cn/HAM.htm.

3.2.2 Different types of optimal methods

At themth-order of approximation, we define the squared residual ofthe governingequation (3.13), i.e.

Em =

∫ +∞

0

(

N

[

m

∑i=0

ui(ξ )

])2

dξ . (3.28)

An optimal homotopy-approximation is given by the minimum of the squared resid-ual. There are different types of optimal methods, dependent upon the number of theconvergence-control parameters.

It is found that much CPU time is needed to calculate the exactsquared residualEm, especially for largem, the order of homotopy-approximation.For example, evenin case that only the basic convergence-control parameterc0 is unknown, by meansof a laptop (MacBook Pro, 2.8 GHz Inter Core 2 Due, 4 GB 1067 MHzDDR3), itneeds 68.13 seconds, 272.7 seconds and 1089.5 seconds to calculate the correspond-ing exact squared residual (3.28) form= 6,8 and 10, respectively. When there aremore than one unknown parameters, the needed CPU time increases exponentiallyso that the exact residual residual (3.28) is often useless in practice. Thus, to greatly


decrease the CPU time, Liao [26] suggested to use the so-called “discrete squaredresidual” defined by

Em ≈ 1(N+1)

N

∑j=0

[

N

(

m

∑i=0

ui(ξ j)

)]2

, (3.29)

whereξ j = j∆ξ , ∆ξ = 0.5 andN=20 for the Blasius boundary-layerflow governedby (3.13). It is found that much less CPU time is needed by means of the discretesquared residual defined above. For example, when only the basic convergence-control parameterc0 is unknown, it takes only 0.40 seconds, 0.87 seconds and1.61 seconds to gain the discrete squared residualE6,E8 andE10 defined by (3.29),respectively, which are only 0.59%, 0.32% and 0.15% of the CPU time for the exactsquared residual defined by (3.28). Therefore, in the following part of this chapter,the discrete squared residual (3.29) is used to find the optimal values of the un-known convergence-control parameters. Besides, for the sake of impartial compar-isons,E10, the discrete squared residual at the 10th-order homotopy-approximation,is employed in this chapter to search for the optimal unknownconvergence-controlparameters.

3.2.2.1 Basic optimal HAM

If only the basic convergence-control parameterc0 is unknown, we have the basicoptimal HAM. In this case, the deformation-functionsα(q) andβ (q), the initialapproximationu0(ξ ) and the auxiliary linear operatorL do not contain any un-known parameters. Since the squared residualEm is dependent uponc0, the optimalhomotopy-approximation is gained by

dEm(c0)

dc0= 0. (3.30)

Without loss of generality, let us choose the simplest deformation-functions,i.e. α(q) = q and β (q) = q. The curves of the exact squared residualEm de-fined by (3.28) versusc0 at different order of approximationm= 6,8 and 10 areshown in Fig. 3.1. Note that the exact squared residual decreases in the region−1.8≤ c0 ≤−0.3 as the order of approximation increases, which indicates that thehomotopy-series converges for an arbitrary valuec0 ∈ [−1.8,−0.3]. More impor-tantly, it indicates that the exact squared residual arrives its minimum atc0 ≈−3/2,which suggests that the optimal value ofc0 is about -3/2. So, the curves of the ex-act squared residual versusc0 provide us not only the effective-regionRc of theconvergence-control parameterc0, but also the optimal value ofc0 which gives theoptimal homotopy-series that converges fastest.

As shown in Fig. 3.2, the discrete squared residual (3.29) also decreases inthe region−1.8≤ c0 ≤ −0.3 as the order of approximation increases, so that thehomotopy-series converges for arbitrary valuec0 ∈ [−1.8,−0.3], too. More impor-


Fig. 3.1 Exact squared resid-ual (3.28) by means of thebasic optimal HAM. Solidline: 10th-order approxima-tion; Dashed line: 8th-orderapproximation; Dash-dottedline: 6th-order approximation.

c0

Em

-2 -1.5 -1 -0.5 010-5

10-4

10-3

10-2

10-1

100

101

102

Fig. 3.2 Discrete squaredresidual (3.29) by means ofthe basic optimal HAM. Solidline: 10th-order approxima-tion; Dashed line: 8th-orderapproximation; Dash-dottedline: 6th-order approximation.

c0

Em

-2 -1.5 -1 -0.5 010-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Fig. 3.3 The absolute error| f ′′(0)−0.332057| at differ-ent order of approximationsby means of the normal andthe optimal HAM. Solidline: the normal HAM withc0 = −1; Dashed line: thebasic optimal HAM withc0 =−7/5.

Order of approximation

|0.3

32

05

7-f

’’(0

)|

10 20 30 4010-7

10-6

10-5

10-4

10-3

10-2

10-1


Table 3.1 Minimum of the discrete squared residualEm and the corresponding optimal value ofc0 given by the basic optimal HAM with the used CPU time

m, order Optimal Minimum |0.33205− f ′′(0)| CPU timeof approx. value ofc0 of Em (seconds)

2 -0.3897 3.46×10−3 1.206 0.344 -1.0800 1.21×10−3 0.0976 1.66 -1.2733 3.23×10−4 0.0116 5.48 -1.3662 4.53×10−5 0.00185 14.110 -1.4002 8.91×10−6 9.71×10−4 31.612 -1.4314 3.16×10−6 2.94×10−4 63.714 -1.4760 4.76×10−7 1.89×10−4 119.916 -1.4823 6.13×10−8 8.84×10−5 218.818 -1.4913 8.37×10−9 1.02×10−5 411.320 -1.4979 1.87×10−9 8.08×10−6 719.922 -1.5180 4.90×10−10 1.06×10−5 1190

Fig. 3.4 Discrete squaredresidual (3.29) at differentorder of approximations bythe normal HAM, the basicoptimal HAM and the three-parameter optimal HAM.Solid line: the normal HAMwith c0 = −1; Symbols: thebasic optimal HAM withc0 = −7/5; Dashed line:the three-parameter optimalHAM with c0 = −1.0801and c1 = c2 = −0.2964;Dash-dotted line: the three-parameter optimal HAM withc0 = −1.7913,c1 = 0.1647andc2 = 0.1075.

m

Em

10 20 30 4010-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

tantly, the optimal value ofc0 is also about -3/2. By means of the computer algebrasystemMathematica , we directly employ the commandNMinimize with

WorkingPrecision->50to get the optimal value of the convergence-control parameterc0. It is found that theoptimal value ofc0 given by the minimum of the discrete squared residual (3.29)is indeed more and more close to−3/2, as shown in Table 3.1. Thus, the discretesquared residualEm defined by (3.29) can give good enough approximations of theoptimal convergence-control parameterc0 and its convergence-region.

According to Table 3.1,E10 has its minimum value 8.91×10−6 atc0 =−1.400.By means ofc0 =−7/5, the value off ′′(0) converges much faster to 0.332057 thanthe corresponding homotopy-series solution given by meansof the normal HAM( [17], whenc0 = −1), as shown in Fig. 3.3. The corresponding discrete squaredresidualEm decreases much more quickly than those given by the normal HAM


with c0 =−1, as shown in Fig. 3.4. So, even the basic optimal HAM can givemuchbetter approximations than the normal HAM. Therefore, it isstrongly suggestedthat at least the basic optimal HAM should be used to gain the optimal homotopy-approximation.

3.2.2.2 Three-parameter optimal HAM

Fig. 3.5 Deformation-function β (q) defined by(3.31) and (3.32). Solidline: c1 = 3/4; Dashed line:c1 = 1/2, Long-dashed line:c1 = 0; Dash-dotted line:c1 = −1/2; Dash-dot-dottedline: c1 =−3/4.

q

β(q

)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

There are many different ways to introduce more convergence-controlparametersin the frame of the HAM. For example, Liao [26] suggested in 2010 the followingone-parameter deformation-functions:

α(q) =+∞

∑m=1

αm(c2) qm, β (q) =+∞

∑m=1

βm(c1) qm, (3.31)

where

α1 = 1−c2, βm= 1−c1, αm= (1−c2) cm−12 , βm= (1−c1) cm−1

1 , m≥ 1 (3.32)

and|c1|< 1 and|c2|< 1 are the so-called convergence-control parameters. The dif-ferent values ofc1 define different deformation-functionβ (q), as shown in Fig. 3.5.In this case, the squared residualEm contains at most three unknown convergencecontrol parametersc0, c1 andc2 at anyorder of approximation. In theory, the morequickly Em decreases to zero, the faster the corresponding homotopy-series solutionconverges. So, at the given order of approximationm, the corresponding optimalconvergence-control parameters are determined by the minimum ofEm, correspond-ing to a set of three nonlinear algebraic equations

∂Em

∂c0= 0,

∂Em

∂c1= 0,

∂Em

∂c2= 0. (3.33)


This provides us a three-parameter optimal HAM. In the special case ofc1 = c2, wehave only two unknown convergence-control parametersc0 andc1, whose optimalvalues are given by

∂Em

∂c0= 0,

∂Em

∂c1= 0,

corresponding to a two-parameter optimal HAM. Whenc1 = c2 = 0, we haveα(q) = q andβ (q) = q, so that only the basic convergence-control parameterc0

is unknown: this is exactly the basic optimal HAM.In case ofc1 = c2, it is found that the discrete squared residualE10 arrives its

minimum 8.91× 10−6 at c0 = −1.0801 andc1 = c2 = −0.2964. Using these twooptimal convergence-control parameters, the corresponding squared residualEm de-creases much faster than that given by the normal HAM, but is almost the same asthe basic optimal HAM, as shown in Fig. 3.4.

In case ofc0 6= c1 6= c2, the discrete squared residualE10 has the minimum 2.53×10−6 at

c0 =−1.7913, c1 = 0.1647, c2 = 0.1075.

Using these three optimal convergence-controlparameters, the corresponding squaredresidual decrease faster than that given by the normal HAM, but is not obviouslybetter than the basic optimal HAM withc0 =−7/5 and the two-parameter optimalHAM with c0 =−1.0801 andc1 = c2 =−0.2964, as shown in Fig. 3.4.

Therefore, all of the optimal HAMs can greatly accelerate the convergence ofseries solution. However, for the considered Blasius flow, the optimal HAM withtwo or three unknown convergence-control parameters arenotobviously better thanthe basic optimal HAM with only one unknown convergence-control parameterc0.So, it is strongly suggested to use the basic optimal HAM firstin practice.

3.2.2.3 Infinite-parameter optimal HAM

If we choose the deformation-functionsα(q) = q and

β (q) =1c0

+∞

∑n=1

cn qn,

where

c0 =+∞

∑n=1

cn 6= 0,

then the zeroth-order deformation equation reads

(1−q)L [φ(ξ ;q)−u0(ξ )] =

(

+∞

∑n=1

cn qn

)

N [φ(ξ ;q)] ,

and the correspondingmth-order deformation equation becomes


L [um(ξ )− χm um−1(ξ )] =m

∑n=1

cn δm−n(ξ ),


um(0) = 0, u′m(0) = 0, u′m(+∞) = 0,

whereχ0= 1 andχm= 1 form≥ 1. Note that themth-order homotopy-approximation

u(ξ )∼ u0(ξ )+m

∑n=1

un(ξ )

contains themunknown convergence-control parameters

c1,c2,c3, · · · ,cm.

Therefore, in theory, there exist aninfinitenumber of unknown convergence-controlparameters

c1,c2,c3, · · · ,asm→ +∞. In this case, the optimalmth-order homotopy-approximation is givenby a set ofm nonlinear algebraic equations

∂Em

∂cn= 0, 1≤ n≤ m. (3.34)

The above optimal HAM was suggested in 2008 by Marinca and Herisanu [29] inthe so-called “optimal homotopy asymptotic method”.

Table 3.2 Minimum of the discrete squared residualEm, the used CPU time (seconds) and theabsolute error|0.332057− f ′′(0)| of the corresponding optimal homotopy-approximations givenby the infinite-parameter optimal HAM suggested by Marinca and Herisanu [29].

m, order Em |0.332057− f ′′(0)| CPU timeof approx. (seconds)

2 1.34×10−3 0.705 0.433 4.94×10−4 0.504 2.24 2.16×10−4 0.305 4.35 9.92×10−5 0.235 8.06 4.22×10−5 0.128 28.47 1.31×10−5 0.103 52.38 2.32×10−6 0.0348 93.29 2.12×10−7 0.0430 235.010 1.19×10−8 0.0192 1387.3

Obviously, at themth-order of approximation, the optimal HAM suggested byMarinca and Herisanu [29] containsmunknown convergence-control parameters. It


Fig. 3.6 Comparison of thediscrete squared residualEm

of the mth-order optimalhomotopy-approximation.Solid line: basic optimalHAM; Dashed line: theinfinite-parameter optimalHAM suggested by Marincaand Herisanu [29].

Order of approximation

Em

0 5 10 15 20 2510-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2


of the mth-order optimalhomotopy-approximation ver-sus the used CPU time. Solidline: basic optimal HAM;Dashed line: the infinite-parameter optimal HAMsuggested by Marinca andHerisanu [29].

CPU times (seconds)

Em

10-1 100 101 102 10310-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fig. 3.8 Comparisonof the absolute error|0.332057− f ′′(0)| of themth-order optimal homotopy-approximation versus the usedCPU time. Solid line: basicoptimal HAM; Dashed line:the infinite-parameter optimalHAM suggested by Marincaand Herisanu [29].

CPU times (seconds)

|f’’(

0)

-0

.33

20

57

|

10-1 100 101 102 10310-6

10-5

10-4

10-3

10-2

10-1

100

101


is found that, as the order of approximation increases, the squared residual of theoptimal homotopy-approximation decreases faster than thebasic optimal HAM thatcontains only one unknown convergence-controlparameterc0, as shown in Table 3.2and Fig. 3.6. Note that, the basic optimal HAM is a special case of the infinite-parameter HAM whenc1 = c0 6= 0 andck = 0 for k> 1. So, it is easy to understandthat, at thesameorder of approximation, the infinite-parameter optimal HAMgivesbetter optimal homotopy-approximation than the basic optimal HAM. However, thenumber of the unknown convergence-controlparameters of optimal HAM suggestedby Marinca and Herisanu [29] increases linearly with the order of approximation,so that the used CPU time increases exponentially, as shown in Table 3.2. So, ifwe consider the minimum ofEm versus the used CPU time, it is found that the dis-crete squared residualEm given by the basic optimal HAM decreases faster thanthe infinite-parameter optimal HAM, as shown in Fig. 3.7. Forexample, using thebasic optimal HAM, we spend 719.9 seconds CPU time to gain the20th-order op-timal homotopy-approximation with the squared residual 1.87×10−9, as shown inTable 3.1. However, using the infinite-parameter optimal HAM, we spend 1387.3seconds CPU time to obtain the 10th-order optimal homotopy-approximation withthe squared residual 1.19×10−8 only, as shown in Table 3.2. Besides, the basic op-timal HAM gives more accurate approximation off ′′(0) than the infinite-parameteroptimal HAM, as shown in Fig. 3.8. So, the basic optimal HAM iscomputation-ally more efficient than the infinite-parameter optimal HAM suggested by Marincaand Herisanu [29]. Therefore, although the infinite-parameter optimal HAM is morerigorous in theory, the basic optimal HAM is computationally more efficient in prac-tice.

The similar conclusion was found by Niu and Wang [34], who solved somedifferential equations by means of the optimal method suggested by Marinca andHerisanu [29] and reported that the CPU time increases exponentially as the or-der of approximation increases, so that the so-called “optimal homotopy asymptoticmethod” is time-consuming [31, 35], especially for high-order approximations. Intheory, this is easy to understand: if we express the solution in the form

u(x) =+∞

∑n=1

an en(x),

whereen(x) is a base function andan is unknown coefficient, and then use themethod of least squares [6], we also had to solve a set of nonlinear algebraic equa-tions with large number of unknowns, which is however rathertime-consuming. Toovercome this disadvantage of “optimal homotopy asymptotic method” suggestedby Marinca and Herisanu [29], Niu and Wang [34] developed the so-called “one-stepoptimal homotopy analysis method”: the optimal value of thefirst convergence-control parameterc1 is approximately determined by solving only one algebraicequation

dE1

dc1= 0,


then keeping this known value ofc1 as a good approximation of its optimal valueforever, one can similarly obtainc2 by solving only one algebraic equation

dE2

dc2= 0,

and so on. In this way, one can obtain the “optimal” valuesc1,c2,c3, · · · up to anyorder of approximation by solving only one algebraic equation

dEk

dck= 0, k= 1,2,3, · · ·

each time. However, rigorously speaking,c1,c2,c3, · · · obtained in this way arenotthe optimal ones in theory.

3.2.2.4 Finite-parameter optimal HAM

As shown above, it is time-consuming if an optimal HAM contains infinite numberof convergence-control parameters. To overcome this disadvantage, we modify theso-called “optimal homotopy asymptotic method” [29] by using only finite numberof convergence-control parameters.

If we chooseα(q) = q and such a special deformation-function

β (q) =1c0

κ

∑n=1

cn qn,

whereκ ≥ 1 is a positive integer and

c0 =κ

∑n=1

cn 6= 0,

then the zeroth-order deformation equation reads

(1−q)L [φ(ξ ;q)−u0(ξ )] =

(

κ

∑n=1

cn qn

)

N [φ(ξ ;q)] ,

and the correspondingmth-order deformation equation becomes

L [um(ξ )− χm um−1(ξ )] =minm,κ

∑n=1

cn δm−n(ξ ),


um(0) = 0, u′m(0) = 0, u′m(+∞) = 0,

whereχ0= 1 andχm= 1 form≥ 1. Note that themth-order homotopy-approximation


u(ξ )∼ u0(ξ )+m

∑n=1

un(ξ )

contains at most theκ unknown convergence-control parameters

c1,c2,c3, · · · ,cκ .

Therefore, in theory, there exist afinite number of unknown convergence-controlparameters

c1,c2,c3, · · · ,cκ

even asm→ +∞. In this case, the optimalmth-order homotopy-approximation isgiven by a set of minm,κ nonlinear algebraic equations

∂Em

∂cn= 0, 1≤ n≤ minm,κ . (3.35)

The above optimal HAM becomes exactly the so-called “optimal homotopyasymptotic method” suggested by Marinca and Herisanu [29], if κ → ∞. Besides,whenc1 = c0 andcn = 0 for n> 1, it becomes the basic optimal HAM. Therefore,this optimal HAM is more general.

Table 3.3 Minimum of the discrete squared residualEm, the used CPU time (seconds) and theabsolute error|0.332057− f ′′(0)| of the corresponding optimal homotopy-approximations givenby the finite-parameter optimal HAM whenκ = 2.


4 6.84×10−4 0.440 2.236 4.74×10−4 0.309 8.698 4.23×10−5 1.85×10−3 27.110 3.38×10−6 2.61×10−3 71.812 1.24×10−6 1.56×10−3 171.014 2.15×10−7 9.55×10−5 382.116 6.07×10−9 2.76×10−4 816.218 2.45×10−9 1.62×10−4 1637.820 1.78×10−9 7.05×10−5 3145.6

Let us first consider the case of two convergence-control parametersc1 andc2,i.e. κ = 2. Regardingc1 andc2 as unknown parameters, we first gain themth-orderhomotopy-approximation and then obtain the optimal convergence-control param-etersc1 andc2 by the minimum ofEm. It is found that, as the order of approxi-mation increases, the used CPU time increases exponentially. The minimum ofEm,the used CPU time and the absolute error| f ′′(0)− 0.332057| of the correspond-ing optimalmth-order homotopy-approximations are given in Table 3.3. Consid-ering the squared residual versus the used CPU time, it is found that the the op-timal homotopy-approximations given by the basic optimal HAM (corresponding


Table 3.4 Minimum of the discrete squared residualEm, the used CPU time (seconds) and theabsolute error|0.332057− f ′′(0)| of the corresponding optimal homotopy-approximations givenby the finite-parameter optimal HAM whenκ = 3.


4 5.21×10−4 0.333 2.706 4.93×10−4 0.508 11.28 2.57×10−5 2.11×10−3 38.010 2.52×10−6 2.82×10−3 107.112 2.50×10−6 4.44×10−4 279.514 1.51×10−7 1.95×10−5 686.216 4.67×10−9 2.86×10−4 1580.518 1.84×10−9 7.07×10−5 3399.2


of the mth-order optimalhomotopy-approximationversus the used CPU time.Solid line: basic optimalHAM (κ = 1); Left-triangle:the finite-parameter optimalHAM when κ = 2; Right-triangle: the finite-parameteroptimal HAM whenκ = 3.

CPU times (seconds)

Dis

cret

esq

uare

dre

sidu

al

10-1 100 101 102 10310-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fig. 3.10 Comparisonof the absolute error|0.332057− f ′′(0)| of themth-order optimal homotopy-approximation versus the usedCPU time. Solid line: basicoptimal HAM (κ = 1); Left-angle: the finite-parameteroptimal HAM whenκ = 2;Right-angle: the finite-parameter optimal HAMwhenκ = 3.

CPU times (seconds)

|f’’(

0)

-0

.33

20

57

|

10-1 100 101 102 10310-6

10-5

10-4

10-3

10-2

10-1

100

101


to κ = 1) converges faster than those by the finite-parameter optimal HAM whenκ = 2, as shown in Figs. 3.9 and 3.10. The same conclusion is obtained in caseof κ = 3, as shown in Figs. 3.9 and 3.10. It is found that the squared residualEm given by two-parameter optimal HAM (κ = 2) decreases faster than those bythree-parameter optimal HAM (κ = 3), as shown in Fig. 3.9. Notice thatκ = 1corresponds to the basic optimal HAM. So, this example illustrates that the basicoptimal HAM with only one convergence-control parameterc0 is computationallymore efficient than other optimal HAM with more unknown convergence-controlparameters. Therefore, it is strongly suggested that the basic optimal HAM withonly one convergence-control parameterc0 should be used first in practice.

Fig. 3.11 Discrete squaredresidual versus used CPUtime. Solid line: the basicoptimal HAM (κ = 1) withc0 = −7/5; Left-triangle:two-parameter optimal HAM(κ = 2) with c1 = −1.4572and c2 = −0.0795; Right-triangle: three-parameteroptimal HAM (κ = 3) withc1 =−1.4870,c2 =−0.0826andc3 =−0.0126.

CPU time

Dis

cret

esq

uare

dre

sidu

al

100 101 102 10310-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

Fig. 3.12 Absolute error| f ′′(0)− 0.332057| versusused CPU time. Solid line: thebasic optimal HAM (κ = 1)with c0 =−7/5; Left-triangle:two-parameter optimal HAM(κ = 2) with c1 = −1.4572and c2 = −0.0795; Right-triangle: three-parameteroptimal HAM (κ = 3) withc1 =−1.4870,c2 =−0.0826andc3 =−0.0126.

CPU time

|f’’(

0)-

0.3

32

05

7|

100 101 102 10310-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

In practice, we can choose the optimal convergence-controlparameters at a rea-sonably high order of approximation. For example, whenκ = 2, the discrete squaredresidual at the 10th-order of approximation arrives its minimum 3.38×10−6 with

3.3 Systematic description 121

the optimal convergence-control parametersc1 =−1.457 andc2 =−0.0795. Simi-larly, whenκ = 3, the discrete squared residual at the 10th-order of approximationarrives its minimum 2.52×10−6 with the optimal convergence-control parametersc1 =−1.4870,c2 =−0.0826 andc3 =−0.0126. It is found that the discrete squaredresidual of the homotopy-series given by the two-parameteroptimal HAM (κ = 2)decreases a little more quickly than the basic optimal HAM (κ = 1) and even thanthe three-parameter optimal HAM (κ = 3), as shown in Fig.3.11. However, it isfound that f ′′(0) given by the basic optimal HAM (κ = 1) converges to the exactvalue 0.332057 more quickly than the two-parameter (κ = 2) and three-parameter(κ = 3) optimal HAM, as shown in Fig. 3.12. Note that the approximations given bythe three-parameter optimal HAM (κ = 3) are often worse than those given by thetwo-parameter optimal HAM (κ = 2), as shown in Figs. 3.11 and 3.12. Therefore,although the used CPU time is nearly the same for different optimal approaches(κ = 1,2,3) to gain results at the same order of approximation, the optimal HAMswith more convergence-control parameters do not give obviously better homotopy-approximations than the basic optimal HAM in general. Thus,the basic optimalHAM is strongly suggested to use in practice.

In summary, the approximations given by an optimal HAM converge much fasterthan the normal HAM in general. The example considered in this chapter suggeststhat the optimal HAMs with one or two convergence-control parameters are com-putationally most efficient and can give accurate enough approximations, but theoptimal HAMs with too many convergence-control parametersare time-consuming.

3.3 Systematic description

The definition of the discrete squared residual (3.29) and the above-mentioned opti-mal HAMs have general meanings. Thus, they can be applied to solve different typesof nonlinear equations with strong nonlinearity. Here, we give a brief description ingeneral.

Given a nonlinear differential equation

N [u(x, t)] = 0, (3.36)

whereu(x, t) is a unknown funciton,x andt denote respectively spatial and temporalindependent variables, respectively, we can choose a proper initial guessu0(x, t)and a proper auxiliary linear operatorL to construct the zeroth-order deformationequation

[1−α(q)]L [φ(x, t;q)−u0(x, t)] = c0 β (q) N [φ(x, t;q)] , (3.37)

whereq ∈ [0,1] is an embedding parameter,α(q) andβ (q) are two deformation-functions withκ unknown convergence-control parameters

c= c1,c2, · · · ,cκ ,


whereκ may be an infinity, and the Maclaurin series ofα(q),β (q) read

α(q)∼+∞

∑n=1

αn qn, β (q)∼+∞

∑n=1

βn qn.

Assuming that the initial guessu0(x, t), the auxiliary linear operatorL , and the(κ+1) convergence-controlparametersc0,c1, · · · ,cκ are so properly chosen that thehomotopy-Maclaurin series

φ(x, t;q) = u0(x, t)++∞

∑m=1

um(x, t) qm (3.38)

converges atq= 1, we have the homotopy-series solution

u(x, t) = u0(x, t)++∞

∑m=1

um(x, t). (3.39)

Substituting (3.38) into the zeroth-order deformation equation (3.37) and then equat-ing the coefficients of the like-power of the embedding parameterq, we have4 themth-order deformation equation:

L

[

um(x, t)−m−1

∑n=1

αm−n un(x, t)

]

= c0

m

∑n=1

βn δm−n(x, t), (3.40)

where

δ j(x, t) = D j

N

[

+∞

∑n=0

un(x, t) qn

]

. (3.41)

The definition of thejth-order homotopy-derivative operatorD j is given in Chap-ter 4. The special solutionu∗m(x, t) of (3.40) is given by

u∗m(x, t) =m−1

∑n=1

αm−n un(x, t)+ c0

m

∑n=1

βn Sm−n(x, t), (3.42)

whereSn(x, t) = L

−1 [δn(x, t)] (3.43)

andL −1 is the inverse operator ofL . Then,um(x, t) is uniquely determined by thecorresponding boundary/initial conditions.

To avoid time-consuming computation for the exact squared residual, at themth-order of approximation, we define a kind of discrete squared residualEm in a similarway to (3.29).

4 Using the so-calledmth-order homotopy-derivative operatorDm defined in Chapter 4, one canobtain exactly the samemth-order deformation equation.

3.3 Systematic description 123

Definition 3.1. Let

(x j , t j) ∈ Ω , j = 0,1,2, · · · ,N,

denote the properly chosenN+1 points in the domainΩ on which a nonlinearequationN u(x, t) = 0 is defined. Then, the integral

Em =1

(N+1)

N

∑j=0

[

N

(

m

∑n=0

un(x j , t j)

)]2

(3.44)

is calledthe discrete squared residualof N (u) = 0 on the domainΩ .

Assume that themth-order homotopy-approximation containsκ ′ + 1 unknownconvergence-control parameters, whereκ ′ ≤ κ . Then,Em containsκ ′+1 unknownconvergence-controlparametersc0,c1, · · · ,cκ ′ , whose optimal values are determinedby the minimum ofEm, corresponding to a set ofκ ′+1 nonlinear algebraic equa-tions

∂Em

∂c j= 0, 0≤ j ≤ κ ′ ≤ κ .

The above approach gives the so-called finite-parameter optimal HAM whenκ ≥ 1is a fixed finite integer. It gives the so-called infinite-parameter optimal HAM whenthere are an infinite number of convergence-control parameters, i.e.κ → +∞, say,the number of convergence-control parameter linearly increases with the order ofapproximation.

The above optimal HAMs are based on the deformation-functions with someunknown convergence-controlparameters. There are an infinite number of such kindof deformation-functions satisfying the properties (3.7)and (3.8). For example, Liao[26] suggested the one-parameter deformation-function

β (q;c) = (1− c)+∞

∑n=1

cn−1qn, |c|< 1 (3.45)

in the so-called three-parameter optimal HAM, which contains the special caseβ (q;0) = q (whenc1 = 0) that is mostly used in the frame of the HAM. Besides,one can define the following one-parameter deformation-function

β (q;c) =1

ζ (c)

+∞

∑n=1

qn

nc , c> 1, (3.46)

whereζ (c) is Riemann zeta function, andc is a convergence-control parameter.We callβ (q;c) andβ (q;c) the first and second-type of one-parameter deformationfunctions, respectively.

Besides, any two different deformation-functions may create a new one. For ex-ample,


α(q) = β (q;c1) β (q;c2) (3.47)

defines a deformation-function with two convergence-control parametersc1 andc2.Zhao and Wong [41] defines a family of deformation-functions

Am+1(q;cm+1) =cm+1 Am(q;cm)

1+(cm+1−1) Am(q;cm), (3.48)

wherec j 6= 0 ( j = 1,2,3, · · · ,m+1) is convergence-controlparameter, andAm(q;cm)is a m-parameter deformation-function with them convergence-control parame-terscm = c1,c2, · · · ,cm. To avoid singularity of the deformation-function definedabove, we had to add such a restriction

1+(cm+1−1) Am(q;cm) 6= 0, q∈ [0,1]. (3.49)

In general, given any a convergent series

Π =+∞

∑n=1

cn 6= 0,

we can always define a corresponding deformation-function

β (q;c∞) =1Π

+∞

∑n=1

cn qn,

wherec∞ = c1,c2, · · · .

Especially, whenΠ =κ∑

n=1cn 6= 0, we have aκ-parameter deformation-function

β (q;cκ) =1Π

+κ

∑n=1

cn qn.

In a special case ofα(q) = q and

β (q) =1c0

κ

∑n=1

ck qk, c0 =κ

∑n=1

cn 6= 0,

whereκ ≥ 1 is either a finite integer or the infinity. Then, the corresponding zeroth-order deformation equation reads

(1−q)L [φ(x, t;q)−u0(x, t)] =

(

κ

∑n=1

cn qn

)

N [φ(x, t;q)] , (3.50)

and the corresponding high-order deformation equation becomes


L [um(x, t)− χm um−1(x, t)] = c0

minm,κ∑n=1

βn δm−n(x, t). (3.51)

The corresponding optimal homotopy-approximation contains minm,κ conver-gence control parameters. Especially, whenκ = 1, we have only one convergence-control parameterc1 = c0, which is exactly the basic optimal HAM. Whenκ = ∞, itis the optimal HAM suggested by Marinca and Herisanu [29]. Whenκ = 2 orκ = 3,we have a finite-parameter optimal HAM first proposed in this chapter. Thus, thisoptimal HAM logically contains the basic optimal HAM and theoptimal approachsuggested by Marinca and Herisanu [29], and therefore is more general.

3.4 Concluding remarks and discussions

Based on the homotopy in topology, the HAM provides us extremely large free-dom to choose the initial approximation, the auxiliary linear operatorL and thedeformation-functions to construct the so-called zeroth-order deformation equation.Especially, the so-called convergence-control parameterc0 introduced by Liao [16]in 1997 provides us a convenient way to control and adjust theconvergence of thehomotopy-series: unlike other analytic techniques, the HAM can guarantee the con-vergence of solution series of nonlinear equations. In fact, it is the convergence-control parameterc0 that differs the HAM from other analytic techniques. So, theinduction of the convergence-control parameterc0 in the zeroth-order deformationequation (3.5) by Liao [16] in 1997 is a milestone of the HAM. In essence, theconvergence-control parameterc0 provides us one more “artificial” degree of free-dom to guarantee the convergence of homotopy-series solution. Therefore, Liao [17]in 1999 further proposed a more generalized zeroth-order deformation equation(3.6), which provides us extremely large freedom to introduce more convergence-control parameters in theory. Then, the optimal values of the convergence-controlparameters are determined by the minimum of the squared residual of governingequations, which give the fastest convergent series solution of a given nonlinearequation in general.

In this chapter, using the Blasius flow as an example, we describe the basic ideasof the different types of optimal HAM, and compare them. Using the deformation-functions

α(q) = q, β (q) =1c0

κ

∑n=1

cn qn, c0 =κ

∑n=1

cn 6= 0, κ ≥ 1,

we propose a more generalized optimal HAM: it becomes the basic optimal HAMwhenκ = 1, Marinca and Herisanu’s optimal HAM [29] whenκ → ∞, and an finite-parameter optimal HAM whenκ is a finite positive integer. Based on these compu-tations, we have the following conclusions:


1. the homotopy-approximations given by the optimal convergence-control param-eters are much more accurate in general;

2. the basic optimal HAM with one convergence-controlparameterc0 can give goodenough approximations in most cases, and therefore is strongly suggested to firstuse in practice;

3. more convergence-controlparameters might give better approximations, but needmuch more CPU time. So, considering the accuracy versus CPU time, the basicoptimal HAM with one convergence-control parameter and theoptimal HAMswith two or three convergence-control parameters are strongly suggested in prac-tice. In other words, we should balance the rigorousness in theory against thecomputational efficiency in practice.

Note that the computational efficiency of different types ofoptimal HAMs de-pends strongly on the method of searching for the minimum of squared residual.In this book, the commandNMinimize with WorkingPrecision->50 ofthe computer algebra systemMathematica is used to find out the minimum ofsquared residual and the corresponding optimal convergence-control parameters.The optimal HAMs mentioned in this chapter should be more powerful, if bettermethods of finding out minimum of squared residual are proposed in future.

Since there exist no rigorous theories up to now to guide us how to choose agood enough zeroth-order deformation equation for a given nonlinear equation ingeneral, it is important in practice to provide a convenientway to guarantee the fastconvergence of the homotopy-series. Our strategy is to introduce some unknownauxiliary-parameters without physical meanings (i.e. theconvergence-control pa-rameters) and then to determine their optimal values by the minimum of squaredresidual of governing equations. It should be emphasized that, in the frame of theHAM, we can introduce such kind of unknown auxiliary-parameters in many differ-ent ways. For example, if we choose such an initial approximation

u0(ξ ) = ξ +(1+ µ) e−ξ − µ2

e−2ξ −(

1+µ2

)

(3.52)

for the Blasius boundary-layer flow considered in this chapter, we can regardµas one convergence-control parameter, too. Indeed, the optimal HAMs provide usextremely large freedom to introduce different types of convergence-control param-eters to gain accurate enough approximation.

Appendix 3.1 Mathematica code for Blasius flow 127

Appendix 3.1Mathematica code for Blasius flow

Blasius boundary-layer flow equation

f ′′′+ 12 f f ′′ = 0, f (0) = 0, f ′(0) = 0, f ′(+∞) = 1

is solved by means of the different types of optimal HAM. Thiscode is free availableat http://numericaltank.sjtu.edu.cn/HAM.htm

A Simple Users GuideInput data:c0 : Basic convergence-control parameterc0;c1,c2 : Convergence-control parametersc1 andc2;

Control parameter:OHAM: Basic or three-parameter optimal HAM whenOHAM = 0;

Infinite or finite-parameter optimal HAM whenOHAM = 1;Nstep : The value of the integerκ in (3.50);PRN: Show the used CPU time whenPRN = 1;

Calculated results:f[k] : f (η);fx[k] : f ′(η);fxx0[k] : f ′′(0);delta[k] : δk(ξ ) defined by (3.25);S[k] : Sk(ξ ) defined by (3.27);alpha[n] : αn, the coefficient of Maclaurin series ofα(q);beta[n] : βn, the coefficient of Maclaurin series ofβ (q);cc[n] : Set of parametersc1,c2, · · · ,cn, wherem= minn,κ;u[k] : uk(ξ );

U[k] : kth-order approximation ofu(ξ ), i.e.k∑

n=0un(ξ );

Err[k] : Residual error squareEk defined by (3.29);

Main codes:ham[1,11] : First, gain 1st to 10th-order homotopy-approximation;ham[12,25] : Then, gain 12th to 25th-order homotopy-approximation;GetErr[k] : Gain squared residualEk defined by (3.29);


Mathematica code for Blasius flowby Shijun LIAO



( *************************************************** *********** )( * Define the initial guess * )( *************************************************** *********** )u[0] = y - 1 + Exp[-y];U[0] = u[0];f[0] = U[0]/lambda /. y-> lambda * x ;fxx0[0] = D[f[0],x,2] /.x->0 ;lambda = N[4,100];

( *************************************************** *********** )( * Define auxiliary linear operator L * )( *************************************************** *********** )L[f_] := D[f,y,3] + D[f,y,2];

( *************************************************** *********** )( * Define inverse operator of auxiliary linear operator Linv * )( * Gain the solution u of equation: L[u ] = f * )( *************************************************** *********** )Linv[f_] := Module[temp,temp = DSolve[ L[w[y]] == f, w[y], y];temp[[1,1,2]] /. C[_]->0 // Expand];

( *************************************************** *********** )( * The property of the inverse operator Linv * )( * Linv[f_+g_] := Linv[f] + Linv[g] * )( *************************************************** *********** )Linv[p_Plus] := Map[Linv,p];Linv[c_ * f_] := c * Linv[f] /; FreeQ[c,y];

( *************************************************** *********** )( * Define Getdelta[m] * )( *************************************************** *********** )Getdelta[k_] := Module[temp,uyy[k] = D[u[k],y,2] //Expand ;uyyy[k] = D[uyy[k],y] //Expand ;temp[1] = Sum[u[k-n] * uyy[n],n,0,k]/2/lambda/lambda//Expand;temp[2] = uyyy[k] + temp[1] //Expand;delta[k] = Collect[temp[2], yˆ_. * Exp[_.]];];

Appendix 3.1 Mathematica code for Blasius flow 129

( *************************************************** *********** )( * Define GetErr[m] * )( * This module gives averaged squared residual * )( *************************************************** *********** )GetErr[k_] := Module[temp,dy,G,x,s,Uyy[k] = D[U[k],y,2];Uyyy[k] = D[Uyy[k],y];error[k] = Uyyy[k] + U[k] * Uyy[k]/2/lambda/lambda;Ymax = 10;Nmax = 20;dy = N[Ymax/Nmax,100];s = 0;For[j = 0, j <= Nmax, j = j + 1,

x = j * dy;G[j] = error[k] /. y -> x //Expand;temp[2] = G[j]ˆ2;s = s + temp[2];

];Err[k] = s/(Nmax+1) /. AA->1-c1, BB-> 1-c2;If[NumberQ[Err[k]],Err[k]//N//Print];];

( *************************************************** *********** )( * Main Code * )( *************************************************** *********** )ham[begin_,end_]:=Block[uSpecial,B0,B2,temp,z,s,time[0] = SessionTime[];For[k=begin,k<=end,k=k+1,

If[k==1,Print["-------------------------------------------- --"]];

If[k == 1 && OHAM == 0,Nstep = Infinity;If[NumberQ[c1], beta[1] = 1-c1, beta[1] = AA ];If[NumberQ[c2], alpha[1] = 1-c2, alpha[1] = BB ];Print["FINITE-parameter optimal HAM"];Print[" c0 = ",c0];Print[" c1 = ",c1];Print[" c2 = ",c2];If[c1 ==0 && c2 ==0,

Print["This is the BASIC optimal HAM ! "]];

];If[ k > 1 && OHAM == 0 ,

beta[k] = c1 * beta[k-1];alpha[k] = c2 * alpha[k-1];];

If[k == 1 && OHAM > 0,If[Nstep == Infinity,

Print["INFINITE-parameter optimal HAM"],Print["FINITE-parameter optimal HAM"]];

alpha[1] = 1 - c2;c0 = 1;


Print[" Nstep = ",Nstep];Print[" c0 = ",c0];Print[" c2 = ",c2];];

If[k > 1 && OHAM > 0, alpha[k] = c2 * alpha[k-1] ];If[k==1,

Print["-------------------------------------------- ---"]];

If[OHAM > 0 && k == 1, cc = beta[1] ];If[OHAM > 0 && k > 1 && k <= Nstep,

temp = Union[cc,beta[k]];cc = temp];

Print["k = ",k];Getdelta[k-1];temp = Linv[delta[k-1]];S[k-1] = Collect[temp, yˆ_. * Exp[_.]];temp = Sum[alpha[k-n] * u[n],n,1,k-1] ;uSpecial = temp+c0 * Sum[beta[n] * S[k-n],n,1,Min[k,Nstep]];B2 = D[uSpecial,y] /. y->0 ;B0 = -B2 - uSpecial /. y->0 ;temp = uSpecial + B0 + B2 * Exp[-y] // Expand;u[k] = Collect[temp, yˆ_. * Exp[_.] ];U[k] = Expand[U[k-1] + u[k]];f[k] = U[k]/lambda /. y -> lambda * x,AA->1-c1,BB->1-c2;fx[k] = D[f[k],x];fxx0[k] = D[fx[k],x] /. x->0 ;If[NumberQ[fxx0[k]], Print[" f’’(0) = ", fxx0[k]//N ] ];If[IntegerQ[k/5] && PRN == 1,

time[k] = SessionTime[];temp = time[k]-time[0];Print["Used CPU times = ",temp, " (seconds) "];];

];Print["successful "];];

( *************************************************** *********** )( * Define physical and control parameters * )( *************************************************** *********** )OHAM = 0;Nstep = Infinity;PRN = 1;c0 = -7/5;c1 = 0;c2 = 0;

( * Gain the 10th-order homotopy-approximation * )ham[1,10];

Problems 131

Problems

3.1. Effect of convergence-control parameter in initial approximationHow to find the optimal homotopy-approximations if one uses the initial approxi-mationu0(ξ ) defined by (3.52) and regardsµ as one of convergence-control param-eters? How to introduce more such kind of convergence-control parameters in initialapproximations?

3.2. Combination of the optimal HAM with iterationHow to combine the optimal HAMs with the iteration approach described in Chap-ter 2 so as to further accelerate the convergence of series solution of a nonlineardifferential equation?


References


2. Adomian, G.: A review of the decomposition method and somerecent results for nonlinearequations. Comput. Math. Appl.21, 101 – 127 (1991)


4. Adomian, G., Adomian, G.E.: A global method for solution of complex systems. Math.Model.5, 521 – 568 (1984)

5. Akyildiz, F.T., Vajravelu, K. Magnetohydrodynamic flow of a viscoelastic fluid. Phys. Lett.A. 372, 3380 – 3384 (2008)

6. Bjorck, A.: Numerical Methods for Least Squares Problems. SIAM (1996)7. Awrejcewicz, J., Andrianov, I.V., Manevitch, L.I.: Asymptotic Approaches in Nonlinear Dy-

namics. Springer-Verlag, Berlin (1998)8. Cherruault, Y.: Convergence of Adomian’s method. Kyberneters.8, 31 – 38 (1988)9. Cole, J.D.: Perturbation Methods in Applied Mathematics. Blaisdell Publishing Company,

Waltham (1992)10. He, J.H.: Homotopy perturbation technique. Comput. Method. Appl. M. 178, 257 – 262

(1999)11. Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge

(1953)12. Karmishin, A.V., Zhukov, A.T., Kolosov, V.G.Methods ofDynamics Calculation and Testing

for Thin-walled Structures (in Russian). Mashinostroyenie, Moscow (1990)13. Kevorkian, J., Cole, J.D.: Multiple Scales and SingularPerturbation Methods. Springer-

Verlag, New York (1995)14. Liang, S.X., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy pertur-









22. Liao, S.J.: Notes on the homotopy analysis method – some definitions and theorems. Com-mun. Nonlinear Sci. Numer. Simulat.14, 983 – 997 (2009)

23. Liao, S.J.: A general approach to get series solution of non-similarity boundary layer flows.Commun. Nonlinear Sci. Numer. Simulat.14, 2144 – 2159 (2009)

24. Liao, S.J.: Series solution of deformation of a beam witharbitrary cross section under an axialload. ANZIAM J. 51, 10–33 (2009)



References 133

27. Lindstedt, A.: Under die integration einer fur die storungstheorie wichtigen differentialgle-ichung. Astron. Nach.103, 211 – 222 (1882)

28. Lyapunov, A.M.: General Problem on Stability of Motion (English translation). Taylor &Francis, London (1992)



31. Marinca, V., Herisanu, N.: Comments on “A one-step optimal homotopy analysis method fornonlinear differential equations”. Commun. Nonlinear Sci. Numer. Simulat.15, 3735 – 3739(2010).

32. Murdock, J.A.: Perturbations – Theory and Methods. JohnWiley & Sons, New York (1991)33. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (2000)34. Niu, Z., Wang, C.: A one-step optimal homotopy analysis method for nonlinear differential

equations. Commun. Nonlinear Sci. Numer. Simulat.15, 2026 – 2036 (2010).35. Niu, Z., Wang, C.: Reply to “Comments on ‘A one-step optimal homotopy analysis method

for nonlinear differential equations’ ”. Commun. Nonlinear Sci. Numer. Simulat.15, 3740 –3743 (2010).

36. Rach, R.: A new definition of Adomian polymonial. Kybernetes.37, 910 – 955 (2008)37. Sen, S.: Topology and Geometry for Physicists. AcademicPress, Florida (1983)38. Von Dyke, M.: Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford

(1975)39. Yabushita, K., Yamashita, M., Tsuboi, K.: An analytic solution of projectile motion with the

quadratic resistance law using the homotopy analysis method. J. Phys. A – Math. Theor.40,8403 – 8416 (2007)

40. Yang, C., Liao, S.J.: On the explicit, purely analytic solution of Von Karman swirling viscousflow. Commun. Nonlinear Sci. Numer. Simulat.11, 83 – 39 (2006)

41. Zhao, J., Wong, H.Y.: A closed-form solution to Americanoptions under general diffussionprocesses. Quant. Financ. (Accepted)

Chapter 4Systematic descriptions and related theorems

Abstract In this chapter, the homotopy analysis method (HAM) is systematicallydescribed in details as a whole. Mathematical theorems related to the so-calledhomotopy-derivative operator and deformation equations are proved, which arehelpful to gain high-order approximations. Some theorems of convergence areproved, and the methods to control and accelerate convergence are generally de-scribed. A few of open questions are discussed.

4.1 Brief frame of the homotopy analysis method

In Chapter 2, the basic ideas of the homotopy analysis method(HAM) [16, 20–34,44] are described by means of two simple examples. In this chapter, we systemati-cally describe the HAM in a general way.

The starting-point of the homotopy analysis method is to construct the so-calledzeroth-order deformation equation. Given an original nonlinear equation

N [u(x, t)] = 0

denoted byE1, which has at least one solutionu(x, t), whereN denotes a nonlinearoperator,x is a vector of all spatial independent-variables,t denotes the temporalindependent-variable, respectively. Assume that we can choose an initial equationE0 whose solutionu0(x, t) is easy to know, and that we can construct such a homo-topy [6, 43] of equationsE (q) : E0 ∼ E1 that, as the homotopy-parameterq∈ [0,1]increases from 0 to 1,E (q) deforms (or varies)continuouslyfrom the initial equa-tion E0 to the original equationE1, while its solution exists forq∈ [0,1] and besidesvariescontinuouslyfrom the known solutionu0(x, t) of the initial equationE0 to theunknown solutionu(x, t) of the original equationE1. i.e.N [u(x, t)] = 0. Such kindof homotopy of equations is calledthe zeroth-order deformation equation, whosedefinition is given in Chapter 2. Given an original nonlinearequationE1, we haveextremely large freedom to construct many different zeroth-order deformation equa-

135

136 4 Systematic descriptions and related theorems

tions, as pointed out in Chapter 2. In essence, it is such kindof freedom that differsthe HAM from other analytic approximation techniques [7,11,12,39–41].

Assume that, for a given original equationN [u(x, t)] = 0, a zeroth-order de-formation equation is constructed so properly that the solution φ(x, t;q) existsinq∈ [0,1] and isanalyticatq= 0, therefore its Maclaurin series about the homotopy-parameterq exists. Atq= 0, according to the definition of the zeroth-order defor-mation equation, we have the auxiliary equationE0, whose solutionu0(x, t) is easyto known, i.e.

φ(x, t;0) = u0(x, t). (4.1)

At q= 1, E (q) is equivalent to the original equationN [u(x, t)] = 0 so that we have

φ(x, t;1) = u(x, t). (4.2)

Using (4.1), the Maclaurin series ofφ(x, t;q) with respect toq reads

φ(x, t;q)∼ u0(x, t)++∞

∑k=1

uk(x, t) qk, (4.3)

where

uk(x, t) =1k!

∂ kφ(x, t;q)∂qk

∣

∣

∣

∣

q=0= Dk [φ(x, t;q)] .

Here, (4.3) is called the homotopy-Maclaurin series ofφ(x, t;q), Dk [φ(x, t;q)]is called thekth-order homotopy-derivative ofφ(x, t;q), Dk is called thekth-order homotopy-derivative operator, respectively. Especially, we have atq = 1 thehomotopy-series

φ(x, t;1)∼ u0(x, t)++∞

∑k=1

uk(x, t). (4.4)

If the above homotopy-series is convergent toφ(x, t;1), then, according to (4.2), wehave the homotopy-series solution

u(x, t) = u0(x, t)++∞

∑k=1

uk(x, t). (4.5)

In practice, only finite terms can be obtained, which give us theMth-order homotopy-approximation

u(x, t)≈ u0(x, t)+M

∑k=1

uk(x, t). (4.6)

Note that the unknown termuk(x, t) is governed by the so-called high-order de-formation equation, which is completely determined by the zeroth-order deforma-tion equation. In§ 4.3, different types of zeroth-order deformation equations andtheir high-order deformation equations are given. All of these high-order deforma-tion equations are linear with respect to the unknownuk(x, t), and thus are easyto solve by mean of computer algebra system such as Mathematica, Maple and so

4.2 Properties of homotopy-derivative 137

on. In § 4.2, some theorems for the operatorDk of the homotopy-derivative areproved. Using these theorems, it is easy to gain the corresponding high-order de-formation equations of different types of zeroth-order deformation equations. In§ 4.4, it is proved that all homotopy-series satisfy the original equation and thusare homotopy-series solution, as long as the zeroth-order deformation equation is soproperly constructed that its solution exists and is analytic for q in the whole domainq∈ [0,1].

Therefore, the key point of the HAM is to construct a proper zeroth-order defor-mation equation so that its solution exists and is analytic in the domainq∈ [0,1]

4.2 Properties of homotopy-derivative

As mentioned in Chapter 2, the so-calledhomotopy-derivativeis used in the frameof the HAM. Here, we first give a rigorous definition of the homotopy-derivativeand then prove some of its properties in general. By means of these properties, it iseasy to deduce the corresponding high-order deformation equations of any a givenzeroth-order deformation equation.

Definition 4.1. Let φ be a function of the homotopy-parameterq, then

Dm(φ) =1m!

dmφdqm

∣

∣

∣

∣

q=0(4.7)

is calledthe mth-order homotopy-derivativeof φ , wherem≥ 0 is an integer, andDm is calledthe operator of the mth-order homotopy-derivative.

Theorem 4.1.For two arbitrary homotopy-Maclaurin series

φ =+∞

∑k=0

uk qk, ψ =+∞

∑k=0

wk qk,

whereφ andψ are analytic in q∈ [0,a), it holds

(a) Dm(φ) = um, (4.8)

(b) Dm(qkφ) = Dm−k(φ) =

um−k, when0≤ k≤ m,0, otherwise,

(4.9)

(c) Dm(φ ψ) =m

∑k=0

Dk(φ) Dm−k(ψ) =m

∑k=0

uk wm−k

=m

∑k=0

Dm−k(φ) Dk(ψ) =m

∑k=0

um−k wk, (4.10)


(d) Dm(φn+1) =m

∑k=0

Dk(φ) Dm−k(φn) =m

∑k=0

Dm−k(φ) Dk(φn), (4.11)

where m≥ 0, n≥ 1, l ≥ 0 and0≤ k≤ m are integers.

Proof. (1) According to Taylor theorem [8], the unique coefficientum of the Maclau-

rin seriesφ =+∞∑

m=0um qm with respect to the homotopy-parameterq is given by

um =1m!

∂ mφ∂qm

∣

∣

∣

∣

q=0,

which gives (4.8) by means of the definition (4.7) ofDm(φ).

(2) It holds

qkφ = qk+∞

∑j=0

u j q j =+∞

∑j=0

u j q j+k =+∞

∑m=k

um−k qm,

which gives by means of (4.8) that

Dm(qkφ) = um−k = Dm−k(φ), when 0≤ k≤ m,

andDm(q

kφ) = 0, whenk> m.

(3) According to Leibnitz’s rule for derivatives of product, it holds

∂ m(φ ψ)

∂qm =m

∑k=0

m!k!(m− k)!

∂ kφ∂qk

∂ m−kψ∂qm−k

=m

∑k=0

m!k!(m− k)!

∂ kψ∂qk

∂ m−kφ∂qm−k

,

which gives according to (4.7) and (4.8) that

Dm(φ ψ) =1m!

∂ m(φ ψ)

∂qm

∣

∣

∣

∣

q=0=

m

∑k=0

(

1k!

∂ kφ∂qk

∣

∣

∣

∣

q=0

) (

1(m− k)!

∂ m−kψ∂qm−k

∣

∣

∣

∣

q=0

)

=m

∑k=0


∑k=0

uk wm−k.

Similarly, it holds

Dm(φ ψ) =m

∑k=0

Dk(ψ)Dm−k(φ) =m

∑k=0

um−k wk.

(4) Writeψ = φn. According to (4.10), it holds


Dm(φn+1) = Dm(φ ψ) =m

∑k=0


∑k=0

Dk(φ) Dm−k(φn).

Similarly, it holds

Dm(φn+1) =m

∑k=0

Dk(φn) Dm−k(φ).

This ends the proof. ⊓⊔

Theorem 4.2.If φ =+∞∑

k=0uk qk andψ =

+∞∑

k=0wk qk are two homotopy-Maclaurin se-

ries, whereφ and ψ are analytic in q∈ [0,a), f and g are independent of thehomotopy-parameter q∈ [0,1], then it holds

Dm( f φ +g ψ) = f Dm(φ)+g Dm(ψ) = f um+g wm. (4.12)

Proof. BecauseDm defined by (4.7) is a linear operator, and besidesf andg areindependent ofq, it obviously holds

Dm( f φ +g ψ) = Dm( f φ)+Dm(g ψ) = f Dm(φ)+g Dm(ψ).

According to (4.8), we haveDm(φ) = um andDm(ψ) = wm. This ends the proof.⊓⊔

Theorem 4.3.LetL denote a linear operator independent of the homotopy-parameterq∈ [0,1]. For two homotopy-Maclaurin series

φ =+∞

∑k=0

uk qk, ψ =+∞

∑k=0

wk qk,

whereφ andψ are analytic in q∈ [0,a), it holds

Dm(L φ) = L [Dm(φ)] = L um, (4.13)

and

Dm(ψ L φ) =m

∑n=0

Dm−n(ψ) L [Dn(φ)] =m

∑n=0

wm−n L un. (4.14)

where m≥ 0 is an integer.

Proof. SinceL is independent ofq, using Theorem 4.2, we have

L φ = L

[

+∞

∑k=0

uk qk

]

=+∞

∑k=0

L (uk) qk.

Using the statement (4.8), one hasDm(L φ) =L (um). On the other side, accordingto statement (4.8), it holdsL [Dm(φ)] = L (um). Thus,


Dm(L φ) = L [Dm(φ)] = L um

holds. Then, according to Theorem 4.1, we have

Dm(ψ L φ) =m

∑n=0

(Dm−nψ) Dn [L (φ)]

=m

∑n=0

Dm−n(ψ) L [Dn(φ)] =m

∑n=0

wm−n L un.


Theorem 4.2 and Theorem 4.3 indicate the linear superposition and commutativ-ity property of the operatorDm defined by (4.7), respectively.

Theorem 4.4.For two homotopy-Maclaurin series

φ =+∞

∑i=0

ui qi , ψ =+∞

∑j=0

wj q j ,

whereφ andψ are analytic in q∈ [0,a), if φ = ψ in q∈ [0,a), then um = wm andDm(φ) = Dm(ψ) for any integer m≥ 0 and a real number a> 0.

Proof. Sinceφ = ψ , it holds

+∞

∑k=0

(uk−wk)qk = 0.

The above expression holds at all pointsq∈ [0,a), if and only if

um = wm, m≥ 0,

which gives, due to (4.8), that

Dm(φ) = Dm(ψ).


Theorem 4.5.Let f(φ),g(ψ) denote two smooth functions. For two homotopy-Maclaurin series

φ =+∞

∑i=0

ui qi , ψ =+∞

∑j=0

wj q j ,

if f (φ) = g(ψ) in a domain q∈ [0,a), then

Dm[ f (φ)] = Dm[g(ψ)]

for any integer m≥ 0 and a real number a> 0.


Proof. WriteΦ = f (φ), Ψ = g(ψ).

Then, using Theorem 4.4, we have

Dm(Φ) = Dm(Ψ),

which givesDm[ f (φ)] = Dm[g(ψ)].


Theorem 4.4 and Theorem 4.5 indicate the uniqueness of the homotopy-derivatives

of a homotopy-Maclaurin seriesφ =+∞∑

k=0uk qk and a functionf (φ). According to the

fundamental theorems in calculus [8], these theorems are obvious. Therefore, givena zeroth-order deformation equation, itsmth-order high-order deformation equa-tions is unique, no matter how one obtains it.

Theorem 4.6.For an arbitrary homotopy-Maclaurin seriesφ =+∞∑

k=0uk qk, it holds

(a) Dm(

φ2)=m

∑n=0

um−n un, (4.15)

(b) Dm(

φ3)=m

∑n=0

um−n

n

∑k=0

un−k uk, (4.16)

(c) Dm(

φ4)=m

∑n=0

um−n

n

∑k=0

un−k

k

∑j=0

uk− j u j , (4.17)

(d) Dm

(

φ5)

=m

∑n=0

um−n

n

∑k=0

un−k

k

∑j=0

uk− j

j

∑i=0

u j−i ui, (4.18)

(e) Dm(φσ ) =m

∑r1=0

um−r1

r1

∑r2=0

ur1−r2

r2

∑r3=0

ur2−r3 · · ·rσ−2

∑rσ−1=0

urσ−2−rσ−1 urσ−1, (4.19)

where m≥ 0 andσ ≥ 2 are positive integer.

Proof. (1) According to (4.10) and (4.8) , it holds

Dm(

φ2)=m

∑n=0

(Dm−nφ) (Dnφ) =m

∑n=0

um−n un.

(2) According to (4.15), we have

Dn(φ2) =n

∑k=0

un−k uk.


Then, according to (4.10), it holds

Dm(

φ3)=m

∑n=0

(Dm−nφ)(

Dnφ2)=m

∑n=0

um−n

n

∑k=0

un−k uk.


Dn(φ3) =n

∑k=0

un−k

k

∑j=0

uk− j u j .


Dm(

φ4)=m

∑n=0

(Dm−nφ)(

Dnφ3)=m

∑n=0

um−n

n

∑k=0

un−k

k

∑j=0

uk− j u j .


Dn(φ4) =n

∑k=0

un−k

k

∑j=0

uk− j

j

∑i=0

u j−i ui .


Dm

(

φ5)

=m

∑n=0

(Dm−nφ)(

Dnφ4)=m

∑n=0

um−n

n

∑k=0

un−k

k

∑j=0

uk− j

j

∑i=0

u j−i ui .

(5) This statement can be proved by the method of mathematical induction.(i) According to (4.15 ), it is obvious that (4.19 ) holds whenσ = 2.(ii) Assume that the statement (4.19 ) holds forσ = κ , i.e.

Dm(φκ ) =m

∑r1=0

um−r1

r1

∑r2=0

ur1−r2

r2

∑r3=0

ur2−r3 · · ·rκ−2

∑rκ−1=0

urκ−2−rκ−1 urκ−1,

wherem≥ 0 andκ ≥ 2 are integers. Replacingr j by r ′j+1 andm by r ′1, the aboveexpression reads

Dr ′1(φκ ) =

r ′1

∑r ′2=0

ur ′1−r ′2

r ′2

∑r ′3=0

ur ′2−r ′3

r ′3

∑r ′4=0

ur ′3−r ′4· · ·

r ′κ−1

∑r ′κ=0

ur ′κ−1−r ′κ ur ′κ .

Using the above expression and by means of (4.11) and (4.8), it holds


Dm(

φκ+1)=m

∑r ′1=0

Dm−r ′1(φ)Dr ′1

(φκ )

=m

∑r ′1=0

um−r ′1

r ′1

∑r ′2=0

ur ′1−r ′2

r ′2

∑r ′3=0

ur ′2−r ′3

r ′3

∑r ′4=0

ur ′3−r ′4· · ·

r ′κ−1

∑r ′κ=0

ur ′κ−1−r ′κ ur ′κ .

Therefore, (4.19 ) holds forσ = κ +1.(iii) According to (i) and (ii), the statement (4.19 ) holds for any positive integerσ ≥ 2. This ends the proof. ⊓⊔

Theorem 4.7.For a homotopy-Maclaurin series

φ =+∞

∑k=0

uk qk,

it holds the recursion formulas

D0(

eα φ) = eα u0,

Dm(

eα φ) = αm−1

∑k=0

(

1− km

)

Dm−k(φ) Dk(

eα φ)

= αm−1

∑k=0

(

1− km

)

um−k Dk(

eα φ) ,

where m≥ 1 is an integer, andα 6= 0 is independent of the homotopy-parameter q.

Proof. According to the definition (4.7) of the operatorDm, it holds obviously

D0(

eα φ)= eα u0.

Besides, sinceα is independent ofq, one has

∂eαφ

∂q= α eαφ ∂φ

∂q.

Thus, according to Leibnitz’s rule for derivatives of product, it holds

1m!

∂ meαφ

∂qm =1m!

∂ m−1

∂qm−1

(

αeα φ ∂φ∂q

)

=αm

m−1

∑k=0

1k!(m−1− k)!

∂ keαφ

∂qk


= αm−1

∑k=0

(m− k)m

[

1(m− k)!


] [

1k!

∂ keαφ

∂qk

]

.

Settingq= 0 in above expression and using the definition (4.7) and the statement(4.8), one has


Dm(

eαφ)= αm−1

∑k=0

(

1− km

)

Dm−k(φ)Dk(

eαφ)= αm−1

∑k=0

(

1− km

)

um−k Dk(

eαφ) ,

wherem≥ 1 is an integer. This ends the proof. ⊓⊔Theorem 4.8.For a homotopy-Maclaurin series

φ =+∞

∑k=0

uk qk,

it holds the recursion formulas

D0(sinφ) = sin(u0), D0(cosφ) = cos(u0),

Dm(sinφ) =m−1

∑k=0

(

1− km

)

Dm−k(φ) Dk(cosφ)

=m−1

∑k=0

(

1− km

)

um−k Dk(cosφ),

Dm(cosφ) = −m−1

∑k=0

(

1− km

)

Dm−k(φ) Dk(sinφ)

= −m−1

∑k=0

(

1− km

)

um−k Dk(sinφ),

where m≥ 1 is an integer.

Proof. According to the definition (4.7), it holds obviously

D0(sinφ) = sin(u0), D0(cosφ) = cos(u0).

Write i =√−1. Using Euler formula and Theorem 4.2, it holds for an integer m≥ 1

that

Dm(sinφ) = Dm

(

ei φ −e−i φ

2i

)

=12i

[

Dm(eiφ )−Dm(e

−iφ )]

(4.20)

and

Dm(cosφ) = Dm

(

ei φ +e−i φ

2

)

=12

[

Dm(eiφ )+Dm(e

−iφ )]

. (4.21)

Using Theorem 4.7 and then Theorem 4.2, we have

Dm(ei φ ) =

m−1

∑k=0

(

1− km

)

Dk(eiφ ) Dm−k(iφ)

= im−1

∑k=0

(

1− km

)

Dk(eiφ ) Dm−k(φ)


and similarly,

Dm(e−i φ ) = −i

m−1

∑k=0

(

1− km

)

Dk(e−iφ ) Dm−k(φ).

Substituting the above two expressions into (4.20) and (4.21), then using Theo-rem 4.2 and Euler formula, we have

Dm(sinφ) =12

m−1

∑k=0

(

1− km

)

Dm−k(φ)[

Dk(eiφ )+Dk(e

−iφ )]

=m−1

∑k=0

(

1− km

)

Dm−k(φ)Dk

(

eiφ +e−iφ

2

)

=m−1

∑k=0

(

1− km

)

Dm−k(φ) Dk(cosφ)

=m−1

∑k=0

(

1− km

)

um−k Dk(cosφ),

and similarly

Dm(cosφ) =i2

m−1

∑k=0

(

1− km

)

Dm−k(φ)[

Dk(eiφ )−Dk(e

−iφ )]

= −m−1

∑k=0

(

1− km

)

Dm−k(φ) Dk

(

eiφ −e−iφ

2i

)

= −m−1

∑k=0

(

1− km

)

Dm−k(φ) Dk(sinφ)

= −m−1

∑k=0

(

1− km

)

um−k Dk(sinφ).


The statements (4.11 ), (4.15 ) and (4.19 ) were proved by Molabahrami andKhani [38] in 2009. Most of the others were proved by Liao [30]in 2009. Us-ing these theorems, one can calculate the homotopy-derivatives of any a givensmooth function of a homotopy-Maclaurin series. For example, given a homotopy-

Maclaurin seriesφ =+∞∑

k=0uk qk, we have using Theorem 4.2 and the statement (4.10)

that

Dm

(

3φ2+4e−5φ sinφ)

= 3m

∑k=0

uk um−k+4m

∑k=0

Dk

(

e−5φ)

Dm−k (sinφ) ,


whereDk(

e−5φ) andDm−k (sinφ) are given by Theorem 4.7 and Theorem 4.8, re-spectively. Following Molabahrami & Khani [38] and Liao [30], Turkyilmazoglu[46] proved in 2010 the following theorem:

Theorem 4.9.Define an operator

Dmφ =1m!

∂ mφ∂qm .

For a smooth function f∈C∞(a,b) and a homotopy-Maclaurin series

φ =+∞

∑k=0

uk qk,

it holds

D0 [ f (φ)] = f (φ), (4.22)

Dm[ f (φ)] =m−1

∑k=0

(

1− km

)

Dm−k(φ)∂

∂φ

Dk [ f (φ)]

, (4.23)

andDm[ f (φ)] =

Dm[ f (φ)]∣

∣

q=0 . (4.24)

Proof. It is obvious thatD0 [ f (φ)] = f (φ) holds. In case ofm≥ 1, we have byLeibnitz’s rule for derivatives of product that

Dm[ f (φ)] =1m!

∂ m f (φ)∂qm =

1m!

∂ m−1

∂qm−1

[

∂φ∂q

∂ f (φ)∂φ

]

=1m!

m−1

∑k=0

(m−1)!k! (m−1− k)!

∂ m−1−k

∂qm−1−k

(

∂φ∂q

)

∂ k

∂qk

[

∂ f (φ)∂φ

]

=m−1

∑k=0

(m− k)m

[

1(m− k)!


]

1k!

∂ k

∂qk

[

∂ f (φ)∂φ

]

=m−1

∑k=0

(

1− km

)

Dm−k(φ) Dk

[

∂ f (φ)∂φ

]

. (4.25)

Since

Dk

[

∂ f (φ)∂φ

]

=∂

∂φ

Dk[ f (φ)]

,

we have

Dm[ f (φ)] =m−1

∑k=0

(

1− km

)

Dm−k(φ)∂

∂φ

Dk [ f (φ)]

for m≥ 1. Then, according to the definition ofDm, it obviously holds


Dm[ f (φ)] =

Dm[ f (φ)]∣

∣

q=0 .


The above theorem contains Theorem 4.7 and Theorem 4.8, and thus is moregeneral, although it is not straightforward. Using the above theorem, we prove herethe following theorem with the explicit expressions:

Theorem 4.10.For a smooth function f(u) and a homotopy-Maclaurin serie

φ =+∞

∑k=0

uk qk,

it holds

D0 [ f (φ)] = f (u0), (4.26)

Dm[ f (φ)] =m−1

∑k=0

(

1− km

)

um−k∂ Dk [ f (φ)]

∂u0, (4.27)

and

Dm[ f (φ)] =m−1

∑k=0

(

1− km

)

um−k Dk[

f ′(φ)]

, (4.28)

for m≥ 1.

Proof. According to the definition (4.7), it is obvious that the statement (4.32) istrue. Atq= 0, we have

Dm−k(φ) = Dm−k(φ) = um−k.

Besides, atq= 0, we haveφ = u0 and therefore∂/∂φ = ∂/∂u0. Then, accordingto Theorem 4.9, we have

Dm[ f (φ)] =m−1

∑k=0

(

1− km

)

um−k∂ Dk [ f (φ)]

∂u0.

Settingq= 0 in (4.25), we obtain

Dm[ f (φ)] =m−1

∑k=0

(

1− km

)

um−k Dk[

f ′(φ)]

.


Note that, Theorem 4.7 and Theorem 4.8 are special cases of the statement (4.28)of Theorem 4.10. By means of the recursion formula (4.27) of Theorem 4.10, it iseasy to get the high-order homotopy-derivatives of anarbitrary smooth function

of a given homotopy-Maclaurin seriesφ =+∞∑

k=0uk qk. For example, it gives when


f (φ) = φα that

D0 (φα ) = uα0 ,

D1 (φα ) = α uα−10 u1,

D2 (φα ) =12

α (α −1)uα−20 u2

1+α uα−10 u2,

D3 (φα ) =16

α (α −1) (α −2) uα−30 u3

1+α(α −1) uα−20 u1 u2+α uα−1

0 u3,

...

whereα 6= 0 is independent of the homotopy-parameterq. Similarly, it gives whenf (φ) = αφ that

D0(

αφ ) = αu0,

D1(

αφ ) = (lnα) αu0 u1,

D2(

αφ ) =12(lnα)2 αu0 u2

1+(lnα) αu0 u2,

D3(

αφ ) =16(lnα)3 αu0 u3

1+(lnα)2 αu0 u1 u2+(lnα)αu0 u3,

...

whereα > 0 is independent of the homotopy-parameterq. These results are exactlythe same as those given by Theorem 4.9. In general, by means ofthe recursionformula (4.27) of Theorem 4.10, we have foranygiven smooth functionf (φ) that

D0[ f (φ)] = f (u0),

D1[ f (φ)] = f ′(u0) u1,

D2[ f (φ)] =12

f ′′(u0) u21+ f ′(u0) u2,

D3[ f (φ)] =16

f ′′′(u0) u31+ f ′′(u0) u1 u2+ f ′(u0) u3,

D4[ f (φ)] =124

f ′′′′(u0) u41+

12

f ′′′(u0) u21 u2+ f ′′(u0)

(

u1 u3+u2

2

2

)

+ f ′(u0) u4,

...

In practice, by means of computer algebra system like Mathematica and Maple, itis easy to getDm[ f (φ)] for rather largem. For example, using the following Mathe-matica commands:

GetD[0] := f[u[0]];GetD[m_] := Sum[(1-k/m) * u[m-k] * D[GetD[k],u[0]],k,0,m-1];


we can obtainDm[ f (φ)] for any a given smooth functionf (φ) of a homotopy-

Maclaurin seriesφ =+∞∑

k=0uk qk.

Therefore, foranygiven smooth functionf (φ), whereφ is a homotopy-Maclaurinseries, we can always obtain itsmth-order derivativeDm[ f (φ)] by means of abovetheorems. Besides, the following theorems can be proved in asimilar way.

Theorem 4.11.For a smooth function f(u,w) and two homotopy-Maclaurin serie

φ =+∞

∑k=0

uk qk, ψ =+∞

∑k=0

wk qk,

it holds

D0 [ f (φ ,ψ)] = f (u0,w0), (4.29)

Dm[ f (φ ,ψ)] =m−1

∑k=0

(

1− km

)

um−k∂ Dk [ f (φ ,ψ)]

∂u0

+m−1

∑k=0

(

1− km

)

wm−k∂ Dk [ f (φ ,ψ)]

∂w0, (4.30)

and

Dm[ f (φ ,ψ)] =m−1

∑k=0

(

1− km

)

um−k Dk

[

∂ f (φ ,ψ)

∂φ

]

+m−1

∑k=0

(

1− km

)

wm−k Dk

[

∂ f (φ ,ψ)

∂ψ

]

(4.31)

for m≥ 1.

Theorem 4.12.For a smooth function f(u,u′,u′′) and a homotopy-Maclaurin serie

φ =+∞

∑k=0

uk(x) qk

with the definition

φ ′ =+∞

∑k=0

u′k(x) qk, φ ′′ =+∞

∑k=0

u′′k(x) qk,

where the prime denotes the differentiation with respect tox, it holds

D0[

f (φ ,φ ′,φ ′′)]

= f (u0,u′0,u

′′0), (4.32)

Dm[

f (φ ,φ ′,φ ′′)]

=m−1

∑k=0

(

1− km

)

um−k∂ Dk [ f (φ ,φ ′,φ ′′)]

∂u0


+m−1

∑k=0

(

1− km

)

u′m−k∂ Dk [ f (φ ,φ ′,φ ′′)]

∂u′0

+m−1

∑k=0

(

1− km

)

u′′m−k∂ Dk [ f (φ ,φ ′,φ ′′)]

∂u′′0, (4.33)

and

Dm[

f (φ ,φ ′,φ ′′)]

=m−1

∑k=0

(

1− km

)

um−k Dk

[

∂ f (φ ,φ ′ ,φ ′′)∂φ

]

+m−1

∑k=0

(

1− km

)

u′m−k Dk

[

∂ f (φ ,φ ′ ,φ ′′)∂φ ′

]

+m−1

∑k=0

(

1− km

)

u′′m−k Dk

[

∂ f (φ ,φ ′ ,φ ′′)∂φ ′′

]

(4.34)

for m≥ 1.

Note that the above theorems do not hold for functions which explicitly containthe homotopy-parameter, such asf (φ ,ψ ,q). For such kind of complicated func-tions, it is efficient to directly expand it into a homotopy-Maclaurin series and thenuse the following theorem.

Theorem 4.13.LetN (φ ,ψ ,q) denote a nonlinear operator, where q∈ [0,1] is thehomotopy-parameter,

φ ∼+∞

∑m=0

um qm, ψ ∼+∞

∑n=0

wn qn,

are two homotopy-Maclaurin series, respectively, which are analytic in q∈ [0,a] fora> 0. Let

N (φ ,ψ ,q)∼+∞

∑k=0

δk qk

denote the homotopy-Maclaurin series ofN (φ ,ψ ,q). Then, it holds

Dk [N (φ ,ψ ,q)] = δk =

1k!

dk

dqk N

(

+∞

∑m=0

um qm,+∞

∑n=0

wn qn,q

)∣

∣

∣

∣

∣

q=0

. (4.35)

Proof. The statement (4.35) holds obviously, according to the definition (4.7), ⊓⊔

By means of Theorem 4.13, we can get homotopy-derivative ofanya given non-

linear operator. For example, letφ ∼+∞∑

n=0un qn denote a homotopy-Maclaurin series.

By means of computer algebra system such as Mathematica, it is easy to obtain thehomotopy-Maclaurin series of sin(qφ)/q, i.e.


sin(q φ)q

∼ u0+u1 q+

(

u2−16

u30

)

q2+

(

u3−12

u20 u1

)

q3

+

(

u4−12

u20 u2−

12

u0 u21+

1120

u50

)

q4+ · · ·

Thus, we have

D0

[

sin(qφ)q

]

= u0, D1

[

sin(qφ)q

]

= u1, D2

[

sin(qφ)q

]

= u2−16

u30, (4.36)

and so on. For details, please refer to Liao [31].Using above-mentioned theorems, we can derive explicit formulas of homotopy-

derivative of some complicated functions. For example, thealternative way to getDm[sin(qφ)/q] is given by the following theorem.

Theorem 4.14.For a homotopy-Maclaurin series

φ =+∞

∑k=0

uk qk,

whereφ is analytic in q∈ [0,a) for a> 0, it holds

Dm

[

sin(q φ)q

]

= Dm+1 [sin(q φ)] , m≥ 0,

where

D0 [sin(q φ)] = 0, D0 [cos(q φ)] = 1,

Dn [sin(q φ)] =n−1

∑k=0

(

1− kn

)

Dn−1−k(φ) Dk [cos(q φ)]

=n−1

∑k=0

(

1− kn

)

un−1−k Dk [cos(q φ)] ,

Dn [cos(q φ)] = −n−1

∑k=0

(

1− kn

)

Dn−1−k(φ) Dk [sin(q φ)]

= −n−1

∑k=0

(

1− kn

)

un−1−k Dk [sin(q φ)]

for n≥ 1.

Proof. According to the definition (4.7), it holds

D0 [sin(q φ)] = sin(0) = 0, D0 [cos(q φ)] = cos(0) = 1.

According to the definition (4.7), we have


sin(q φ) =+∞

∑k=1

Dk [sin(q φ)] qk,

which gives

sin(q φ)q

=+∞

∑k=1

Dk [sin(q φ)] qk−1 =+∞

∑m=0

Dm+1 [sin(q φ)] qm

so that

Dm

[

sin(q φ)q

]

= Dm+1 [sin(q φ)] .

According to Theorem 4.8, it holds


∑k=0

(

1− kn

)

Dn−k(q φ) Dk [cos(q φ)]

which gives according to Theorem 4.1 that


∑k=0

(

1− kn

)

Dn−1−k(φ) Dk [cos(q φ)]

=n−1

∑k=0

(

1− kn

)

un−1−k Dk [cos(q φ)]

for n≥ 1. Similarly, it holds

Dn [cos(q φ)] = −n−1

∑k=0

(

1− kn

)

un−1−k Dk [sin(q φ)] , n≥ 1.


Using Theorem 4.14, we gain exactly the same results as (4.36). Using Mathe-matica, it is easy to gainDn [sin(q φ)/q], where 0≤ n≤ 4, by the following com-mands

Dsin[0] = 0;Dcos[0] = 1;Dsin[n_] := Sum[(1 - k/n) * u[n - 1 - k] * Dcos[k],k,0,n-1];Dcos[n_] := -Sum[(1 - k/n) * u[n - 1 - k] * Dsin[k],k,0,n-1];For[k = 0, k <= 4, k = k + 1, Dsin[k+1] // Print]

4.3 Deformation equations 153

4.3 Deformation equations

4.3.1 A brief history

Nonlinear differential equations are much more difficult tosolve than linear ones.Perturbation techniques are based on the expansion of small/large physical parame-ters, called perturbation quantity. Unfortunately, many nonlinear problems have nosuch kind of small/large physical parameters. More importantly, perturbation tech-niques can not guarantee the convergence of approximation series. To overcome thedependence of perturbation techniques on the small/large physical parameters, theso-called “artificial small parameter” [35] (denoted here by ε) was introduced insome different ways: whenε = 0, one has a much simpler equation whose solutionis easy to obtain; whenε = 1, one has exactly the same original nonlinear equationwhose solution is unknown. Then, expanding the unknown solution in a perturba-tion series with respect to the artificial parameterε and then substituting it into theoriginal nonlinear equation, one can get a power series ofε. Finally, settingε = 1,one has an approximation of the solution of the original equation. For example, if anonlinear differential equation

L0[u(x, t)]+N0[u(x, t)] = 0

does not contain any small/large physical parameter, whereL0 andN0 are linearand nonlinear differential operators, corresponding to the linear and nonlinear partsof a given nonlinear equation, andu(x, t) denotes the unknown function with theindependent spatial and temporal variables, respectively, then, one can introduce“an artificial small parameter”ε to construct such a family of equation

L0[u(x, t)]+ ε N0[u(x, t)] = 0,

and in addition, gives its “perturbation approximations” by first regardingε as aperturbation quantity and then settingε = 1. This method, called “the artificial smallparameter method ” [35], was first proposed in 1892 by the Russian mathematicianAleksandr Mikhailovich Lyapunov (1857-1918), but unfortunately did not becomea mainstream of perturbation techniques, possibly becausethe meaning of the so-called “artificial small parameter” has not been understoodin details and besides theuse of the so-called “artificial small parameter” needs a rigorous theory.

On the other side, Jules Henri Poincare (1854 - 1912), a French mathematician,proposed the concept of homotopy in topology [6,43]. Topology becomes an impor-tant field of mathematics, and the concept of homotopy has been widely applied bypure mathematicians to prove the existence, uniqueness of solutions of equations,and so on. Besides, based on the concept of homotopy, some numerical techniquessuch as the homotopy continuation method [5, 17–19] have been developed, whichare rather powerful to find out zeros of a set of nonlinear algebraic equations. It is apity that the concept of the homotopy was not introduced to gain analytic approxi-mations of nonlinear differential equations.


The concept of homotopy was first introduced to gain analyticapproximations ofnonlinear differential equations by Shijun Liao [20] in 1992. For a given nonlineardifferential equation

N [u(x, t)] = 0,

whereN is a nonlinear differential operator,u(x, t) is a unknown function,x de-notes a vector of spatial independent-variables,t denotes the temporal independent-variable, respectively, Liao [20] applied the concept of homotopy in topology toconstruct such a kind ofone-parameterfamily of equations in the embedding pa-rameterq∈ [0,1], namely the zeroth-order deformation equation

(1−q)L [φ(x, t;q)−u0(x, t)]+q N [φ(x, t;q)] = 0, q∈ [0,1], (4.37)

whereL is an auxiliary linear operator andu0(x, t) is an initial approximation, that,as the embedding parameterq∈ [0,1] increases from 0 to 1,φ(x, t;q) varies from theinitial approximationu0(x, t) to the unknown exact solutionu(x, t) of the originalequation. This work reveals the relationship between Lyapunov’s “artificial smallparameter” [35] and Poincare’s concept of homotopy [6,43]. Since the embedding-parameterq ∈ [0,1] has no physical meanings, one can always constructs such akind of zeroth-order deformation equation, no matter whether there exist small/largephysical parameters or not. So, this approach is more general than perturbation tech-niques in theory. On the other hand, the concept of the homotopy, i.e. the continuousvariation or deformation, is more general in theory, which provides much larger free-dom to construct the so-called zeroth-order deformation equations than Lyapunov’sartificial small parameter method [35]: for example, one hasextremely large free-dom to choose the auxiliary linear operatorL , which is unnecessary to beL0 thatcorresponds to the whole linear part of a given nonlinear differential equation.

Unfortunately, it was found that the homotopy-series givenby the zeroth-orderdeformation equation (4.37) are sometimes divergent, especially for equations withstrong nonlinearity. This is mainly because, like perturbation techniques, the earlyHAM based on (4.37) can not guarantee the convergence of the homotopy-series.Realizing that one has extremely large freedom to constructa homotopy of equa-tions, Liao [21] introduced in 1997 a non-zero auxiliary parameterc0 to construct amore generalized zeroth-order deformation equation

(1−q)L [φ(x, t;q)−u0(x, t)] = c0 q N [φ(x, t;q)], q∈ [0,1], (4.38)

such that, the corresponding homotopy-series is dependentupon not only the phys-ical independent-variablesx and t but also the non-zero auxiliary parameterc0,althoughc0 has no physical meanings at all. In essence, the use of the auxiliaryparameterc0 introduces us one more “artificial” degree of freedom, whichhas nophysical meaning but can greatly improve the early HAM: the non-zero auxiliaryparameterc0 indeed provides a convenient way to control the convergenceof thehomotopy-series, say, one can guarantee the convergence ofhomotopy series bymeans of choosing a proper value ofc0. For example, by choosing a proper auxiliaryparameterc0, Abbasbandy [1] applied the HAM to obtain accurate approximations


of a nonlinear heat transfer equation forall possible values of a physical param-eter, even if the corresponding perturbation approximations are divergent for thelarge physical parameter. Especially, Liang & Jeffrey [15]illustrated by a simpleexample that, the HAM can give convergent series solution bychoosing a propervalue ofc0, even when the series given by the other analytic method is divergentin the whole domain except at the initial/boundary condition. For this reason, wecall c0 the convergence-control parametertoday, which provides us a new concept:the convergence-control of series. In essence, it is the so-called convergence-controlparameterc0 that differs the HAM from other analytic techniques, such asperturba-tion techniques [7,11,12,39–41], Lyapunov’s artificial small parameter method [35],Adomian decomposition method [2–4], theδ -expansion method [13] and so on. So,the use of the convergence-control parameterc0 is the most important step in thedevelopment of the HAM.

The use of the convergence-controlparameterc0 is indeed a great progress, whichprovides us one more “artificial” degree of freedom in essence. It seems that more“artificial” degrees of freedom imply larger possibility toget better approximationsby means of the homotopy analysis method. Thus, Liao [22] in 1999 further intro-duced more “artificial” degrees of freedom by constructing the zeroth-order defor-mation equation in a more general form:

[1−α(q)]L [φ(x, t;q)−u0(x, t)] = c0 β (q) N [φ(x, t;q)], q∈ [0,1], (4.39)

whereα(q) andβ (q) are the analytic functions satisfying

α(0) = β (0) = 0, α(1) = β (1) = 1, (4.40)

whose Maclaurin series

α(q)∼+∞

∑m=1

αm qm, β (q)∼+∞

∑m=1

βm qm, (4.41)

exist and are convergent for|q| ≤ 1, i.e.

α(1) =+∞

∑k=1

αk = 1, β (1) =+∞

∑k=1

βk = 1, (4.42)

whereαk andβk are constants. This kind of generalization provides us larger possi-bility to guarantee the convergence of homotopy-series of original nonlinear equa-tions.

In addition, realizing the extremely large freedom of constructing a homotopyof equations, Liao [26] introduced in 2003 a non-zero auxiliary functionH(x, t) tofurther generalize the zeroth-order deformation equation, i.e.

[1−α(q)]L [φ(x, t;q)−u0(x, t)] = c0 H(x, t) β (q) N [φ(x, t;q)], (4.43)


where the analytic functionsα(q) andβ (q) satisfy (4.40) and their Maclaurin se-ries satisfy (4.42), respectively. Besides, Liao [26] gavein 2003 a more generalizedzeroth-order deformation equation

[1−α(q)]L [φ(x, t;q)−u0(x, t)]

= c0 H(x, t) β (q) N [φ(x, t;q)]+A [φ(x, t;q),x, t;q], (4.44)

whereA is a nonlinear operator, which equals to zero whenq= 0 andq= 1, i.e.

A [φ(x, t;q),x, t;q] = 0, whenq= 0 andq= 1. (4.45)

Using the generalized zeroth-order deformation equations(4.43) and (4.44), Liao[26] proved that Lyapunov’s artificial small parameter method [35], Adomian de-composition method [2–4] and theδ -expansion method [13] are only special casesof the HAM. Besides, Sajid and Hayat [42] proved that the so-called “homotopyperturbation method” [9, 10] proposed in 1998 is exactly thesame as the earlyHAM [20] developed by Liao in 1992, and therefore the “homotopy perturbationmethod” contains “nothing new except its name” [42]. This reveals the generalityand validity of the HAM from another view-point. Obviously,there exist an infinitenumber of analytic functionsα(q) andβ (q) satisfying (4.40) and (4.42). Besides,there exist an infinite number of nonlinear operatorsA satisfying (4.45). So, thezeroth-order deformation equation (4.44) is indeed rathergeneral. Thus, the ho-motopy approximations given by the HAM contain so many “artificial” degrees offreedom that we have a lots of ways to guarantee the convergence of homotopy-series.

In 2008, Marinca and Herisanu [37] combinedc0 andβ (q) in the zeroth-orderdeformation equation (4.39) as a functionβ(q) with β (0) = 0 but β(1) = c0 6= 1,and constructed such a homotopy of equations

(1−q)L [φ(x, t;q)−u0(x, t)] = β (q) N [φ(x, t;q)], q∈ [0,1], (4.46)

where the Taylor series

β (q) =+∞

∑k=1

βk qk

converges atq= 1. Note that the above equation is a special case of (4.39), ifonechoosesα(q) = q and

β (q) = c0 β (q), i.e. βk = c0 βk.

So, the so-called “homotopy asymptotic method” proposed byMarinca and Herisanu[37] is in essence still in the frame of the HAM. Even so, Marinca and Herisanu [37]suggested an interesting approach, which has the advantagethatβ (1) = 1 is unnec-essary so that one has more freedom to choose the parametersβk: each of them canbe unknown now and thus can be regarded as a convergence-control parameter.


In summary, the development of the HAM is strongly related tothe generalizationof the so-called zeroth-order deformation equations. In the early zeroth-order defor-mation equation (4.37), the concept of homotopy in topologywas first introducedinto the analytic approximation methods. In the zeroth-order deformation equation(4.38), the so-called convergence-controlparameter was first introduced, which pro-vides us not only a convenient way to control the convergenceof homotopy series,but also a new concept of “convergence-control of series”. By means of extremelylarge freedom on constructing a homotopy of equations, we can construct rathergeneralized zeroth-order deformation equations. The moregeneral a zeroth-orderdeformation equation, the more larger freedom it provides us to guarantee the con-vergence of homotopy series. However, this does not mean that more complicatedzeroth-order deformation equations are always better thansimple ones. In general,the simple types of the zeroth-order deformation equations, such as (4.38), (4.39)and (4.43), can often give rather accurate approximations,as shown later in thisbook. For a few complicated nonlinear problems, it might be necessary to applymore generalized type of the zeroth-order deformation equations.

In § 4.3.2, different types of the zeroth-order deformation equations and the cor-responding high-order deformation equations are given. In§ 4.3.3, some examplesare given to illustrate how to gain the high-order deformation equation by means ofthese theorems.

4.3.2 High-order deformation equations

In this section, the high-order deformation equations for various types of zeroth-order deformation equations are given.

Lemma 3.1Let

φ =+∞

∑m=0

um qm

denote a homotopy-Maclaurin series, where q∈ [0,1] is the homotopy-parameter,andL denotes an auxiliary linear operator which has the propertyL [0] = 0 andis independent of q. It holds

Dm(1−q)L (φ −u0)= L (um− χm um−1)

where

χm =

0, m≤ 1,1, m> 1.

(4.47)

Proof. SinceL is a linear operator independent ofq, it holds

(1−q)L (φ −u0) = L (φ −qφ +u0 q−u0) .


According to Theorem 4.3, Theorem 4.2 and the statement (4.8) of Theorem 4.1 ,we have

Dm[(1−q)L (φ −u0)]

= Dm[L (φ −qφ +u0 q−u0)]

= L [Dm(φ −qφ +u0 q−u0)]

= L [Dm(φ)−Dm(qφ)+u0Dm(q)]

= L [um−um−1+u0Dm(q)] ,

which equals toL um whenm= 1, andL (um−um−1) whenm> 1, respectively.Thus, according to the definition (4.47) ofχm, it holds

Dm(1−q)L (φ −u0) = L (um− χm um−1) .


Theorem 4.15.Let L denote an auxiliary linear operator which has the propertyL [0] = 0and is independent of the homotopy-parameter q∈ [0,1], N denote a non-linear operator, u0(x, t) an initial approximation of the original equationN u= 0,c0 the convergence-control parameter independent of q, and H(x, t) an auxiliaryfunction independent of q, respectively, wherex denotes a vector of the spatial in-dependent variable, and t is temporal independent variable. If

φ =+∞

∑m=0

um(x, t) qm

is the homotopy-Maclaurin series of the zeroth-order deformation equation

(1−q)L (φ −u0) = q c0 H(x, t) N (φ) , (4.48)

then the corresponding mth-order deformation equation reads

L [um(x, t)− χm um−1(x, t)] = c0 H(x, t) Dm−1 [N (φ)] , (4.49)

whereDm−1 is defined by (4.7) andχm is defined by (4.47).

Proof. According to Theorem 4.5, we have

Dm[(1−q)L (φ −u0)] = Dm[q c0 H(x, t) N (φ)] .

According to Lemma 3.1, it holds

Dm(1−q)L (φ −u0)= L (um− χm um−1) .

According to Theorem 4.2 and (4.8), it holds

Dm[q c0 H(x, t) N (φ)] = c0 H(x, t) Dm−1 [N (φ)] .


Thus, we have themth-order deformation equation

L (um− χm um−1) = c0 H(x, t) Dm−1 [N (φ)] .


Due to the extremely large freedom on constructing the zeroth-order deforma-tion equation, we can introduce an infinite number of constant convergence-controlparametersc0,c1,c2, · · · to construct a more general zeroth-order deformation equa-tion, as proved in the following theorem.

Theorem 4.16.Let L denote an auxiliary linear operator which has the propertyL [0] = 0 and is independent of the homotopy-parameter q∈ [0,1], N denote anonlinear operator, u0(x, t) denote an initial approximation of the original equationN u= 0, and H(x, t) be an auxiliary function independent of q, respectively, wherex denotes a vector of the spatial independent variable, and t is temporal independentvariable. If the series

+∞

∑k=0

ck qk = c0+ c1 q+ c2 q2 · · ·

converges at q= 1, where c0,c1, · · · are constant convergence-control parameters

with c0 6= 0 and+∞∑

k=0ck 6= 0, besides if

φ =+∞

∑m=0

um(x, t) qm


(1−q)L (φ −u0) = H(x, t) q

(

+∞

∑k=0

ck qk

)

N (φ) , (4.50)


L [um(x, t)− χm um−1(x, t)] = H(x, t)m

∑k=1

ck−1 Dm−k [N (φ)] , (4.51)

whereDm−k is defined by (4.7) andχm is defined by (4.47)

Proof. Writing

ψ =+∞

∑k=0

ck qk+1,

we have according to (4.8) that

D0(ψ) = 0, Dk(ψ) = ck−1

for k≥ 1. According to Theorem 4.5, we have


Dm(1−q)L (φ −u0)= Dm[H(x, t) ψ N (φ)] .


Dm(1−q)L (φ −u0)= L (um− χm um−1) .

Then, using Theorem 4.2 and (4.10) of Theorem 4.1, we have

Dm[H(x, t) ψ N (φ)] = H(x, t)m

∑k=0

Dk(ψ) Dm−k [N (φ)]

= H(x, t)m

∑k=1

Dk(ψ) Dm−k [N (φ)] = H(x, t)m

∑k=1

ck−1 Dm−k [N (φ)] .

Thus, the corresponding high-order deformation equation reads

L [um(x, t)− χm um−1(x, t)] = H(x, t)m

∑k=1

ck−1 Dm−k [N (φ)] .

This ends the proof. ⊓⊔Note that the zeroth-order deformation equation (4.48) is aspecial case of the

zeroth-order deformation equation (4.50) in case ofck = 0 for k ≥ 1. Note that theconvergence-control parametersc0,c1,c2 and so on areunnecessaryto be a con-stant. So, combining the auxiliary functionH(x, t) with the convergence-controlparameterck, we have a more general zeroth-order deformation equation.

Theorem 4.17.Let L denote an auxiliary linear operator which has the propertyL [0] = 0 and is independent of the homotopy-parameter q∈ [0,1], N denote anonlinear operator, u0(x, t) denote an initial approximation of the original equationN u= 0, respectively, wherex denotes a vector of the spatial independent variable,and t is temporal independent variable. If the series

+∞

∑k=0

βk(x, t) qk · · ·

converges at q= 1 to a nonzero function, whereβ0(x, t),β1(x, t), · · · are calledconvergence-control functions, and besides if

φ =+∞

∑m=0

um(x, t) qm


(1−q)L (φ −u0) = q

[

+∞

∑k=0

βk(x, t) qk

]

N (φ) , (4.52)



L [um(x, t)− χm um−1(x, t)] =m

∑k=1

βk−1(x, t) Dm−k [N (φ)] , (4.53)

whereDm−k is defined by (4.7) andχm is defined by (4.47).

Proof. Writing

ψ =+∞

∑k=0

βk(x, t) qk+1,


D0(ψ) = 0, Dk(ψ) = βk−1(x, t)


Dm(1−q)L (φ −u0)= Dm[ψ N (φ)] .


Dm(1−q)L (φ −u0)= L (um− χm um−1) .


Dm[ψ N (φ)] =m

∑k=0


=m

∑k=1

Dk(ψ) Dm−k [N (φ)] =m

∑k=1

βk−1(x, t) Dm−k [N (φ)] .


L [um(x, t)− χm um−1(x, t)] =m

∑k=1

βk−1(x, t) Dm−k [N (φ)] .


Note that the zeroth-order deformation equation (4.50) is aspecial case of thezeroth-order deformation equation (4.52) in case ofβk(x, t) = ck H(x, t) for k ≥ 0.So, the concept of the so-called convergence-control function is more general: itcontains not only the constant convergence-controlparameters but also the so-calledauxiliary functionH(x, t). This reveals the essence of the auxiliary functionH(x, t).


Definition 4.2. An analytic functionf of the homotopy-parameterq ∈ [0,1] iscalled adeformation-function, if f = 0 whenq= 0, f = 1 whenq= 1, and itsMaclaurin series

f ∼+∞

∑k=1

αk qk,

absolutely converges atq= 1, i.e.

+∞

∑k=1

αk = 1,

whereαk can be a constant or a function of spatial and temporal independentvariables.

Lemma 3.2Let

φ =+∞

∑m=0

um qm

denote a homotopy-Maclaurin series, where q∈ [0,1] is the homotopy-parameter,and L be an auxiliary linear operator which has the propertyL [0] = 0 and isindependent of q. Ifα(q) is a deformation-function, i.e.α(0) = 0,α(1) = 1 and itsMaclaurin series

α(q) =+∞

∑k=1

αk qk,

exists and absolutely converges at q= 1, whereαk is constant, then it holds

Dm[1−α(q)]L (φ −u0)= L

(

um−m−1

∑n=1

αn um−n

)

Proof. Write

Φ = φ −u0 =+∞

∑k=1

uk qk, Ψ = α(q) =+∞

∑k=1

αk qk.

According to (4.8) of Theorem 4.1, we have

D0Φ = D0Ψ = 0, DnΦ = un, DnΨ = αn, n≥ 1.

Then, using Theorem 4.2, Theorem 4.1 and Theorem 4.3, we have

Dm[1−α(q)]L (φ −u0)= Dm[(1−Ψ)L Φ]

= Dm(L Φ −ΨL Φ) = Dm(L Φ)−m

∑n=0

Dn (Ψ) Dm−n (L Φ)


= L (DmΦ)−m

∑n=0

Dn (Ψ) L (Dm−nΦ)

= L (DmΦ)−m−1

∑n=1

Dn (Ψ ) L (Dm−nΦ)

= L um−m−1

∑n=1

αn L um−n

which gives, sinceαk is constant, that

Dm[1−α(q)]L (φ −u0)= L

(

um−m−1

∑n=1

αn um−n

)

.


Theorem 4.18.Let L denote an auxiliary linear operator which has the propertyL [0] = 0 and is independent of the homotopy-parameter q∈ [0,1], N denote anonlinear operator, u0(x, t) denote an initial approximation of the original equationN u= 0, respectively, wherex denotes a vector of the spatial independent variable,and t is temporal independent variable. Letα(q) denote a deformation-function, i.e.α(0) = 0,α(1) = 1 and its Maclaurin series

α(q)∼+∞

∑k=1

αk qk,

exists and absolutely converges at q= 1, whereαk is constant. If the series

+∞

∑k=0

βk(x, t) qk · · ·


φ =+∞

∑m=0

um(x, t) qm


[1−α(q)]L (φ −u0) = q

[

+∞

∑k=0

βk(x, t) qk

]

N (φ) , (4.54)


L

[

um(x, t)−m−1

∑n=1

αn um−n(x, t)

]

=m

∑k=1

βk−1(x, t) Dm−k [N (φ)] , (4.55)


whereDm−k is defined by (4.7).

Proof. Writing

ψ =+∞

∑k=0

βk(x, t) qk+1,


D0(ψ) = 0, Dk(ψ) = βk−1(x, t)


Dm[1−α(q)]L (φ −u0)= Dm[ψ N (φ)] .


Dm[1−α(q)]L (φ −u0)= L

(

um−m−1

∑n=1

αn um−n

)

.


Dm[ψ N (φ)] =m

∑k=0


=m

∑k=1

Dk(ψ) Dm−k [N (φ)] =m

∑k=1

βk−1(x, t) Dm−k [N (φ)] .


L

[

um(x, t)−m−1

∑n=1

αn um−n(x, t)

]

=m

∑k=1

βk−1(x, t) Dm−k [N (φ)] .


Note that the zeroth-order deformation equation (4.52) andthe correspondinghigh-order deformation equation (4.53) are special cases of (4.54) and (4.55) whenα1 = 1 andαk = 0 for k≥ 2, respectively.

Definition 4.3. An operator A (φ ,x, t,q) is called deformation-operator, ifA (φ ,x, t,q) = 0 atq= 0 andq= 1, whereq∈ [0,1] is the homotopy-parameter,

φ =+∞∑

k=0uk qk is a homotopy-Maclaurin series,x denotes a vector of the spatial

independent variable, andt is temporal independent variable, respectively.

Theorem 4.19.Let L denote an auxiliary linear operator which has the propertyL [0] = 0 and is independent of the homotopy-parameter q∈ [0,1], N denote a


nonlinear operator, u0(x, t) denote an initial approximation of the original equationN u= 0, respectively, wherex denotes a vector of the spatial independent variable,and t is temporal independent variable. Letα(q) denote a deformation-function, i.e.α(0) = 0,α(1) = 1 and its Maclaurin series

α(q)∼+∞

∑k=1

αk qk,

exists and converges at q= 1, whereαk is constant. LetA be a deformation-operator, i.e.

A (φ ,x, t,q) = 0, when q= 0 and q= 1.

If the series+∞

∑k=0

βk(x, t) qk · · ·


φ =+∞

∑m=0

um(x, t) qm


[1−α(q)]L (φ −u0) = q

[

+∞

∑k=0

βk(x, t) qk

]

N (φ)+A (φ ,x, t,q) , (4.56)


L

[

um(x, t)−m−1

∑n=1

αn um−n(x, t)

]

=m

∑k=1

βk−1(x, t) Dm−k [N (φ)]+Dm[A (φ ,x, t,q)] , (4.57)


Proof. Writing

ψ =+∞

∑k=0

βk(x, t) qk+1,


D0(ψ) = 0, Dk(ψ) = βk−1(x, t)


Dm[1−α(q)]L (φ −u0)= Dm[ψ N (φ)+A (φ ,x, t,q)] .



Dm[1−α(q)]L (φ −u0)= L

(

um−m−1

∑n=1

αn um−n

)

.


Dm[ψ N (φ)] =m

∑k=0

Dk(ψ) Dm−k [N (φ)]+Dm[A (φ ,x, t,q)]

=m

∑k=1

Dk(ψ) Dm−k [N (φ)]+Dm[A (φ ,x, t,q)]

=m

∑k=1

βk−1(x, t) Dm−k [N (φ)]+Dm[A (φ ,x, t,q)] .


L

[

um(x, t)−m−1

∑n=1

αn um−n(x, t)

]

=m

∑k=1

βk−1(x, t) Dm−k [N (φ)]+Dm[A (φ ,x, t,q)] .


Note that the zeroth-order deformation equation (4.54) andits high-order defor-mation equation (4.55) are special cases of (4.56) and (4.57) in case of

A (φ ,x, t,q) = 0,

respectively. Although (4.56) and (4.57) are rather general, a even more generalizedzeroth-order deformation equation can be given by the following theorem.

Lemma 3.3 LetL be an auxiliary linear operator which has the propertyL [0] = 0and is independent of the homotopy-parameter q∈ [0,1], and

φ =+∞

∑m=0

um qm

denote a homotopy-Maclaurin series. Ifα(x, t,q) is a deformation-function, i.e.α(x, t,q) = 0 when q= 0 andα(x, t,q) = 1 when q= 1, and its Maclaurin series

α(x, t,q)∼+∞

∑k=1

αk(x, t) qk,


exists and converges at q= 1, whereαk is a function of the vectorx of the spatialindependent variables and the temporal independent variable t, then it holds

Dm[1−α(x, t,q)]L (φ −u0) = L (um)−m−1

∑n=1

αn(x, t) L (um−n)

Lemma 3.3 can be proved similarly as Lemma 3.2.

Theorem 4.20.Let L denote an auxiliary linear operator which has the propertyL [0] = 0 and is independent of the homotopy-parameter q∈ [0,1], N denote anonlinear operator, u0(x, t) denote an initial approximation of the original equationN u= 0, respectively, wherex denotes a vector of the spatial independent variable,and t is temporal independent variable. Letα(x, t,q) denote a deformation-function,i.e.α(x, t,q) = 0 when q= 0 andα(x, t,q) = 1 when q= 1, and its Maclaurin series

α(x, t,q)∼+∞

∑k=1

αk(x, t) qk,

exists and converges at q= 1, whereαk(x, t) is dependent onx and t. LetB be anoperator dependent ofφ ,x, t and the homotopy-parameter q∈ [0,1], which satisfies

B (φ ,x, t,q) = 0, when q= 0,

B (φ ,x, t,q) = γ(x, t) N (φ), when q= 1,

whereγ(x, t) 6= 0 is a non-zero function. If

φ =+∞

∑m=0

um(x, t) qm


[1−α(x, t,q)]L (φ −u0) = B (φ ,x, t,q) , (4.58)


L [um(x, t)]−m−1

∑n=1

αn(x, t) L [um−n(x, t)] = Dm[B (φ ,x, t,q)] , (4.59)


Proof. According to Theorem 4.5, it holds

Dm[1−α(x, t,q)]L (φ −u0)= Dm[B (φ ,x, t,q)] .



Dm[1−α(x, t,q)]L (φ −u0)= L (um)−m−1

∑n=1

αn(x, t) L (um−n) .

Thus, the correspondingmth-order deformation equation reads

L [um(x, t)]−m−1

∑n=1

αn(x, t) L [um−n(x, t)] = Dm[B (φ ,x, t,q)] .


Note that the zeroth-order deformation equations (4.48), (4.50), (4.52), (4.54),(4.56) and (4.58) are more and more general, so are the corresponding high-orderdeformation equations (4.49), (4.51), (4.53), (4.55), (4.57) and (4.59). This is mainlybecause that we have extremely large freedom to construct the so-called zeroth-order deformation equation, whose general definition is given in Chapter 2. Thehigh-order deformation equations have some common properties:

1. The high-order deformation equations are always linear with respect to the un-knownum;

2. The terms on the left-hand side of the high-order deformation equation are rathersimilar, i.e.

L (um− χm um−1), whenα(q) = q,

or

L (um−m−1

∑n=1

αm−n un), whenα(q) =+∞∑

n=1αk qk,

whereαk is constant, or

L (um)−m−1

∑n=1

αm−n(x, t) L (un) , whenα(x, t,q) =+∞∑

n=1αk(x, t) qk,

whereαk(x, t) is dependent on the vectorx of spatial independent variables andthe temporal independent variablet, respectively. All of them are completelydetermined by the auxiliary linear operatorL and the deformation-functionα.Note that we have great freedom to choose the auxiliary linear operatorL andthe deformation-functionα(q) or α(x, t,q);

3. For themth-order deformation equation,u0,u1,u2, · · · ,um−1 are known and thusthe term on the right-hand side of the high-order deformation equation is oftenregarded to be known in essence. So, it is often convenient tosuccessively solvethe linear high-order deformation equation, especially bymeans of computer al-gebra systems like Mathematica, Maple and so on.

4. Give the type of a zeroth-order deformation, the corresponding high-order defor-mation equation is given directly by means of the theorems proved above. Mostly,it is necessary to calculate the homotopy-derivativeDm[N (φ)]. As pointed outin § 4.2, foranygiven smooth functionf (φ), whereφ is a homotopy-Maclaurinseries, we can always obtain itsmth-order homotopy-derivativeDm[ f (φ)] by


means of the theorems proved in§ 4.2, especially by means of the recursionformula (4.27). Similarly, foranya given original nonlinear equationN u= 0,we can always obtain the termDm[N (φ)], no matter how complicated the non-linear operatorN is. This indicates the importance of the so-called homotopy-derivative operatorDm defined by (4.7), and reveals the reason why the theoremsaboutDm are proved in§ 4.2.

Note that, as shown in Chapter 2, using the extremely large freedom on con-structing a homotopy of equations, we can express a unknown physical parameter asa homotopy-Maclaurin series in the zeroth-order deformation equation. Therefore,all above theorems hold when we replaceDk [N (φ)] by Dk [N (φ ,Ω ,q)], whereq∈ [0,1] is the homotopy-parameter and

Ω ∼+∞

∑k=0

ωk qk

is a homotopy-Maclaurin series of a physical parameterω . Besides, we can treat theinitial/boundary conditions of a nonlinear equation in a similar way as mentionedabove.

4.3.3 Examples

Here, we use some simple examples to show how to apply the above theorems todeduce high-order deformation equations of nonlinear problems.

Example 3.1Let us consider a nonlinear heat transfer problem [1]:

(1+ εu)u′+u= 0, u(0) = 1,

where the prime denotes the differentiation with respect tothe timet. ChoosingL u= u′+u as the auxiliary linear operator, and defining the nonlinearoperator

N [φ(t;q)] = (1+ εφ)φ ′+φ ,

where

φ =+∞

∑k=0

uk(t) qk

is a homotopy-Maclaurinseries, we construct such a zeroth-order deformation equa-tion

(1−q)L [φ(t;q)−u0(t)] = q c0 N [φ(t;q)],

subject to the initial condition

φ(t;q) = 1, whent = 0,


whereu0(t) is an initial guess satisfying the initial conditionu(0) = 1. According toTheorem 4.15, the correspondingmth-order deformation equation reads

L [um(t)− χm um−1(t)] = c0 Dm−1N [φ(t;q)] ,

subject to the initial guessum(0) = 0.

According to Theorem 4.1, Theorem 4.2 and Theorem 4.3, one has

Dm−1N [φ(t;q)] = Dm−1(φ ′)+Dm−1(φ)+ ε Dm−1(φ φ ′)

= u′m−1+um−1+ εm−1

∑n=0

um−1−n u′n.

The corresponding homotopy-series solution is given by

u(t) =+∞

∑k=0

uk(t),

which is convergent forany physical parameter 0≤ ε < +∞ if one chooses theconvergence-control parameterc0 =−(1+ ε)−1. For details, please refer to Abbas-bandy [1].

Example 3.2Let us consider a nonlinear oscillation equation [34]:

u′′(t)+λu(t)+ εu3(t) = 0, u(0) = 1,u′(0) = 0,

where the prime denotes the differentiation with respect tothe timet, λ andε arephysical parameters. Letω denote the unknown frequency of the periodic solution.Writing τ = ω t, the above equation becomes

γ u′′(τ)+λu(τ)+ εu3(τ) = 0, u(0) = 1,u′(0) = 0,

whereγ = ω2 is a unknown physical parameter. Regardγ as a homotopy-Maclaurinseries and define a nonlinear operator

N [φ(τ;q),Γ (q)] = Γ (q) φ ′′(τ;q)+λ φ(τ;q)+ εφ3(τ;q),

where the prime denotes the differentiation with respect toτ, and

φ(τ;q) =+∞

∑k=0

uk(t) qk, Γ (q) =+∞

∑k=0

γk qk

are two homotopy-Maclaurin series. Choosing the auxiliarylinear operator

L u= u′′+u,


we construct the following zeroth-order deformation equation

(1−q)L [φ(τ;q)−u0(τ)] = q c0 N [φ(τ;q),Γ (q)], q∈ [0,1]


φ(τ;q) = 1, φ ′(τ;q) = 0, at τ = 0,

whereu0(t) is an initial guess satisfying the initial conditions. According to Theo-rem 4.15, the corresponding high-order deformation equation reads

L [um(τ)− χm um−1(τ)] = c0 Dm−1N [φ(τ;q),Γ (q)] ,


um(0) = 0, u′m(0) = 0.

According to Theorem 4.1, Theorem 4.2, Theorem 4.3, and Theorem 4.6, we have

Dm−1N [φ(τ;q),Γ (q)]= Dm−1(Γ φ ′′)+λ Dm−1(φ)+ ε Dm−1(φ3)

=m−1

∑n=0

γm−1−n u′′n +λ um−1+ εm−1

∑n=0

um−1−n

n

∑k=0

un−k uk.

The corresponding homotopy-series solutions are given by

u(t) =+∞

∑m=0

um(ω t), ω =

(

+∞

∑m=0

γm

)1/2

,

which are convergent if the convergence-control parameterc0 is chosen properly.For example, whenλ = 0, the homotopy-series solutions are convergent foranyaphysical parameter 0≤ ε <+∞ by usingc0 =−(1+ ε)−1. For details, please referto Chapter 2.

Example 3.3Let us consider a nonlinear differential equation with variable coeffi-cients:

A(x) u′′(x)+A′(x) u′(x)+ γ sinu(x) = 0, u′(0) = 0, u′(π) = 0, u(0) = a,

whereA(x) is a given function,γ is an unknown eigenvalue,a is a given constant.Chooseu0 = a cosx as the initial approximation,L u= u′′+u as the auxiliary linearoperator, respectively, and define a nonlinear operator

N (φ ,Γ ,q) = A(x) φ ′′(x;q)+A′(x) φ ′(x;q)+Γ (q)

[

sin(qφ)q

]

,


where the prime denotes the differentiation with respect tox. Construct the zeroth-order deformation equation

(1−q)L (φ −u0) = c0 q N (φ ,Γ ,q), φ ′(0;q) = 0, φ ′(π ;q) = 0, φ(0;q) = a.

According to Theorem 4.15, the corresponding high-order deformation equationreads

L (um− χm um−1) = c0 Dm−1 [N (φ ,Γ ,q)] , u′m(0) = 0, u′m(π) = 0, um(0) = 0.

According to Theorem 4.1, Theorem 4.2 and Theorem 4.3, we have

Dm−1 [N (φ ,Γ ,q)]

= Dm−1[

A(x) φ ′′(x;q)]

+Dm−1[

A′(x) φ ′(x;q)]

+Dm−1

Γ (q)

[

sin(qφ)q

]

= A(x) u′′m−1(x)+A′(x) u′m−1(x)+m−1

∑n=0

γm−1−n Dn

[

sin(qφ)q

]

,

where the termDn[sin(qφ)/q] is given by (4.36), which is obtained by means ofTheorem 4.13. For details, please refer to Liao [31].

4.4 Convergence theorems

In Chapter 2, two theorems about the convergence of the homotopy series of a non-linear oscillation equation are given. Qualitatively speaking, these theorems havegeneral meanings. Here, two similar theorems are proved in general.

Theorem 4.21.Let L denote an auxiliary linear operator which has the propertyL [0] = 0 and is independent of the homotopy-parameter q∈ [0,1], N denote anonlinear operator, u0(x, t) denote an initial approximation of the original equationN [u(x, t),γ] = 0, respectively, wherex denotes a vector of the spatial independentvariable, t is temporal independent variable, andγ is a unknown physical param-eter, respectively. Letα(x, t;q) denote a deformation-function, i.e.α = 0 at q= 0andα = 1 at q= 1, and assume that its Maclaurin series

α(x, t;q) =+∞

∑k=1

αk(x, t) qk,

absolutely converges at q= 1 so that+∞∑

k=1αk(x, t;q) = 1. Let

+∞

∑k=0

βk(x, t) qk+1

4.4 Convergence theorems 173

be a series that is absolutely convergent at q= 1 to a nonzero functionβ (x, t), i.e.

β (x, t) =+∞

∑k=0

βk(x, t) 6= 0,

whereβ0(x, t),β1(x, t), · · · are the so-called convergence-control functions. Let

Γ ∼+∞

∑m=0

γm qm

be a homotopy-Maclaurin series of a unknown physical parameter γ, and

φ ∼+∞

∑m=0

um(x, t) qm

be the homotopy-Maclaurin series of the zeroth-order deformation equation

[1−α(x, t;q)]L (φ −u0) =

[

+∞

∑k=0

βk(x, t) qk+1

]

N (φ ,Γ ) , (4.60)

where um is governed by the corresponding mth-order deformation equation

L [um(x, t)]−m−1

∑n=1

αn(x, t) L [um−n(x, t)] =m

∑k=1

βk−1(x, t) δm−k(x, t), (4.61)

whereδk(x, t) = Dk [N (φ ,Γ )]

with the definition (4.7) ofDk. If the homotopy-series

u(x, t)∼+∞

∑m=0

um(x, t), γ ∼+∞

∑m=0

γm

converge, and besides+∞∑

m=0L [um(x, t)] also converges, then

+∞

∑k=0

δk(x, t) =+∞

∑k=0

Dk [N (φ ,Γ )] = 0. (4.62)

Proof. According to (4.61), we have

+∞

∑m=1

L [um(x, t)]−m−1

∑n=1

αn(x, t) L [um−n(x, t)]

=+∞

∑m=1

m

∑k=1

βk−1(x, t) δm−k(x, t).


Since+∞∑

k=1αk(x, t) is absolutely convergent and

+∞∑

m=0L [um(x, t)] is convergent, we

have due to the theorem of Cauchy product that

+∞

∑m=1

L [um(x, t)]−m−1

∑n=1


=+∞

∑m=1

L [um(x, t)]−+∞

∑m=1

m−1

∑n=1


=+∞

∑m=1

L [um(x, t)]−+∞

∑n=1

+∞

∑m=n+1


=+∞

∑m=1

L [um(x, t)]−+∞

∑n=1

+∞

∑k=1

αn(x, t) L [uk(x, t)]

=+∞

∑m=1

L [um(x, t)]−[

+∞

∑n=1

αn(x, t)

]

+∞

∑k=1

L [uk(x, t)]

=

[

1−+∞

∑n=1

αn(x, t)

]

+∞

∑m=1

L [um(x, t)]

,

which gives, due to+∞∑

n=1αn(x, t) = 1, that

+∞

∑m=1

L [um(x, t)]−m−1

∑n=1


= 0.

Similarly, since+∞∑

k=1βk−1(x, t) is absolutely convergent, we have due to the theorem

of Cauchy product that

+∞

∑m=1

m

∑k=1

βk−1(x, t) δm−k(x, t)

=+∞

∑k=1

+∞

∑m=k

βk−1(x, t) δm−k(x, t)

=+∞

∑k=1

+∞

∑n=0

βk−1(x, t) δn(x, t)

=+∞

∑k=1

βk−1(x, t)+∞

∑n=0

δn(x, t)

=

[

+∞

∑m=0

βm(x, t)

][

+∞

∑n=0

δn(x, t)

]

.

Thus, it holds

4.4 Convergence theorems 175

[

+∞

∑m=0

βm(x, t)

][

+∞

∑n=0

δn(x, t)

]

= 0,

which gives, since+∞∑

m=0βm(x, t) 6= 0, that

+∞

∑n=0

δn(x, t) = 0,

i.e.+∞

∑n=0

Dn [N (φ ,Γ )] = 0.


Theorem 4.22.Let φ(x, t;q) andΓ (q) denote the solution of the zeroth-order de-formation equation (4.60), whose homotopy-Maclaurin series read

φ ∼+∞

∑m=0

um(x, t) qm, Γ ∼+∞

∑m=0

γm qm.

Assume that the homotopy-series

u(x, t)∼+∞

∑m=0

um(x, t), γ ∼+∞

∑m=0

γm

converge, and besides+∞∑

m=0L [um(x, t)] also converges, whereL is an auxiliary lin-

ear operator, so that Theorem 4.21 holds. IfN (φ ,Γ ) is analytic about q in q∈ [0,1],

then+∞∑

m=0um(x, t) and

+∞∑

m=0γm satisfy the original equationN (u,γ) = 0.

Proof. Since the two homotopy-series

u(x, t)∼+∞

∑m=0

um(x, t), γ ∼+∞

∑m=0

γm

converge and besides+∞∑

m=0L [um(x, t)] is also convergent, we have by Theorem 4.21

that+∞

∑n=0

δn(x, t) =+∞

∑n=0

Dn [N (φ ,Γ )] = 0.

Note thatD0 [N (φ ,Γ )] = N (u0,γ0)

denotes the residual of the original governing equation forthe initial approximationsu0 andγ0. So,N (φ ,Γ ) can be regarded as the residual of the governing equation


in q∈ [0,1]. According to Theorem 4.10, its homotopy-Maclaurin seriesreads

N (φ ,Γ )∼+∞

∑k=0

δk(x, t)qk,

say,

N

[(

+∞

∑m=0

um qm

)

,

(

+∞

∑n=0

γn qn

)]

∼+∞

∑k=0

δk(x, t) qk.

SinceN (φ ,Γ ) is analytic aboutq in q∈ [0,1], then the above series converges toN (φ ,Γ ) in q∈ [0,1]. Thus, it holds

N

[(

+∞

∑m=0

um qm

)

,

(

+∞

∑n=0

γn qn

)]

=+∞

∑k=0

δk(x, t) qk.

Settingq= 1 and using+∞∑

n=0δn(x, t) = 0, we have

N

[(

+∞

∑m=0

um

)

,

(

+∞

∑n=0

γn

)]

=+∞

∑k=0

δk(x, t) = 0.

Therefore, the two convergent homotopy-series satisfy theoriginal equation. Thisends the proof. ⊓⊔

Theorem 4.22 reveals the importance of the convergence of homotopy-series.Due to this theorem, it is enough for us to guarantee the convergence of everyhomotopy-series given by the HAM. Theorem 4.21 indicates a necessary conditionof the convergence of homotopy-series. It is very interesting that the homotopy-Maclaurin series of the residual of the original governing equation reads

N (φ ,Γ )∼+∞

∑k=0

δk(x, t) qk,

whereδk(x, t) = Dk [N (φ ,Γ ,q)]. Define

∆m =

∫

Ω

[

m

∑k=0

δk(x, t)

]2

dΩ . (4.63)

According to Theorem 4.22, if each homotopy-series converges, then it holds

limm→+∞

∆m = 0.

Therefore,∆m defined above indicates the accuracy of themth-order homotopy-approximation. Obviously, the smaller the value of∆m, the better the correspond-ing homotopy-approximation. When the homotopy-approximations contain one

4.5 Solution expression 177

convergence-controlparameterc0, ∆m is a function ofc0, so that the optimal value ofc0 is determined by the minimum of∆m. This provides us a simple way to determinethe optimal convergence-control parameters. Note that theterm

δk(x, t) = Dk [N (φ ,Γ )]

is on the right-hand side of the high-order deformation equation, thus we need notspend additional CPU time to calculate it. So, it is more efficient to calculate∆m

than to directly calculate the squared residual of the original equation, i.e.

∆m =

∫

Ω

N

[(

m

∑n=0

um

)

,

(

m

∑k=0

γk

)]2

dΩ . (4.64)

4.5 Solution expression

For a given nonlinear equation, the key of the HAM is to construct a good enoughzeroth-order deformation equation by means of choosing a proper initial approx-imation u0 and a proper auxiliary linear operatorL . As shown above, we haveextremely large freedom to choose the initial approximation u0 and the auxiliarylinear operatorL : it is such kind of fantastic freedom that differs the HAM fromother analytic techniques.

However, anything has its bright and dark sides. The extremely large freedom ofconstructing a zeroth-order deformation equation makes itdifficult for a beginnerto apply the HAM. Thus, for multifarious applications of theHAM in science andengineering, we need some rules to guide the choice of the initial approximationu0

and the auxiliary linear operatorL .It is well-known that the starting-point of perturbation techniques is the so-called

small/large physical parameter, i.e. perturbation quantity. However, the HAM hasnothing to do with any small/large physical parameters (this is one of the advantagesof the HAM). So, we need a new but different starting-point for the HAM.

In essence, to analytically approximate a functionf (t) in a domaint ∈ Ω is toexpress it by a complete set of base functions, say,

f (t)∼+∞

∑k=1

ak ek(t),

whereek(t) denotes the base function andak is a coefficient. The choice of theproper base functions are determined not only by the property of f (t) but also bythe domainΩ . For example, if f (t) is periodic, it is convenient to choose peri-odic base functionsek(t). Besides, according to Weierstrass’s Theorem [36], for anygiven f (t) in C[a,b] and for any givenε > 0, there exists a polynomialpn for somesufficiently largen such that‖ f (t)− pn‖< ε. Therefore, for a given functionu(x, t),the key-point is to find a proper set of base functionsek(x, t) to fit it, i.e.


u(x, t)∼+∞

∑k=1

ak ek(x, t),

whereek(x, t) denotes the base function.

Definition 4.4. Let ek(x, t) denote the base function,u(x, t) be a solution of anonlinear equationN u= 0, respectively. Then,

u(x, t)∼+∞

∑k=1

ak ek(x, t), (4.65)

is calledthe solution expressionof u(x, t), if

limm→∞

∥

∥

∥

∥

∥

u(x, t)−m

∑k=1

ak ek(x, t)

∥

∥

∥

∥

∥

→ 0. (4.66)

Note that a function can be expressed by different base functions. For example,an arbitrary continuous functionf (t) ∈C[−1,1] has a best approximation by meansof Chebyshev series, i.e.

f (t)∼+∞

∑n=0

bn Tn(t),

where Chebyshev polynomialTn(t) of the first kind is a polynomial int of degreen,defined by the relation [36]:

Tn(t) = cos(nθ )

with the recursion formula

T0(t) = 1, T1(t) = t,

Tn(t) = 2t Tn−1(t)−Tn−2(t), n= 2,3,4, · · ·Besides, the solution-expression off ∈ C[a,b] can be a polynomial, according toWeierstrass’s Theorem, or a Fourier series [36].

In fact, finding the so-called solution-expression of a given equationN u= 0 isthe destination (or goal) of solving the equation. In other words, it is the end-pointfor us. However, such kind of end-point is used as the starting-point of the HAM.Given a nonlinear differential equationN u = 0, one should first ask himself animportant question: what kinds of base functions can be usedto approximate theunknown solutionu? For some types of equations, such as equations with continu-ous solutions defined in a finite domain, it is easy to answer this question. However,the answer of this question is not always obvious, especially for some new types ofequations whose physical backgrounds are not very clear. Inthis case, it is helpfulto gain some asymptotic properties of solutions. The more, the better. These asymp-


totic properties often provide us valuable information about the solution expressionof a nonlinear equation.

Some rules are given here for the choice of the solution-expression of a givenequationN u= 0:

1. Equations defined in a finite domain:

• Polynomials and Fourier series can be always used as the solution-expression;• Chebyshev series gives the best approximation for arbitrary continuous solu-

tions ofN u= 0.

2. Equations defined in an infinite domain:

• Periodic base functions should be used, if the solution is periodic;• Non-periodic base functions should be used, if the solutionis not periodic;• The solution-expression should satisfy as many asymptoticproperties of so-

lution as possible, if the solution is not periodic.

Note that the above rules are not absolute, especially when the unknown solutionis non-periodic and defined in an infinite domain. Fortunately, as mentioned before,even a unique solution ofN u= 0 can be often expressed by different types of basefunctions. So, even if one has little information about the solution-expression of agiven equationN u= 0, one can always guess some forms of its solution-expressionand then check whether these guesses are correct or not.

The concept of solution-expressions of nonlinear equations is easy to understandfor applied mathematicians who have rich physical knowledge. For example, it iswell-known that oscillations of a conservative dynamic system are mostly periodic,although its period and amplitude of oscillations are unknown. Besides, it is a well-known knowledge that laminar viscous flows vary greatly neara solid boundary(i.e. the boundary-layer flow) but tend to the uniform flow exponentially at infinity.So, solution-expressions of nonlinear differential equations related to laminar vis-cous flows in fluid mechanics contain the exponential functions which exponentiallytend to zero at infinity. Furthermore, all periodic traveling waves can be expressedby periodic base functions. All of these physical knowledge, if possible, are ratherhelpful for the choice of solution-expression of a given nonlinear equation.

The concept of the so-called solution-expression mentioned above is importantin the frame of the HAM, because it is the start-point for us tochoose the initialapproximationu0 and the auxiliary linear operatorL , as shown below.

4.5.1 Choice of initial approximation

Assume that a solution expression (4.65) is chosen for a given equationN u = 0.Our aim is to find a proper initial approximationu0 and a proper auxiliary linearoperatorL such that the corresponding homotopy-series converge.

Obviously, the initial approximationu0 must obey the so-called solution expres-sion (4.65). Since we have freedom to choose the initial approximation, we could


choose such a kind of initial approximation

u0(x, t)∼n0

∑k=1

ak ek(x, t), (4.67)

whereek(x, t) is the base function, ¯ak is unknown constant, andn0 is equal to orgreater than the number of linear boundary/initial conditions1, denoted byκ . Then,enforcing the above initial approximation to satisfy theκ boundary/initial linearconditions, we haven0 − κ unknown coefficients left. So, whenn0 = κ , then allcoefficients of the initial approximation are known so that it is completely deter-mined. However, whenn1 = n0−κ > 0, we haven1 unknown coefficients, denotedby b1,b2, · · · ,bn1. To gain an optimal initial approximation, we define the squaredresidual of the governing equation

E0(b1,b2, · · · ,bn1) =

∫

Ω

[N (u0)]2d Ω . (4.68)

It is well-known that the minimum of the squared residualE0(b1,b2, · · · ,bn1) isdetermined by a set of nonlinear algebraic equations

∂E0

∂bk= 0, 1≤ k≤ n1,

whose solution gives the optimal valuesb∗1,b∗2, · · · ,b∗n1

of the unknown coefficients.In this way, we obtain an optimal approximationu0. Obviously, the larger the num-ber n1 is, i.e. there are more unknown coefficients, the better the optimal initialapproximation, but it needs more CPU time to solve the set of more complicatednonlinear algebraic equations. In case of very largen1, the set of nonlinear algebraicequations become difficult to solve, and it becomes the method of least squares. So, abalance is needed. In practice, it is often suggested to haveone unknown coefficientin the initial approximation (i.e.n1 = 1), whose optimal value is then determinedby the minimum of the squared residual∆ defined by (4.68). Mostly, such kindof optimal initial approximations with one optimal coefficient are good enough, asillustrated in Chapter 2.

Therefore, as long as a solution-expression of a given equationN u=0 is known,it is straight-forward to gain an optimal initial approximation u0 in the way men-tioned above.

1 For simplicity, we assume here that all boundary/initial conditions are linear. The HAM alsoworks for nonlinear boundary/initial conditions, as shownin Chapter 15 and 16


4.5.2 Choice of auxiliary linear operator

To obey the so-called solution-expression (4.65) of a givenequationN u= 0, theinitial guessu0 and the auxiliary linear operatorL must be chosen so that the solu-tion um of the high-order deformation equation exists and obeys (4.65), and besidesthe homotopy-series

u0++∞

∑m=1

um

converges.The choice of the auxiliary linear operatorL is mainly determined by the

solution-expression (4.65), but sometimes also by the boundary/initial conditions.For example, if the solutionu(t) is a periodic function with a known frequencyω ,then the auxiliary linear operator should be

L u= u′′+ω2 u,

where the prime denotes the differentiation with respect tot. If the solutionu(t) isa periodic function with unknown frequencyω , we first use the transformτ = ω tand then2 choose such an auxiliary linear operator

L u= u′′+u,

where the prime denotes the differentiation with respect toτ. If the solution-expression ofu(t) is polynomial, then one can simply choose the auxiliary linearoperator

L u=dσ udtσ

,

where the positive integerσ > 0 denotes the highest order of derivative ofN u= 0.In general, let the integerσ > 0 denote the highest-order of derivative of an ordi-

nary differential equationN u(t) = 0, u∗m(t) a special solution of the correspondingmth-order deformation equation, respectively. Let

L u= u(σ′)+

σ ′

∑k=1

µk(t) u(σ′−k) (4.69)

denote the unknown auxiliary linear operator, whereu(k) denotes thekth-orderderivative ofu(t), σ ′ is the highest order of derivative, and the unknown coefficientµi(t) is determined later. Let

um(t) = u∗m(t)+σ ′

1

∑k=1

Ak ek(t)+σ ′

2

∑k=1

Bk ek(t), (4.70)

2 In this case, the unknownω in the governing equation is often replaced by its homotopy-

Maclaurin seriesΩ(q) = ω0++∞∑

k=1ωk qk.


denote the common solution of themth-order deformation equation, whereσ ′1 +

σ ′2 = σ ′, Ak andBk are unknown coefficients,ek(t) is the base functions, but ¯ek(t)

is not, i.e.ek(t) /∈ e1(t),e2(t),e3(t), · · · .

Obviously, to obey the solution expression, it must hold

Bk = 0,1≤ k≤ σ ′2. (4.71)

Thus, the common solution reads

um(t) = u∗m(t)+σ ′

1

∑k=1

Ak ek(t). (4.72)

The unknown auxiliary linear operatorL must be so chosen that theσ ′1 unknown

coefficientsAk of the above expression areuniquelydetermined by means ofallrelated boundary/initial conditions of themth-order deformation equation, so thatthe solutionum(t) uniquely exists and besides obey the solution expression. If thisis not true, then we had to change the numberσ ′

1 until it is satisfied. If this comestrue, then the unknown coefficientsµk(t) (1≤ k≤σ ′) of the corresponding auxiliarylinear operatorL is determined by solving the set of linear algebraic equations

L ek(t) = 0, 1≤ k≤ σ ′1

andL ek(t) = 0, 1≤ k≤ σ ′

2.

In this way, we obtain the auxiliary linear operatorL defined by (4.69). For exam-ple, please refer to Liao [34]. Note that, although one can chooseσ ′ = σ in mostcases, this is however not absolutely necessary, as shown inChapter 2, mainly be-cause we have extremely large freedom to choose the auxiliary linear operatorL .Similarly, the above method can be used to find a proper auxiliary linear operatorfor nonlinear partial differential equations, too.

It is interesting that the zeroth-order deformation equation

(1−q)L (φ −u0) = c0 q N (φ), q∈ [0,1], c0 6= 0

can be rewritten in the form

(1−q)L (φ −u0) = q N (φ),

whereL = L /c0, andc0 is the convergence-control parameter. So, in essence,choosing the convergence-control parameterc0 is a part of choosing the auxiliarylinear operator. Therefore, the optimal convergence-control parameter correspondsto the optimal auxiliary linear operator! Let‖L −1‖ denote the norm of the inverseoperatorL −1. Then, we have‖L −1‖= |c0|‖L −1‖. Thus, we can adjust the norm

4.6 Convergence control and acceleration 183

of L −1, and this is the essential reason why we can guarantee the convergence ofhomotopy-series by means of choosing a proper convergence-control parameterc0.

Similarly, the more generalized zeroth-order deformationequation

(1−q)L (φ −u0) = q

(

+∞

∑k=0

c0 qk

)

N (φ), q∈ [0,1], c0 6= 0

can be rewritten in the “basic” form

(1−q)L (φ −u0) = q N (φ),

whereck is convergence-control parameter and

L =

(

+∞

∑k=0

c0 qk

)−1

L .

Therefore, choosing the convergence-control parametersck is essentially a part ofchoosing the auxiliary linear operatorL . Obviously, the norm‖L −1‖ is deter-mined by the convergence-control parametersck, and the optimal convergence-control parametersck correspond to an optimal auxiliary linear operatorL . In otherwords, we choose the auxiliary linear operator in two steps:the“basic” auxiliarylinear operatorL is first chosen according to the so-called solution-expression,then the auxiliary linear operator as a whole is modified by choosing optimalconvergence-control parameters. Therefore, even if the “basic” auxiliary linear op-eratorL is not perfect, we can still guarantee the convergence of homotopy-seriesby choosing optimal convergence-control parametersck.

In summary, thesolution-expressionis an important concept of the HAM, whichprovides us a start-point to choose the initial approximation and the “basic” auxiliarylinear operatorL .

4.6 Convergence control and acceleration

Another important concept of the HAM is theconvergence-control: the convergenceof homotopy-series is guaranteed by choosing optimal convergence-control param-eters.

According to Theorem 4.22, it is very important to guaranteethe convergenceof homotopy-series. However, it is a pity that there do not exist any mathematicaltheorems which can guide us in details how to construct a goodenough homotopyof equations for any a given nonlinear equation so as to gain its convergent seriessolution. This is mainly because nonlinear equations differ in thousands ways sothat it seems rather difficult to give a common approach for all of them.

In theory, the convergence of homotopy-series is strongly dependent upon theinitial approximation and the auxiliary linear operator asa whole. As shown in


§ 4.5.1, it is easy to choose an optimal initial approximationthat obeys the givensolution-expression. Then, we use such a strategy to choosethe auxiliary linear op-erator as a whole: a “basic” auxiliary linear operatorL is chosen first by meansof the given solution-expression, and then some unknown convergence-control pa-rameters are introduced into the zeroth-order deformationequation, whose optimalvalues are determined by the minimum of the squared residualof the original equa-tion N u = 0. In this way, the convergence of the homotopy-series is “controlled”by the so-called convergence-control parameters.

The convergence-control of homotopy-series is similar to the control of a dy-namic system [14]. Here, “the desired output” is the minimumof the squared resid-ual of a given equationN u= 0, “the inputs” are the unknown convergence-controlparameters, which have no physical meanings at all, as mentioned in Chapter 2.It is the concept of convergence-control that differs the HAM from all other an-alytic techniques, such as perturbation techniques [7, 11,12, 39–41], Lyapunov’sartificial small parameter method [35], Adomian decomposition method [2–4], theδ -expansion method [13] and so on.

4.6.1 Optimal convergence-control parameter

Our aim is to construct a good enough zeroth-order deformation equation for a givennonlinear equation

N u= 0

so that the corresponding homotopy-series converge, i.e. the squared residual of themth-order homotopy-approximation

Em(c0,c1, · · · ,cκ) =

∫

Ω

[

N

(

m

∑n=0

un

)]2

d Ω (4.73)

tends to zero asm → +∞, whereci (0 ≤ i ≤ κ) is the convergence-control pa-rameter. Obviously, at a given orderm of approximation, the optimal homotopy-approximation is given by the minimum of the squared residual Em, and the cor-responding optimal convergence-control parametersc∗n are determined by a set of(κ +1) nonlinear algebraic equations

∂Em

∂cn= 0, 0≤ n≤ κ . (4.74)

In theory, the more convergence-control parameters we have, the better the corre-sponding homotopy-approximation. Mostly, even one optimal convergence-controlparameter can greatly accelerate the convergence of homotopy-series solution, asshown in Chapter 2. In general, one or two convergence-control parameters areenough to give accurate homotopy-approximations, as illustrated by Liao [33].

4.6 Convergence control and acceleration 185

According to Theorem 4.21, the approximate squared residual

Eδm(c0,c1, · · · ,cκ) =

∫

Ω

(

m

∑k=0

δk

)2

dΩ , (4.75)

tends to zero asm→+∞, if the homotopy-Maclaurin seriesφ =+∞∑

n=0un qn converges

at q= 1, whereδk = Dk [N (φ)]. Thus, as an alternative, the optimal convergence-control parametersc∗n (0≤ n≤ κ) are determined, approximately, by the minimumof Eδ

m, i.e. a set of the(κ +1) nonlinear algebraic equations

∂Eδm

∂cn= 0, 0≤ n≤ κ . (4.76)

Note that the termDk [N (φ)] is on the right-hand side of the high-order deforma-tion equation and thus can be regarded as known, i.e. we need not additional CPUtime to calculate it. So, it is more efficient to useEδ

m to gain the optimal convergence-control parameterscn, where 0≤ n ≤ κ . It is found that the optimal convergence-control parameters given by the above approach are rather close to those given bythe exact squared residualEm of governing equations.

4.6.2 Optimal initial approximation

As illustrated in Chapter 2, the optimal initial approximation can greatly acceler-ate the convergence of homotopy-series. It is relatively simple to get an optimalinitial approximation, as shown in§ 4.5.1. First, we introducen1 unknown coeffi-cientb1,b2, · · · ,bn1 into the initial approximationu0, which satisfies as many bound-ary/initial conditions as possible. The optimal values of these unknown coefficientsare determined by the minimum ofE0, which gives a set of the correspondingn1

nonlinear algebraic equations

∂E0

∂bk= 0, 1≤ k≤ n1.

Note that the optimal initial approximation has an influenceon the choice of theoptimal convergence-control parameters. In most cases, ifboth of the optimal ini-tial approximation and the optimal convergence-control parameters are used, thehomotopy-series converges rather fast, as shown in Chapter2. So, the optimal initialapproximation is strongly suggested to use in the frame of the HAM to acceleratethe convergence.


4.6.3 Homotopy-iteration technique

Obviously, a better initial approximation gives a better homotopy-approximation.Since we have great freedom to choose the initial approximation in zeroth-orderdeformation equations, we can replace the initial approximationu0 by means of themth-order homotopy-approximation

u∼ u0+m

∑n=1

un,

which is mostly better than the initial approximationu0. The above expression iscalled themth-order homotopy-iteration formula. For a renewed initial approxima-tion, we can choose the corresponding optimal convergence-control parameters in asimilar way as mentioned above. As shown in Chapter 2, such kind of homotopy-iteration method can greatly accelerate the convergence ofhomotopy-series.

The key-point of the homotopy-iteration method is the truncation of the solutionexpression, i.e.

um(x, t)≈M

∑k=1

am,k ek(x, t), (4.77)

whereek(x, t) is the base function,M > 0 is a large enough integer, respectively. Inother words, the homotopy-approximation contains at most the firstM base func-tions. For this reason, we often rewrite the right-hand sideterm of the high-orderdeformation equation in a truncated form

Dm−1 [N (φ)]≈M

∑k=1

cm,k ek(x, t), (4.78)

whereek(x, t) is the base function. Note that the coefficientcm,k in above expressionis easy to calculate when the base functionsek(x, t) are orthogonal, i.e.

cm,k =〈Dm−1 [M (φ)] ,ek(x, t)〉

〈ek(x, t),ek(x, t)〉, (4.79)

where< x,y> is an inner product ofx andy.

4.6.4 Homotopy-Pade technique

The so-called Pade approximant developed by French mathematician Henri EugenePade (1863 -1953) is widely applied, which gives the “best”approximation of agiven function by a rational function of given order. For a power series

+∞

∑n=0

cn zn,

4.7 Discussions and open questions 187

the corresponding[m,n] Pade approximant is expressed by

m∑

k=0am,k zk

n∑

k=0bm,k zk

,

wheream,k,bm,k are determined by the coefficientsc j ( j = 0,1,2,3, · · · ,m+n).The so-called homotopy-Pade technique is a combination ofthe traditional Pade

technique with the homotopy analysis method. Regarding a homotopy-Maclaurinseries

u(x, t;q)∼ u0(x, t)++∞

∑n=1

un(x, t) qn

of a given nonlinear equationN u = 0 as a power series ofq, we first employ thetraditional[m,n]Pade technique about the homotopy-parameterq to obtain the[m,n]Pade approximant

m∑

k=0Am,k(x, t) qk

n∑

k=0Bm,k(x, t) qk

, (4.80)

where the coefficientsAm,k(x, t) and Bm,k(x, t) are determined by the firstm+ nterms

u0(x, t),u1(x, t),u2(x, t), · · · ,um+n(x, t)

of the homotopy-Maclaurin series. Since the homotopy-approximation is obtainedat q = 1, settingq = 1 in (4.80), we have the so-called[m,n] homotopy-Pade ap-proximation

u(x, t;q)≈

m∑

k=0Am,k(x, t)

n∑

k=0Bm,k(x, t)

. (4.81)

In general, the homotopy-Pade technique can greatly accelerate the convergenceof homotopy-series, as shown in Chapter 2.

4.7 Discussions and open questions

In this chapter, the HAM is systematically described in details as a whole. Mathe-matical theorems related to the so-called homotopy-derivative operator and defor-mation equations are proved, which are helpful to gain high-order approximations.Some theorems of convergence are proved, and the methods to control and acceler-ate convergence are generally described.


Note that, based on the homotopy of topology, the HAM provides us extremelylarge freedom to construct zeroth-order deformation equation. Such kind of fantasticfreedom provides us great flexibility to choose initial approximation and auxiliarylinear operatorL so as to construct zeroth-order deformation equation in a quitegeneral form, as shown in§ 4.3. Especially, it is due to this kind of freedom that wecan introduce the so-called convergence-control parameters into the zeroth-orderdeformation equation, which provide us a convenient way to guarantee the conver-gence of homotopy-series solution.

The “solution-expression” and “convergence-control” aretwo important con-cepts in the frame of the HAM. The solution-expression provides a start-point anda guide to choose the initial approximation and the “basic” auxiliary linear operatorL . The so-called convergence-control parameters are used tocontrol and accel-erate the convergence of homotopy-series solution. In essence, it is the so-calledconvergence-control parameter that differs the HAM from all other analytic tech-niques. In the frame of the HAM, the convergence of homotopy-approximation canbe controlled and greatly accelerated by means of optimal initial approximation,and/or optimal convergence-control parameters, and/or iteration approach. There-fore, unlike other analytic approximation methods, the HAMis valid even for highlynonlinear problems.

However, nonlinear problems are often hard to understand inessence. Naturally,there are some open questions for the HAM. First, for a given nonlinear equationN u = 0 in general, there are no mathematical theorems up to now, which canclearly guide us in details how to construct a good enough homotopy of equations soas to gain a convergent series solution. It seems very difficult to give such a kind ofgeneral theorems, because nonlinear equations differ in thousand ways. But, sucha theorem, if it is indeed existed, could heighten the HAM in theory and greatlysimplify its applications, especially for beginners of theHAM. If such a theoremindeed exists, it would be possible to develop a code, which can automatically solvemost of nonlinear equations by computer algebra system likeMathematica, Mapleand so on.

Secondly, the zeroth-order deformation equation

(1−q)L (φ −u0) = c0 q N (φ)

can be rewritten in the “basic” form

(1−q)L (φ −u0) = q N (φ)

whereL is the “basic” auxiliary linear operator andL = L /c0 is the auxiliarylinear operator as a whole,u0 is the initial approximation andφ is a homotopy-Maclaurin series, respectively. Since

∥

∥L−1∥

∥= |c0|∥

∥L−1∥

∥ , (4.82)

for any a given smallε > 0, we can always find such a convergence-control param-eterc0 that

4.7 Discussions and open questions 189

∥

∥L−1∥

∥< ε, (4.83)

if∥

∥L −1∥

∥ is bounded. So, it seems that, if∥

∥L −1∥

∥ is bounded, and if the initial ap-proximationu0 is close enough to the exact solutionu∗ so that‖u0−u∗‖ is bounded,then we could always find such a convergence-control parameter c0 that the corre-sponding homotopy-series converges. Unfortunately, we can not prove this guess ingeneral up to now.

In addition, although in theory we have extremely large freedom to construct azeroth-order deformation equation for a given nonlinear equationN u= 0, it is notvery clear how to use such kind of freedom, especially for therather general formsof the zeroth-order deformation equations, such as (4.52),(4.54), (4.56) and (4.58).Currently, these rather generalized zeroth-order deformation equations are hardlyused in practice. There are many related open questions. Forexample, how to findthe “best” auxiliary linear operator for a given nonlinear differential equation in gen-eral? How to find the “best” convergence-control functionβk(x, t)? What happens ifthe convergence-control functionsβk(x, t) are orthogonal to the termsDk [N (φ)]?How to obtain the “best” deformation-operatorA in (4.56) for a given nonlinearequation in general? Is the HAM valid for complicated nonlinear problems relatedto chaos and turbulence?

So, there is still a long way ahead for us, although the HAM hasbeen successfullyapplied to so many nonlinear problems in science, finance andengineering. Indeed,nonlinear problems are often hard to understand in essence,especially those relatedto chaos and turbulence. That is the reason why our aim is to develop an analyticapproximation method valid for asmanynonlinear problemsas possible.


References





5. Alexander, J.C., Yorke, J.A.: The homotopy continuationmethod: numerically implementabletopological procedures. Trans Am Math Soc.242, 271-284 (1978)

6. Armstrong, M.A., Basic Topology (Undergraduate Texts inMathematics). Springer, NewYork (1983)


8. Fitzpatrick, P.M.: Advanced Calculus. PWS Publishing Company, New York (1996)9. He, J.H.: An approximate solution technique depending upon an artificial parameter. Com-

mun. Nonlinear Sci. Numer. Simulat.3, 92 – 97 (1998)10. He, J.H.: Homotopy perturbation technique. Comput. Method. Appl. M. 178, 257 – 262

(1999)11. Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge

(1953)12. Hinch, E.J.: Perturbation Methods. In seres of Cambridge Texts in Applied Mathematics ,

Cambridge University Press, Cambridge (1991)13. Karmishin, A.V., Zhukov, A.T., Kolosov, V.G.: Methods of Dynamics Calculation and Testing

for Thin-walled Structures (in Russian). Mashinostroyenie, Moscow (1990)14. Levine, W.S.: The Control Handbook. CRC Press, New York (1996)15. Liang, S.X., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy pertur-



17. Li, T.Y., Yorke, J.A.: Path following approaches for solving nonlinear equations: homotopy,continuous Newton and projection. Functional Differential Equations and Approximation ofFixed Points. 257-261 (1978)

18. Li, T.Y, Yorke, J.A.: A simple reliable numerical algorithm for floowing homotopy paths.Analysis and Computation of Fixed Points. 73 – 91 (1980)

19. Li, T.Y.: Solving polynomial systems by the homotopy continuation methods (Handbook ofNumerical Analysis, 209-304). North-Holland, Amsterdam (1993)







References 191







32. Liao, S.J.: On the relationship between the homotopy analysis method and Eu-ler transform. Commun. Nonlinear Sci. Numer. Simulat.15, 1421 – 1431 (2010).doi:10.1016/j.cnsns.2009.06.008




36. Mason, J.C., Handscomb D.C.: Chebyshev Polynomial. Chapman & Hall/CRC, Boca Raton(2003)



39. Murdock, J.A.: Perturbations: - Theory and Methods. John Wiley & Sons, New York (1991)40. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (1973)41. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (2000)42. Sajid, M., Hayat, T.: Comparison of HAM and HPM methods for nonlinear heat conduction

and convection equations. Nonlin. Anal. B.9, 2296 – 2301 (2008)43. Sen, S.: Topology and Geometry for Physicists. AcademicPress, Florida (1983)44. Xu, H., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy-based solutions of the Navier-



46. Turkyilmazoglu, M., A note on the homotopy analysis method. Appl. Math. Lett.23, 1226 –1230 (2010)

Chapter 5Relationship to Euler transform

Abstract The so-called generalized Taylor series and homotopy-transform are de-rived in the frame of the homotopy analysis method (HAM). Some related theoremsare proved, which reveal in theory the reason why convergence-control parameterprovides us a convenient way to guarantee the convergence ofthe homotopy-seriessolution. Especially, it is proved that the homotopy-transform logically contains thefamous Euler transform that is often used to accelerate convergence of a series or tomake a divergent series convergent. All of these provide us aconner-stone for theconcept of convergence-control and the great generality ofthe HAM.

5.1 Introduction

As mentioned in Chapter 2 and Chapter 3, the convergence-control is an importantconcept in the frame of the homotopy analysis method (HAM) [3–18]: the con-vergence of homotopy-series can be guaranteed by means of a non-zero auxiliaryparameterc0, called today the convergence-control parameter, that wasfirst intro-duced into the zeroth-order deformation equations by Liao [5] in 1997. In Chap-ter 2, a nonlinear oscillation equation is used to illustrate how powerful and effi-cient such kind of convergence-control parameter is to guarantee the convergenceof homotopy-series. Especially, we prove that, by means of introducing a non-zeroauxiliary parameterc0, we can give a power series of(1+ z)−1 convergent in thewholedomain except the singular pointz=−1, i.e.(−∞,−1)∪ (−1,+∞) (see The-orem 2.3). Note that the traditional Taylor series of(1+ z)−1 converges only in abounded domain|1+z|< 1. This shows the validity and great potential of the HAMfrom another view-point.

The function(1+z)−1 considered in Theorem 2.3 is simple and special. In 2009,Liao [15] considered a smooth functionf (z) in general, and derived a more gen-eralized power series off (z) by introducing the convergence-control parameterc0 and two deformation-functionsA(q) andB(q), satisfyingA(0) = B(0) = 0 andA(1) = B(1) = 1. Such kind of power series is called the generalized Taylorseries,

193

194 5 Relationship to Euler transform

since it logically contains the traditional Taylor series of f (z). It is found that theconvergence-region of the generalized Taylor series can befantastically enlargedby the so-called convergence-control parameterc0, and this explains in theory whythe convergence-control parameterc0 can guarantee the convergence of homotopy-series obtained by the HAM. Besides, based on this generalized Taylor series, a kindof transform, called the homotopy-transform, is defined, which is more general thanthe famous Euler transform, as proved by Liao [15]. Especially, Liao [15] illustratedthat the so-called homotopy-transformcan be derived in theframe of the HAM. Thisnot only further confirms the generality and great potentialof the HAM for stronglynonlinear problems, but also provides the HAM a solid base inmathematics.

5.2 Generalized Taylor series

The traditional Taylor series of a functionf (z), i.e.

f (z) ∼ f (z0)++∞

∑k=1

f (k)(z0)

k!(z− z0)

k,

was formally introduced by the English mathematician BrookTaylor FRS (1685– 1731) in 1715, which is called Maclaurin series whenz0 = 0, named after theScottish mathematician Colin Maclaurin (1698 – 1746). The traditional Taylor seriesconverges tof (z) in the domain

∣

∣

∣

∣

z− z0

ζ − z0

∣

∣

∣

∣

< 1,

whereζ is a pole of f (z) closest toz0. It is well known that the traditional Taylorseries of many functions converge in a bounded domain. For example, the traditionalTaylor series of(1+ z)−1 is convergent only in an unit circle|z|< 1.

In Chapter 2, we give a definition of the so-called deformation-functionby meansof the real homotopy-parameterq∈ [0,1]. Here, we give a more general definitionof a deformation function in a complex numberz.

5.2 Generalized Taylor series 195

Definition 5.1. Let q be a complex number. A complex functionA(q) is called adeformation-functionif it satisfies

A(0) = 0, A(1) = 1 (5.1)

and is analytic in the region|q| ≤ 1 so that its Maclaurin series+∞∑

k=1ak qk is con-

vergent in the region|q| ≤ 1, say,

A(q) =+∞

∑k=1

ak qk, |q| ≤ 1.

Lemma 5.1Let A(q) be a deformation function, whose Maclaurin series A(q) =+∞∑

k=1ak qk converges at|q| ≤ 1. Then, it holds

(

+∞

∑k=1

ak qk

)m

=+∞

∑n=m

am,n qn, (5.2)

where

a1,n = an, n≥ 1, (5.3)

am,n =n−1

∑k=m−1

an−k am−1,k, n≥ m≥ 2. (5.4)

Proof. Write

[A(q)]m =

(

+∞

∑k=1

ak qk

)m

=+∞

∑k=m

am,k qk, m≥ 1, (5.5)

wheream,k = ak, whenm= 1.

Assume that ¯am−1,k (k≥ m≥ 2) are known, it follows that

[A(q)]m =+∞

∑k=m

am,k qk =

(

+∞

∑k=1

ak qk

)m−1(+∞

∑k=1

ak qk

)

=

(

+∞

∑n=m−1

am−1,n qn

)(

+∞

∑j=1

a j q j

)

=+∞

∑k=m

qk

(

k−1

∑n=m−1

am−1,n ak−n

)

,

which gives the recurrence formula


am,k =k−1

∑n=m−1

ak−n am−1,n, k≥ m≥ 2.

This ends the proof. ⊓⊔Theorem 5.1.Let q,z,z0 and c0 6= 0 be complex numbers, A(q) and B(q) denotetwo deformation-functions satisfying A(0) = B(0) = 0, A(1) = B(1) = 1, whose

Maclaurin series+∞∑

k=1ak qk and

+∞∑

k=1bk qk are absolutely convergent in the region

|q| ≤ 1, and besides|(1+ c0) B(q)|< 1. Define

Tm,k(c0,A,B) = (−c0)k

m−k

∑n=0

n

∑r=0

(

k+ r −1r

)

(1+ c0)r

n−r

∑s=0

ak,k+sbr,n−s, (5.6)

where m≥ 1 and1≤ k≤ m, and

a1,k = ak, b1,k = bk, b0,0 = 1, b0,k = 0, k≥ 1, (5.7)

am,k =k−1

∑n=m−1

ak−n am−1,n, k≥ m≥ 2, (5.8)

bm,k =k−1

∑n=m−1

bk−n bm−1,n, k≥ m≥ 2. (5.9)

If a complex function f(z) is analytic at z0 but singular atξk (k = 1,2, · · · ,M0),where M0 may be infinity, the series

f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B), (5.10)

converges to f(z) in the region D=M⋂

k=0Sk, where Sk = z : |ζk|> 1, andζk is the

solution of either1− (1+ c0) B(ζk) = 0,

or

1− (1+ c0) B(ζk)+ c0

(

z− z0

ξn− z0

)

A(ζk) = 0, 1≤ n≤ M0.

Proof. According toLemma 5.1, we have

[A(q)]m =

(

+∞

∑k=1

ak qk

)m

=+∞

∑k=m

am,k qk, m≥ 1,

and

[B(q)]m =

(

+∞

∑k=1

bk qk

)m

=+∞

∑k=m

bm,k qk, m≥ 1,


wheream,k, bm,k are defined by (5.7) to (5.9), respectively. For simplicity,we define

b0,0 = 1, b0,k = 0 (k≥ 1).

For simplicity, define

Γk(z) =f (k)(z0)

k!(z− z0)

k, k≥ 1,

τ = z0−c0(z− z0) A(q)

1− (1+ c0) B(q), c0 6= 0, (5.11)

and construct such a related complex function

F(q) = f (τ) = f

[

z0−c0 (z− z0) A(q)

1− (1+ c0) B(q)

]

. (5.12)

SinceA(0) = B(0) = 0 andA(1) = B(1) = 1, it holdsτ = z0 whenq= 0 andτ = zwhenq= 1, respectively. Therefore, we have

F(0) = f (z0), F(1) = f (z). (5.13)

In other words,F(q) is a homotopy, i.e.F(q) : f (z0)∼ f (z). Writing

δτ =− c0 (z− z0) A(q)1− (1+ c0) B(q)

, (5.14)

we haveF(q) = f (τ) = f (z0+δτ). If |δτ| is sufficiently small and|(1+c0)B(q)|<1 holds, then the Maclaurin series ofF(q) atq= 0 reads

f (z0)++∞

∑k=1

f (k)(z0)

k!(δτ)k

= f (z0)++∞

∑k=1

f (k)(z0)

k!

[

− c0 (z− z0) A(q)1− (1+ c0) B(q)

]k

= f (z0)++∞

∑k=1

Γk(z)(−c0)k[A(q)]k [1− (1+ c0) B(q)]−k

= f (z0)++∞

∑k=1

Γk(z) (−c0)k[A(q)]k

+∞

∑r=0

(

k+ r −1r

)

(1+ c0)r [B(q)]r

= f (z0)++∞

∑k=1

+∞

∑r=0

Γk(z)

(

k+ r −1r

)

(−c0)k (1+ c0)

r

(

+∞

∑i=k

ak,i qi

)(

+∞

∑j=r

br, j q j

)

= f (z0)++∞

∑k=1

+∞

∑r=0

Γk(z)

(

k+ r −1r

)

(−c0)k (1+ c0)

r+∞

∑s=k+r

qs

(

s−r

∑i=k

ak,i br,s−i

)


= f (z0)++∞

∑s=1

qs

[

s

∑k=1

Γk(z) (−c0)k

s−k

∑r=0

(

k+ r −1r

)

(1+ c0)r

(

s−r

∑i=k

ak,i br,s−i

)]

= f (z0)++∞

∑n=1

σn(z) qn, (5.15)

where

σn(z) =n

∑k=1

Γk(z) (−c0)k

n−k

∑r=0

(

k+ r −1r

)

(1+ c0)r

(

n−r

∑i=k

ak,i br,n−i

)

. (5.16)

Let ζ denote a singular point ofF(q). According to the definition (5.12), all solu-tions of the equation

1− (1+ c0)B(ζ ) = 0,

are the singular points ofF(q). Besides, each original singular pointξk (1≤ k≤M0)of f (z) gives corresponding singular points ofF(q), governed by the equation

z0−c0 (z− z0) A(ζ )

1− (1+ c0) B(ζ )= ξk, 1≤ k≤ M0,

i.e.

1− (1+ c0) B(ζ )+ c0

(

z− z0

ξk− z0

)

A(ζ ) = 0, 1≤ k≤ M0.

Let ζk (1 ≤ k ≤ M) denote these singular points ofF(q). Note that these singularpoints are dependent uponz,z0 andc0. The Maclaurin series (5.15) converges toF(1) = f (z) at q = 1, if and only if all singular pointsζk of F(q) are out of theregion|q| ≤ 1, say,

|ζk|> 1, 1≤ k≤ M.

In this case, according to (5.15) and (5.16), we have

f (z) = F(1) = f (z0)+ limm→+∞

m

∑n=1

σn

= f (z0)+ limm→+∞

m

∑n=1

n

∑k=1

Γk(z) (−c0)k

n−k

∑r=0

(

k+ r −1r

)

(1+ c0)r

(

n−r

∑i=k

ak,i br,n−i

)

= f (z0)+ limm→+∞

m

∑k=1

Γk(z)

[

(−c0)k

m

∑n=k

n−k

∑r=0

(

k+ r −1r

)

(1+ c0)r

(

n−r

∑i=k

ak,i br,n−i

)]

= f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B)


in the domainD =M⋂

k=0Sk, whereSk = z : |ζk|> 1 and

Tm,k(c0,A,B)

= (−c0)k

m

∑n=k

n−k

∑r=0

(

k+ r −1r

)

(1+ c0)r

(

n−r

∑i=k

ak,i br,n−i

)

= (−c0)k

m−k

∑n=0

n

∑r=0

(

k+ r −1r

)

(1+ c0)r

(

n+k−r

∑i=k

ak,i br,n+k−i

)

= (−c0)k

m−k

∑n=0

n

∑r=0

(

k+ r −1r

)

(1+ c0)r

(

n−r

∑s=0

ak,k+s br,n−s

)

.


Definition 5.2. Let c0 6= 0 denote the convergence-control parameter,A(q) and

B(q) be two deformation-functions whose Maclaurin seriesA(q) =+∞∑

k=1ak qk and

B(q) =+∞∑

k=1bk qk converge atq= 1 so that

+∞∑

k=1ak = 1 and

+∞∑

k=1bk = 1. If a complex

function f (z) is analytic atz= z0, then

f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B) (5.17)

is called the generalized Taylor series off (z), whereTm,k(c0,A,B) is defined by(5.6) and (5.7) – (5.9).

Note that the so-called generalized Taylor series (5.17) ofa complex f (z) atz= z0 is dependent upon one auxiliary parameterc0 and the two complex analyticfunctionsA(q) andB(q) with their Maclaurin series

A(q) =+∞

∑k=1

ak qk, B(q) =+∞

∑k=1

bk qk

under the restriction+∞

∑k=1

ak = 1,+∞

∑k=1

bk = 1.

Here, the two convergent series+∞∑

k=1ak and

+∞∑

k=1bk are derived from the two analytic

functionsA(q) andB(q), the so-called deformation-functions.Alternatively, any two

convergent series+∞∑

k=1ak = 1 and

+∞∑

k=1bk = 1 can be used. For example,


ak = (1− γ) γ k−1, |γ|< 1

bk =6

(k π)2 ,

completely define two deformation-functionsA(q) andB(q), andTm,k(c0,A,B) by(5.6). There are many such kinds of convergent series, whichcan be used to defineTm,k(c0,A,B).

Theorem 5.2.Let c0 denote a convergence-control parameter, A(q) = q and B(q) =q denote two deformation-functions, respectively. If a real function f(x) is analyticin the whole domain−∞ < x < +∞ except at the unique singular point x= ξ , itsgeneralized Taylor series at x0 6= ξ , i.e.

f (x0)+ limm→+∞

m

∑k=1

[

f (k)(x0)

k!(x− x0)

k

]

Tm,k(c0,A,B)

converges to f(x) in the region

1− 2|c0|

<x− x0

ξ − x0< 1, −2< c0 < 1,

which becomes an infinite domain

−∞ <x− x0

ξ − x0< 1 as c0 → 0.

Proof. SinceA(q) = q, B(q) = q and there exists only one singular pointx = ξ ,according to Theorem 5.1, we have two singular pointsζ0 andζ1, governed by

1− (1+ c0) ζ0 = 0

and

1− (1+ c0) ζ1+ c0

(

x− x0

ξ − x0

)

ζ1 = 0,

respectively. Obviously,ζ0 = (1+ c0)

−1

and

ζ1 =

[

1+ c0− c0

(

x− x0

ξ − x0

)]−1

.

According to Theorem 5.1, the generalized Taylor series

f (x0)+ limm→+∞

m

∑k=1

[

f (k)(x0)

k!(x− x0)

k

]

Tm,k(c0,A,B)

converges tof (x) in the region|ζ0|> 1∩|ζ1|> 1. From|ζ0|> 1, we have|1+c0|<1, i.e.


−2< c0 < 0.

Besides,|ζ1|> 1 gives∣

∣

∣

∣

1+ c0− c0

(

x− x0

ξ − x0

)∣

∣

∣

∣

< 1,

i.e.

−2− c0 <−c0

(

x− x0

ξ − x0

)

<−c0.

Since−2< c0 < 0, we have the convergence domain

1− 2|c0|

<x− x0

ξ − x0< 1, −2< c0 < 1,

which becomes−∞ <

x− x0

ξ − x0< 1 asc0 → 0.


Therefore, if a real functionf (x) has only one singular point atx = ξ , then itgeneralized Taylor series (5.17) byA(q) = B(q) = q as c0 → 0 converges in theinfinite domain(−∞,ξ ) whenx0 < ξ , or in the infinite domain(ξ ,+∞) whenx0 >ξ . So, it explaines in theory why the convergence-control parameterc0 can greatlyenlarge the convergence-region of a series.

Theorem 5.3.Let c0 ∈ ℜ denote a real convergence-control parameter, A(q) = qand B(q) = q denote two deformation-functions, respectively. If a complex functionf (z) is analytic in the whole plane except at the unique singular point z= ξ , itsgeneralized Taylor series at z0 6= ξ , i.e.

f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B)

converges to f(z) in the region

ρ =

∣

∣

∣

∣

z− z0

ξ − z0

∣

∣

∣

∣

<(1+ c0) cosθ −

√

1− (1+ c0)2sin2 θc0

, −2< c0 < 0,

which becomes an infinite domain

ρ =

∣

∣

∣

∣

x− x0

ξ − x0

∣

∣

∣

∣

<

1/cosθ , θ ∈(

− π2 ,

π2

)

,+∞, otherwise,

as c0 → 0, where

η =z− z0

ξ − z0= ρ ei θ , θ ∈ [0,2π ], i =

√−1.


Proof. SinceA(q) = B(q) = q and there exists only one singular pointz= ξ , ac-cording to Theorem 5.1, we have two singular pointsζ0 andζ1, governed by

1− (1+ c0) ζ0 = 0

and

1− (1+ c0) ζ1+ c0

(

z− z0

ξ − z0

)

ζ1 = 0,


−1

and

ζ1 =

[

1+ c0− c0

(

z− z0

ξ − z0

)]−1

.


f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B)

converges tof (z) in the region|ζ0|> 1∩|ζ1|> 1. From|ζ0|> 1, we have|1+c0|<1, i.e.

−2< c0 < 0.


∣

∣

∣

1+ c0− c0

(

z− z0

ξ − z0

)∣

∣

∣

∣

< 1.

Writing

η =z− z0

ξ − z0= ρ ei θ , θ ∈ [0,2π ],

We havec2

0 ρ2−2 c0 (1+ c0) cosθ ρ +(1+ c0)2 < 1,

whose solution reads

0≤ ρ <(1+ c0) cosθ −

√

1− (1+ c0)2sin2 θc0

, −2< c0 < 0.

Since√

1− (1+ c0)2 sin2 θ = |cosθ | asc0 → 0, we have whenπ/2≤ θ ≤ 3π/2that

(1+ c0) cosθ −√

1− (1+ c0)2 sin2 θc0

=cosθ −|cosθ |

c0=

2cosθc0

→+∞


asc0 → 0. Sinceρ = 0/0 whenθ ∈ (−π/2,π/2) andc0 → 0, we have by means ofBernoulli’s rule in calculus that

(1+ c0) cosθ −√

1− (1+ c0)2 sin2 θc0

= cosθ +sin2 θ|cosθ | =

1cosθ

.


In case ofA(q) = B(q) = q and there is an unique singular point, the convergencedomain of the generalized Taylor series depends upon the convergence-control pa-rameterc0, denoted byS(c0). Then, according to Theorem 5.3, we have

S(−1) =

(ρ cosθ ,ρ sinθ )∣

∣ ρ < 1, θ ∈ (0,2π)

,

S

(

−12

)

=


∣

∣ρ < 2

√

1− 14

sin2 θ − cosθ , θ ∈ (0,2π)

,

S

(

−15

)

=


∣

∣ρ < 5

√

1− 1625

sin2 θ −4cosθ , θ ∈ (0,2π)

,

...

S(0) =


∣

∣ρ <

1/cosθ , θ ∈(

− π2 ,

π2

)

+∞, otherwise

.

Note thatS(−1)⊂ S(−1/2)⊂ S(−1/5)⊂ S(0), i.e. the convergence-region greatlyenlarges asc0 → 0, as shown in Fig. 5.1. The convergence-domainS(c0) is in theshape of a circle: its centre is at(1+ c−1

0 ), and its radius equals to|c−10 |. Note that

c0 =−1 corresponds to the traditional Taylor series which converges in the domain∣

∣

∣

∣

z− z0

ζ − z0

∣

∣

∣

∣

< 1,

i.e. the traditional Taylor series converges in a circle with the centre atz = z0

and the radiusζ − z0. However, the convergence-domain greatly enlarges as theconvergence-control parameter tends to zero. Especially,whenc0 → 0, the radius|c−1

0 | of the convergence-domainS(c0) tends to infinity so that the generalizedTaylor-series converges in an infinite domain, i.e.

|η |=∣

∣

∣

∣

z− z0

ζ − z0

∣

∣

∣

∣

<

1/cosθ , θ ∈(

− π2 ,

π2

)

,+∞, otherwise,

whereη = ρ exp(i θ ). In theη-plane, it is a half-plane

S(0) =

(x′,y′)∣

∣

∣x′ < 1,−∞ < y′ <+∞


Fig. 5.1 Convergence-domain of the generalizedTaylor series of a complexfunction with unique singularpoint atz= ζ . Solid line:the boundary of convergence-domainS(−1) of a traditionalTaylor series; Dashed-line:the boundary ofS(−1/2);Dash-dotted line: the bound-ary of S(−1/5); Dash-dot-dotted line: the boundary ofS(−1/10); Long-dashed line:the boundary ofS(0).

-20 -15 -10 -5 0-12

-8

-4

0

4

8

12

S(0)

S(-0.1)

S(-0.2)

η - plane

Note that the convergence-control parameterc0 is a real number in Theorem 5.3.If a complex convergence-control parameterc0 is used, we have a more generaltheorem.

Theorem 5.4.Let i=√−1 denote the imaginary unit,

c0 =−1+ ε ei γ , ε ∈ [0,1), γ ∈ [0,2π),

be a complex convergence-control parameter, and A(q) = q and B(q) = q be twodeformation-functions, respectively. If a complex function f(z) is analytic in thewhole z-plane except at the unique singular point z= ξ , its generalized Taylor seriesat z0 6= ξ , i.e.

f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B)

converges to f(z) in the region

ρ =

∣

∣

∣

∣

z− z0

ξ − z0

∣

∣

∣

∣

<ε[ε cosθ − cos(θ − γ)]+

√∆

1−2ε cosγ + ε2 , ε ∈ [0,1),γ ∈ [0,2π), (5.18)

where

∆ = ε2 [ε cosθ − cos(θ − γ)]2+(1− ε2)(

1−2ε cosγ + ε2) (5.19)

andη =

z− z0

ξ − z0= ρ ei θ , θ ∈ [0,2π ].

Especially, in case ofε = 1, i.e. c0 =−1+ei γ , it holds

0≤ ρ < max

cosθ − cos(θ − γ)1− cosγ

,0

, (5.20)


where0≤ θ < 2π , 0≤ γ < 2π .

Proof. SinceA(q) = B(q) = q and there exists only one singular pointz= ξ , ac-cording to Theorem 5.1, we have two singular pointsζ0 andζ1, governed by

1− (1+ c0) ζ0 = 0

and

1− (1+ c0) ζ1+ c0

(

z− z0

ξ − z0

)

ζ1 = 0,


−1

and

ζ1 =

[

1+ c0− c0

(

z− z0

ξ − z0

)]−1

.


f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B)

converges tof (z) in the region|ζ0|> 1∩|ζ1|> 1. |ζ0|> 1 gives|1+c0|< 1, whichis automatically satisfied since

c0 =−1+ ε ei γ , 0≤ ε < 1, 0≤ γ < 2π .


∣

∣

∣

1+ c0− c0

(

z− z0

ξ − z0

)∣

∣

∣

∣

< 1.

Writing

η =z− z0

ξ − z0= ρ ei θ , θ ∈ [0,2π ]

and using the definition ofc0 mentioned above, we have∣

∣

∣εei γ −

(

ε ei γ −1)

ρ eiθ∣

∣

∣< 1,

i.e.[

1−2ε cosγ + ε2] ρ2−2ε [ε cosθ − cos(θ − γ)] ρ + ε2 < 1,

whose solution reads

ε[ε cosθ − cos(θ − γ)]−√

∆1−2ε cosγ + ε2 < ρ <

ε[ε cosθ − cos(θ − γ)]+√

∆1−2ε cosγ + ε2 ,

where


∆ = ε2 [ε cosθ − cos(θ − γ)]2+(1− ε2)(

1−2ε cosγ + ε2) .

Obviously,ε[ε cosθ − cos(θ − γ)]−

√∆

1−2ε cosγ + ε2 < 0.

Sinceρ ≥ 0, we have the solution

0≤ ρ <ε[ε cosθ − cos(θ − γ)]+

√∆

1−2ε cosγ + ε2 .

Especially, whenε = 1, i.e.c0 =−1+exp(i γ), the above expression gives

0≤ ρ < max

cosθ − cos(θ − γ)1− cosγ

,0

.


Obviously, Theorem 5.3 is a special case of Theorem 5.4 whenγ = 0. Thus,as mentioned above, whenγ = 0 andε → 1, i.e. c0 → 0, the generalized Taylorseries converges tof (z) in the halfη-plane Re(η) < 1, as shown in Fig.5.1. Now,regardingc0 as a complex number, there are an infinite number of differentwaysfor c0 = 0 to approach zero. For example, let us consider a special case: ε = 1andγ → 0, i.e. c0 tends to zero along the circle|c0 + 1| = 1. Note that there aretwo different ways to approachc0 = 0 even along the circle: from above (γ > 0)or below (γ < 0). LetS0(γ) denote the convergence-domain given by (5.20),S0(0+)andS0(0−) denote the convergence-domainasγ tends to zero from above and below,respectively. It is found that, asγ tends to zero from above, the convergence-domainS0(γ) enlarges, for example,

S0

(π4

)

⊂ S0

(π9

)

⊂ S0

( π18

)

⊂ S0

( π36

)

⊂ S0(

0+)

,

as shown in Figs.5.2. Especially, asγ → 0 from above, the convergence-domainS0(0+) becomes the below half-plane Im(η)< 0, i.e.

S0(

0+)

=

(x′,y′)∣

∣

∣y′ < 0,−∞ < x′ <+∞

in the planeη = (x′,y′). Similarly, asγ tends to zero from below, the convergence-domainS0(γ) also enlarges, for example,

S0

(

−π4

)

⊂ S0

(

−π9

)

⊂ S0

(

− π18

)

⊂ S0

(

− π36

)

⊂ S0(

0−)

,

as shown in Figs.5.3. Besides, asγ → 0 from below, the convergence-domainS0(0−)becomes the above half-plane Im(η)> 0, i.e.

S0(

0−)

=

(x′,y′)∣

∣

∣y′ > 0,−∞ < x′ <+∞


in the planeη = (x′,y′). It should be emphasized thatS0(0+)∪S0(0−) nearly coversthewholeη-plane except the real axis Im(η) = 0.

Fig. 5.2 Convergence-domain of the generalizedTaylor series of a complexfunction f (z) with uniquesingular point atz= ζ whenc0 = 1+ei γ asγ → 0 fromabove. Solid line: the bound-ary of convergence-domainS0(π/4); Dashed-line: theboundary ofS0(π/9); Dash-dotted line: the boundaryof S0(π/18); Dash-dot-dotted line: the boundaryof S0(π/36); Long-dashedline: the boundary ofS0(0+).

-15 -10 -5 0 5 10 15

-25

-20

-15

-10

-5

0

S0(π/18)

S0(π/36)

S+0(0)

η - plane

Fig. 5.3 Convergence-domain of the generalizedTaylor series of a complexfunction f (z) with uniquesingular point atz= ζ whenc0 = 1+ ei γ as γ → 0 be-low. Solid line: the bound-ary of convergence-domainS0(−π/4); Dashed-line: theboundary ofS0(−π/9); Dash-dotted line: the boundaryof S0(−π/18); Dash-dot-dotted line: the boundary ofS0(−π/36); Long-dashedline: the boundary ofS0(0−).

-15 -10 -5 0 5 10 15-5

0

5

10

15

20

25

S0(−π/18)

S0(−π/36)

S-0(0)

η - plane

S-0(0)

We can prove this by means of Theorem 5.4. According to (5.20), asγ → 0, ρhas an expression of 0/0. Then, using Bernoulli’s rule in calculus, we have

ρ < max

−sin(θ )sinγ

,0

.

Thus, asγ > 0 tends to zero,ρ tends to infinity when sinθ < 0, i.e.−π < θ < 0,so thatS0(0+) is the below half-plane Im(η)< 0 in theη-plane. Similarly, asγ < 0tends to zero,ρ tends to infinity when sinθ > 0, i.e. 0< θ < π , so thatS0(0−) isthe above half-plane Im(η)> 0 in theη-plane.


Since there are an infinite number of different ways to approach c0 = 0 in thez-plane, we further consider such an approach

ε =sinδ

sin(δ + γ), γ → 0,

whereδ γ > 0, δ ∈ (−π/2,π/2),δ+γ ∈ (−π/2,π/2). In this case, the convergence-domain of the generalized Taylor series is dependent uponδ and γ, denoted byΩ(δ ,γ). Without loss of generality, let us consider two cases:δ =±π/4. It is foundthat, in case ofδ = π/4, the convergence-domain enlarges asγ → 0 from above, i.e.

Ω(π

4,

π9

)

⊂ Ω(π

4,

π18

)

⊂ Ω(π

4,

π36

)

⊂ Ω(π

4,

π72

)

⊂ Ω(π

4,0)

,

as shown in Fig. 5.4. Especially, asγ → 0 from above, the convergence-domainbecomes a half plane

Ω(π

4,0)

=

(x′,y′)∣

∣

∣y′ <−x′+1, −∞ < x′ <+∞

in the η-plane! Similarly, in case ofδ = −π/4, the convergence-domain enlargesasγ → 0 from below, i.e.

Ω(

−π4,−π

9

)

⊂ Ω(

−π4,− π

18

)

⊂ Ω(

−π4,− π

36

)

⊂ Ω(

−π4,− π

72

)

⊂ Ω(

−π4,0)

,

as shown in Fig. 5.5. Especially, asγ → 0 from below, the convergence-domainbecomes a half plane

Ω(

−π4,0)

=

(x′,y′)∣

∣

∣y′ < x′−1, −∞ < x′ <+∞

in theη-plane!Besides, the convergence-domains of the generalized Taylor series in cases of

δ = ±π/4 andδ = ±35π/36 are as shown in Fig. 5.6. It is easy to prove that,asδ → π/2 andγ → 0 from above, the convergence-domain becomes the belowhalf-plane Im(η)< 0, i.e.

Ω(π

2,0)

=

(x′,y′)∣

∣

∣y′ < 0, −∞ < x′ <+∞

in the η = (x′,y′) plane. Similarly, asδ → −π/2 and γ → 0 from below, theconvergence-domain becomes the above half-plane Im(η)> 0, i.e.

Ω(π

2,0)

=

(x′,y′)∣

∣

∣y′ > 0, −∞ < x′ <+∞


Fig. 5.4 Convergence-domain of the generalizedTaylor series of a com-plex function f (z) withunique singular point atz= ζ whenc0 = 1+ εei γ ,ε = sinδ/sin(δ + γ) asγ → 0. Solid line: theboundary of convergence-domain Ω(π/4,π/9);Dashed-line: the boundaryof Ω(π/4,π/18); Dash-dotted line: the boundary ofΩ(π/4,π/36); Dash-dot-dotted line: the boundaryof Ω(π/4,π/72); Long-dashed line: the boundary ofΩ(π/4,0).

-35 -30 -25 -20 -15 -10 -5 0 5 10-35

-30

-25

-20

-15

-10

-5

0

5

10

Ω(π/4,π/36)

Ω(π/4,π/72)

Ω(π/4,0)

η - Plane

Fig. 5.5 Convergence-domain of the generalizedTaylor series of a com-plex function f (z) withunique singular point atz= ζ whenc0 = 1+ εei γ ,ε = sinδ/sin(δ + γ) asγ → 0. Solid line: theboundary of convergence-domainΩ(−π/4,−π/9);Dashed-line: the boundaryof Ω(−π/4,−π/18); Dash-dotted line: the boundary ofΩ(−π/4,−π/36); Dash-dot-dotted line: the boundary ofΩ(−π/4,−π/72); Long-dashed line: the boundary ofΩ(−π/4,0).

-35 -30 -25 -20 -15 -10 -5 0 5 10-10

-5

0

5

10

15

20

25

30

35

Ω(−π/4,−π/36)

Ω(−π/4,−π/72)

Ω(−π/4,0)

η - Plane

in the planeη =(x′,y′). All of these show that, as the convergence-controlparameterc0 → 0, the convergence-domainΩ is dependent upon its approach-curve alongwhich c0 tends to zero: the convergence-domain is a different half-plane in theη-plane for a different way ofc0 → 0.

Recall thatc0 =−1+ε exp(i γ), wherec0 is a complex number. As shown above,asγ = 0 andε → 1, i.e.c0 → 0 along the parallel-axis, then the convergence-domainof the generalized Taylor series becomes a half-plane Re(η) < 1. In case ofε = 1,i.e. c0 → 0 along the circle|1+ c0| = 1, the convergence domain is a below half-plane Im(η) < 0 whenγ → 0 from above, but is an above half-plane Im(η) > 0whenγ → 0 from below. So, if a complex functionf (z) has an unique singular pointat z= ζ , we can always find a complex convergence-control parameter|1+ c0|< 1such that the corresponding generalized Taylor series converges in the wholeη-


Fig. 5.6 Convergence-domain of the generalizedTaylor series of a com-plex function f (z) withunique singular point atz= ζ whenc0 = 1+ εei γ ,ε = sinδ/sin(δ + γ) asγ → 0. Solid line: theboundary of convergence-domainΩ(−π/4,−π/72);Dashed-line: the boundaryof Ω(π/4,π/72); Dash-dotted line: the boundary ofΩ(−35π/36,−π/72); Dash-dot-dotted line: the boundaryof Ω(35π/36,π/72).

-50 -25 0 25 50

-50

-25

0

25

50

η - Plane

plane except the half of the real axis Im(η) = 0 and Rm(η)≥ 1, i.e.

(x′,y′)∣

∣

∣x′ ≥ 1,y′ = 0

,

whereη = (x′,y′) =

z− z0

ζ − z0.

Thus, we have the following theorem:

Theorem 5.5.Let c0 denote a complex convergence-control parameter, A(q) = qand B(q) = q be two deformation-functions, respectively. If a complexfunction f(z)has a unique pole at z= ζ in the whole z-plane, its generalized Taylor series atz0 6= ζ , i.e.

f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B)

can converge to f(z) in the whole z-plane except the points on a half-line

z= z0+ρ (ζ − z0), ρ ≥ 1.

So, obviously, if f (z) has an unique pole atz= ζ , we can find arbitrary twonon-singular pointsz0 6= ζ andz′0 6= ζ , which are not collinear withζ , so that atany a given point in the wholez-plane except the polez= ζ , at least one of thetwo generalized Taylor series atz= z0 andz= z′0 converges tof (z). So, we haveobviously the following theorem:

Theorem 5.6.Let c0 denote a complex convergence-control parameter, A(q) = qand B(q) = q be two deformation-functions, respectively. Letζ denote an uniquepole of a complex function f(z), z0 6= ζ and z′0 6= ζ are two non-singular points. Ifζ ,z0 and z′0 are not collinear, then for any a given z6= ζ in the whole plane, there


exist such a convergence-control parameter c0 that at least one of the generalizedTaylor series

f (z0)+ limm→+∞

m

∑k=1

[

f (k)(z0)

k!(z− z0)

k

]

Tm,k(c0,A,B),

and/or

f (z′0)+ limm→+∞

m

∑k=1

[

f (k)(z′0)k!

(z− z′0)k

]

Tm,k(c0,A,B),

converges to f(z).

Proof. Choose two arbitrary non-singular pointsz0 6= ζ andz′0 6= ζ , which are notcollinear. Letz 6= ζ be an arbitrary point in thez-plain. If z is not collinear withz0 andζ , then according to Theorem 5.5, there exists such a convergence-controlparameterc0 that the generalized Taylor series atz0 converges atz. If z is collinear withz0 andζ , thenzmust be not collinear withz′0 andζ . Then, according to Theorem 5.5, thereexists such a convergence-control parameterc′0 that the generalized Taylor series atz′0 converges atz. This ends the proof. ⊓⊔

Theorem 5.7.Let c0 be a convergence-control parameter, A(q) and B(q) are defor-

mation functions whose Maclaurin series A(q) =+∞∑

k=1ak qk and B(q) =

+∞∑

k=1bk qk are

convergent at q= 1, i.e.+∞

∑k=1

ak = 1,+∞

∑k=1

bk = 1.

If |1+ c0|< 1, then for any a finite integer k≥ 1, it holds

limm→+∞

Tm,k(c0,A,B) = 1. (5.21)

Proof. According to the definition ofTm,k(c0,A,B), it holds

limm→+∞

Tm,k(c0,A,B)

= limm→+∞

(−c0)k

m−k

∑n=0

n

∑r=0

(

k+ r −1r

)

(1+ c0)r

n−r

∑s=0

ak,k+sbr,n−s

= (−c0)k+∞

∑n=0

n

∑r=0

(

k+ r −1r

)

(1+ c0)r

n−r

∑s=0

ak,k+sbr,n−s

= (−c0)k+∞

∑r=0

+∞

∑n=r

(

k+ r −1r

)

(1+ c0)r

n−r

∑s=0

ak,k+sbr,n−s

= (−c0)k+∞

∑r=0

(

k+ r −1r

)

(1+ c0)r+∞

∑n=r

n−r

∑s=0

ak,k+sbr,n−s.

Since


+∞

∑n=r

n−r

∑s=0

ak,k+sbr,n−s=+∞

∑s=0

+∞

∑n=r+s

ak,k+sbr,n−s

=+∞

∑m=k

+∞

∑j=r

ak,mbr, j =

(

+∞

∑m=k

ak,m

)(

+∞

∑j=r

br, j

)

=

(

+∞

∑m=1

am

)k(+∞

∑j=1

b j

)r

= 1,

and+∞

∑r=0

(

k+ r −1r

)

= [1− (1+ c0)]−k = (−c0)

−k

due to|1+ c0|< 1, we have

limm→+∞

Tm,k(c0,A,B) = (−c0)k+∞

∑r=0

(

k+ r −1r

)

(1+ c0)r = (−c0)

k (−c0)−k = 1.


According to Theorem 5.7, it holds for any afinitepositive numberk≥ 1 that

limm→+∞

f (k)(z0)

k!(z− z0)

k Tm,k(c0,A,B) =f (k)(z0)

k!(z− z0)

k.

Therefore, the firstM terms of the generalized Taylor series, whereM is any afinitepositive number, are the same or very close to the corresponding first M terms ofthe traditional Taylor series. However, as shown above, thegeneralized Taylor seriesconverges in the much larger domain than the traditional one. Why? The key-pointis that Theorem 5.7 holds only for afinite positive integerk. It is well-known thatthe convergence of a series

a0+a1 t +a2 t2+ · · ·+am tm+ · · ·

depends on its property of the termscm asm→ +∞, i.e. the radius of convergenceof the above series is determined by

limm→+∞

∣

∣

∣

∣

am

am+1

∣

∣

∣

∣

.

Therefore, the property of the generalized Taylor series atinfinity is greatly modifiedby the convergence-control parameterc0 and the deformation-functions so that itsconvergence-region is enlarged fantastically.

Finally, we explain, from the another points of view, why theconvergence of thegeneralized Taylor series off (z), i.e.

+∞

∑k=0

Γk(z) Tm,k(c0,A,B),

where


Γk(z) =f (k)(z0)

k!(z− z0)

k,

can be fantastically modified by the convergence-control parameterc0 and thedeformation-functionsA(q) andB(q). Regard

P∗ = (Γ0(z),Γ1(z),Γ2(z),Γ3(z), · · · ,)

as a point of an infinite-dimension spaceR∞ (for example, a Hilbert space). Then,the traditional Taylor series off (z), i.e.

+∞

∑k=0

Γk(z),

corresponds to a limit which tends to the pointP∗ ∈ R∞ along such a traditionalpath:

(Γ0(z),0,0,0,0, · · ·) ∈ R∞ ,

(Γ0(z),Γ1(z),0,0,0, · · ·) ∈ R∞ ,

(Γ0(z),Γ1(z),Γ2(z),0,0, · · ·) ∈ R∞ ,

(Γ0(z),Γ1(z),Γ2(z),Γ3(z),0, · · ·) ∈ R∞ ,

...


+∞

∑k=0

Γk(z) Tm,k(c0,A,B),

corresponds to the same pointP∗ ∈ R∞ as the traditional Taylor series

+∞

∑k=0

Γk(z).

However, the generalized Taylor series corresponds to afamily of limits whichtend to thesamepoint P∗ ∈ R∞ along different paths that are dependent upon theconvergence-control parameterc0 and two deformation-functionsA(q) andB(q),i.e.


(Γ0(z) T0,0(c0,A,B),0,0,0,0, · · ·) ∈ R∞ ,

(Γ0(z) T1,0(c0,A,B),Γ1(z) T1,1(c0,A,B),0,0,0, · · ·) ∈ R∞ ,

(Γ0(z) T2,0(c0,A,B),Γ1(z) T2,1(c0,A,B),Γ2(z) T2,2(c0,A,B),0,0, · · ·) ∈ R∞ ,

...

It is well-known that a limit of a function with multiple variables might be quitedifferent for different paths. For example, the limit

lim(x,y)→(0,0)

√

x2+ y2

|x| =√

1+α2

is dependent on an arbitrary real numberα, if we gain the limit along the differentapproach pathsy= αx. This is exactly the reason why the convergence-domain ofthe generalized Taylor series of a complex functionf (z) with a unique pole atz= ζconverges tof (z) in the whole half-plane Im(η) < 0 asc0 tends to zero along thecircle |1+c0|= 1 from above, but in the whole half-plane Im(η)> 0 asc0 tends tozero below, respectively.

All of these show in theory that the convergence of a series can be indeedmodified fantastically by introducing a non-zero auxiliaryparameterc0, called theconvergence-control parameter. This well explains why theconvergence-control pa-rameterc0 in the frame of the HAM can guarantee the convergence of homotopy-series. Note that the generalized Taylor series depends upon a convergence-controlparameterc0 and two deformation-functionsA(q) andB(q), although most of theabove theorems are given for the simplest deformation functionsA(q) = B(q) = q.Note that, the convergence-control parameter can even be a complex number in thez-plane|1+ c0| < 1, and the two deformation functionsA(q) andB(q) can be alsocomplex functions! Obviously, using better deformation-functions, it is quite pos-sible for us to get even larger convergence-domain of a generalized Taylor series.Similarly, in the frame of the HAM, we have extremely large freedom to choosenot only the convergence-control parameterc0, but also the initial approximation,the auxiliary linear operator, and the deformation-functions to construct differenttypes of zeroth-order deformation equations. Therefore, the HAM provides us moredegrees of freedom to control and adjust the convergence of homotopy-series.

Finally, we point out that the convergence-control is an important concept of theHAM. Here, weprovethat the convergence of an infinite series can be indeed con-trolled and adjusted by means of a non-zero auxiliary parameterc0. This provides acorner-stone for the homotopy- analysis method in theory.

5.3 Homotopy transform 215

5.3 Homotopy transform

Write the sequencesn =n∑

k=0uk for a series

+∞∑

k=0uk. Due to Agnew’s [2] definition,

the Euler transform, denoted byE (λ ), of the sequencesn is the sequencewndefined by

wn =n

∑k=0

(

nk

)

λ k (1−λ )n−k sk. (5.22)

The Euler transform was developed by Leonhard Euler (15 April 1707 – 18 Septem-ber 1783), a pioneering Swiss mathematician. Today, it is widely applied to accel-erate convergence of a sequence or even to make a divergent series convergent.

Using Tm,k(c0,A,B) defined by (5.6) and (5.7) – (5.9), Liao [15] defined a newtransform of a sequence, namely the homotopy-transform, and proved that it logi-cally contains the Euler-transformE (λ ).

Definition 5.3. Let c0 6= 0 denote the convergence-control parameter,A(q) and

B(q) be two deformation-functions whose Maclaurin seriesA(q) =+∞∑

k=1ak qk and

B(q)=+∞∑

k=1bk qk converge atq= 1 so that

+∞∑

k=1ak = 1 and

+∞∑

k=1bk = 1. The so-called

homotopy-transform, denoted byH (c0,A,B), of a series+∞∑

k=0uk, is a sequence

µn defined by

µn = u0+n

∑k=1

uk Tn,k(c0,A,B), (5.23)

whereTn,k(c0,A,B) is given by (5.6) under the definitions (5.7)-(5.9). The series+∞∑

k=0uk is called summable by the homotopy-transformH (c0,A,B) if µn tends to

a bounded value asn→+∞.

Lemma 5.2 Let c0 denote a convergence-control parameter. If A(q) = B(q) = q,then

Tm,k(c0,A,B) = µm,k0 (c0), k≥ m, (5.24)

where Tm,k(c0,A,B) is defined by (5.6) and (5.7) – (5.9), µm,k0 (c0) is defined by

µm,k0 (c0) = (−c0)

km−k

∑n=0

(

n+ k−1n

)

(1+ c0)n, m≥ k≥ 1. (5.25)


The same definition is given in (2.82).

Proof. WhenA(q) = B(q) = q, according to the definitions (5.7) – (5.9), we have

am,n = bm,n =

1, whenm= n,0, when n> m.

Then, according to the definitions (5.6) and (5.25), it holdsfor m≥ k≥ 1 that

Tm,k(c0,A,B) = (−c0)k

m−k

∑n=0

n

∑r=0

(

k+ r −1r

)

(1+ c0)r ak,k br,n

= (−c0)k

m−k

∑n=0

(

k+n−1n

)

(1+ c0)r ak,k bn,n

= (−c0)k

m−k

∑n=0

(

k+n−1n

)

(1+ c0)r

= µm,k0 (c0).

This completes the proof. ⊓⊔

Theorem 5.8.Write the sequence sn =n∑

k=0uk for a series

+∞∑

k=0uk. Letλ be a complex

number and c0 a convergence-control parameter. The Euler transformE (λ ) of thesequencesn is the same as the homotopy-transformH (c0,A,B) of the series+∞∑

k=0uk if A(q) = B(q) = q and c0 =−λ .

Proof. Due to Agnew’s [2] definition (5.22) of Euler transformE (λ ), the sequenceµm given by the Euler transform of the sequencesn reads

µm =m

∑k=0

(

mk

)

λ k (1−λ )m−k sk

=m

∑k=0

(

mk

)

λ k (1−λ )m−kk

∑n=0

un

=m

∑n=0

un

m

∑k=n

(

mk

)

λ k (1−λ )m−k

= u0

m

∑k=0

(

mk

)

λ k (1−λ )m−k+m

∑n=1

un

m

∑k=n

(

mk

)

λ k (1−λ )m−k,

which gives, since

m

∑k=0

(

mk

)

λ k (1−λ )m−k = [λ +(1−λ )]m= 1,

that

5.3 Homotopy transform 217

µm = u0+m

∑n=1

un

m

∑k=n

(

mk

)

λ k (1−λ )m−k (5.26)

According to Lemma 5.2, it holdsTm,n(c0,A,B) = µm,n0 (c0) whenA(q) = B(q) =

q. Therefore, by means of the definition (5.25), the sequenceµm given by thehomotopy-transformH (c0,A,B) in case ofc0 =−λ andA(q) = B(q) = q is givenby

µm = u0+m

∑n=1

un µm,n0 (−λ ) = u0+

m

∑n=1

un λ nm−n

∑k=0

(

n+ k−1k

)

(1−λ )k. (5.27)

Enforcingµm = µm and comparing (5.26) with (5.27), it remains to show that

λ nm−n

∑k=0

(

n+ k−1k

)

(1−λ )k =m

∑k=n

(

mk

)

λ k (1−λ )m−k, 1≤ n≤ m. (5.28)

When 1≤ n≤ m, we have

λ nm−n

∑k=0

(

n+ k−1k

)

(1−λ )k

= λ nm−n

∑k=0

(

n+ k−1k

) k

∑r=0

(

kr

)

(−λ )r

=m−n

∑r=0

(−λ )n+r(−1)nm−n

∑k=r

(

n+ k−1k

)(

kr

)

(5.29)

and

m

∑k=n

(

mk

)

λ k (1−λ )m−k

=m

∑k=n

(

mk

)

λ km−k

∑r=0

(

m− kr

)

(−λ )r

=m

∑k=n

(

mk

)

(−1)km−k

∑r=0

(

m− kr

)

(−λ )k+r

=m−n

∑k=0

(

mk+n

)

(−1)k+nm−n−k

∑r=0

(

m−n− kr

)

(−λ )n+k+r

=m−n

∑s=0

(−λ )n+s(−1)ns

∑k=0

(−1)k(

mk+n

) (

m−n− ks− k

)

=m−n

∑r=0

(−λ )n+r(−1)nr

∑k=0

(−1)k(

mk+n

)(

m−n− kr − k

)

=m−n

∑r=0

(−λ )n+r(−1)nr

∑k=0

(−1)k(

mm−n− r

)(

r +nk+n

)

, (5.30)


where we use such a formula(

mk+n

) (

m− k−nr − k

)

=

(

mm−n− r

) (

r +nk+n

)

,

which holds in the relevant ranges. So, by (5.28), (5.29) and(5.30), we need to show

m−n

∑k=r

(

n+ k−1k

)(

kr

)

=r

∑k=0

(−1)k(

mm−n− r

)(

r +nk+n

)

, (5.31)

where 1≤ n≤ m. Noticing that, forn≥ 1 and sufficiently smallx,

+∞

∑r=0

xr(

n+ r −1r

)

= (1− x)−n,

and that

+∞

∑r=0

xrr

∑k=0

(−1)k(

r +nk+n

)

=+∞

∑k=0

(−1)k+∞

∑r=k

(

r +nk+n

)

xr =+∞

∑k=0

(−1)k xk+∞

∑r=0

(

r + k+nk+n

)

xr

=+∞

∑k=0

(−1)k xk (1− x)−n−k−1 = (1− x)−n−1+∞

∑k=0

(−1)k xk (1− x)−k

= (1− x)−n,

it holds(

n+ r −1r

)

=r

∑k=0

(−1)k(

r +nk+n

)

. (5.32)

According to (5.31) and (5.32), we need to show

m−n

∑k=r

(

n+ k−1k

)(

kr

)

=

(

mm−n− r

)(

n+ r −1r

)

.

Noticing that

m−n

∑k=r

(

n+ k−1k

)(

kr

)

=m−n−r

∑k=0

(

n+ k+ r −1k+ r

)(

k+ rr

)

=m−n−r

∑k=0

(k+ r +n−1)!(n−1)! k! r!

5.4 Relation between homotopy analysis method and Euler transform 219

=

(

n+ r −1r

)m−n−r

∑k=0

(

n+ k+ r −1k

)

,

this reduces to show that(

mm−n− r

)

=m−n−r

∑k=0

(

n+ k+ r −1k

)

.

Writing i = n+ r, the above expression reads

(

mm− i

)

=m−i

∑k=0

(

k+ i −1k

)

=m−i

∑k=0

(

k+ i −1i −1

)

=m−1

∑j=i−1

(

ji −1

)

.

In order to prove this, we use the formula

N

∑j=n

(

jn

)

=

(

N+1n+1

)

in a handbook of mathematics [1]. SettingN= m−1 andn= i−1 in above formulagives

m−1

∑j=r−1

(

ji −1

)

=

(

mi

)

=

(

mm− i

)

.

This completes the proof. ⊓⊔Remark 5.1 According to Theorem 5.8, the Euler transformE (λ ) is only a spe-cial case of the so-called homotopy-transformH (c0,A,B) defined by (5.23) whenc0 = −λ andA(q) = B(q) = q, corresponding toa1 = b1 = 1 andak = bk = 0 fork> 1.

Here, we would like to emphasize two points. First, Euler transformE (λ ) isonly a special case of the so-called homotopy-transformation H (c0,A,B). Thus,the homotopy-transform is more general. Secondly, Euler transform is widely usedto accelerate convergence of a series or to make a divergent series convergent. Thus,the homotopy-transformH (c0,A,B) provides us with a new but more general wayto accelerate convergence of a series or to make a divergent series convergent.

5.4 Relation between homotopy analysis method and Eulertransform

It is very interesting that the so-called homotopy-transformH (c0,A,B) defined by(5.23) can be obtained in the frame of the HAM, as shown by Liao[15].

To illustrate this, let us consider here a nonlinear ordinary differential equation


u′(z)+u(z)

[

1− 12

u(z)

]

= 0, u(0) = 1, (5.33)

where the prime denotes the differentiation with respect toz. This equation has aclosed-form solutionu(z) = 2/(1+ez).

Let c0 6= 0 denote a convergence-control parameter,q ∈ [0,1] the homotopy-parameter,α(q) andβ (q) be two deformation-functions satisfying

α(0) = β (0) = 0, α(1) = β (1) = 1, (5.34)

and their Maclaurin seriesα(q) =+∞∑

k=1αk qk andβ (q) =

+∞∑

k=1βk qk are convergent at

q= 1, respectively. Chooseu0(z) = 1 as the initial approximation andL u= u′ asthe auxiliary linear operator, respectively, and define thenonlinear operator

N u=dudz

+u

(

1− 12

u

)

. (5.35)

Then, we construct the zeroth-order deformation equation

[1−α(q)]L [u(z;q)−u0(z)] = c0 β (q) N [u(z;q)], u(0;q) = 1. (5.36)

Themth-order homotopy-approximation is given by

u(z)≈ u0(z)+m

∑m=1

um(z), (5.37)

whereum(z) is governed by themth-order deformation equation

L

[

um(z)−m−1

∑k=1

αk um−k(z)

]

= c0

m−1

∑k=1

βm−k δk−1(z), um(0) = 0, (5.38)

with the definition

δn(z) = DnN [u(z;q)]= u′n(z)+un(z)−12

n

∑k=0

uk(z) un−k(z). (5.39)

Note that we have great freedom to choose the deformation functionsα(q) andβ (q) in the zeroth-order deformation equation (5.36). LetA(q) andB(q) denote twodeformation-functions satisfyingA(0) = B(0) = 0 andA(1) = B(1) = 1, and their

Maclaurin seriesA(q) =+∞∑

k=1ak qk andB(q) =

+∞∑

k=1bk qk converge atq= 1, i.e.

+∞

∑k=1

ak = 1,+∞

∑k=1

bk = 1.

To derive the so-called homotopy-transform, we define

5.4 Relation between homotopy analysis method and Euler transform 221

α(q) = B(q)+ c0[B(q)−A(q)] =+∞

∑k=1

[(1+ c0) bk− c0 ak] qk,

β (q) = A(q) =+∞

∑k=1

ak qk,

i.e.αk = (1+ c0) bk− c0 ak, βk = ak.

Then, sinceu′0(z) = 0, the zeroth-order deformation equation (5.36) becomes

[1− (1+ c0) B(q)+ c0 A(q)]∂ u(z;q)

∂z

= c0 A(q)

∂ u(z;q)∂z

+ u(z;q)

[

1− 12

u(z;q)

]

(5.40)

and the correspondingmth-order(m≥ 1) deformation equation reads

ddz

um(z)−m−1

∑k=1

[(1+ c0) bm−k− c0 am−k] uk(z)

= c0

m−1

∑k=1

am−k δk−1(z), (5.41)


um(0) = 0. (5.42)

The solution of the above high-order deformation equation reads

um(z) =m−1

∑k=1

[(1+ c0) bm−k− c0 am−k]uk(z)

+c0

m−1

∑k=1

am−k

∫ z

0

[

u′k(z)+uk(z)−12

k

∑i=0

ui(z) uk−i(z)

]

dx. (5.43)

Thus, using the initial approximationu0(z) = 1 and above recurrence formula, wecan getu1(z),u2(z) and so on. It is found that the correspondingmth-order approxi-mation reads

u0(z)+m

∑k=1

uk(z) = 1+m

∑k=1

(

γk zk)

Tm,k(c0,A,B), (5.44)

whereTm,k(c0,A,B) is exactly defined by (5.6) and (5.7) – (5.9), and 1++∞∑

k=1γk zk

is the Taylor series of the closed-form solutionu(z) = 2/(1+ez) of (5.33). This isindeed a surprise!

We can prove the correctness of (5.44) in another way. Noticethat the zeroth-order deformation equation (5.40) can be rewritten as


[

1− (1+ c0) B(q)−c0 A(q)

]

∂ u(z;q)∂z

+ u(z;q)

[

1− 12

u(z;q)

]

= 0, (5.45)

i.e.

dudτ

+ u

(

1− 12

u

)

= 0, (5.46)

whose solution, satisfying the initial condition ˜u= 1 atz= 0, is exactly

u(z;q) =2

1+exp(τ)(5.47)

where

τ =−c0 A(q) z

1− (1+ c0) B(q)

is a special case ofτ whenz0 = 0, as defined by (5.11). Similarly, expanding ˜u(z;q)in Maclaurin series ofq and then settingq= 1, we get the homotopy-series

u(z) = u(z;1) = 1+ limm→+∞

m

∑k=1

(

γk zk)

Tm,k(c0,A,B), (5.48)

whereTm,k(c0,A,B) is exactly defined by (5.6) and (5.7) – (5.9), andγk is the coef-ficient of the Taylor series of the exact solutionu(z) = 2/(1+ez).

Therefore, the so-called homotopy transform described in§ 5.3 can be indeedderived in the frame of the HAM in some special cases. As proved in § 5.3, thefamous Euler transformE (λ ) is only a special case of the homotopy transformH (c0,A,B) in case ofc0 =−λ andA(q)=B(q)= q. Thus, for some special choicesof the initial approximation and the auxiliary linear operator, the HAM in case ofA(q) = B(q) = q are sometimes equivalent to the famous Euler transform.

In theory, this fact explains why the convergence of homotopy-series given bythe HAM can be guaranteed, because the Euler transform is widely applied to ac-celerate the convergence of a series or to make a divergent series convergent. On theother hand, it should be emphasized that the homotopy analysis method is more gen-eral than Euler transform, because we have extremely large freedom to choose notonly different types of deformation functionsA(q) andB(q), but also the auxiliarylinear operatorL and the initial approximation. Note that the homotopy-transformH (c0,A,B) defined by (5.23) is obtained in the frame of the homotopy analysismethod by using a special initial approximationu0(z) = 1 and the special auxil-iary linear operatorL u= u′ for the considered example. However, by means of thehomotopy analysis method, we have great freedom to choose other initial approx-imations and other auxiliary linear operators. For example, if the auxiliary linearoperatorL u= u′+κu and the initial approximationu0(x) = exp(−κ x) are chosenfor the considered simple example, whereκ > 0 is the second auxiliary parameter,we can obtain approximations expressed by exponential basefunctions

5.5 Concluding remarks 223

exp(−κx),exp(−2κx),exp(−3κx), · · ·

Obviously, such kind of homotopy-approximations contain two non-zero auxiliaryparametersc0 andκ , and thus is more general than Euler transformE (λ ) that hasonly one auxiliary parameterλ .

Euler transform is widely used to accelerate the convergence of a sequence oreven to make a divergent series convergent. In this chapter,it is proved that the fa-mous Euler transform is a special case of the so-called homotopy-transform, whichcan be derived in the frame of the HAM by means of special initial approxima-tion and auxiliary linear operator. This further explains in theory why the HAM canguarantee the convergence of homotopy-series, and why it isgenerally valid for somany highly nonlinear equations.

5.5 Concluding remarks

The convergence-control is a key concept of the HAM: it is theconvergence-controlparameterc0 that provides us a convenient way to control and adjust the convergenceof homotopy-series, so that the HAM is valid even for strongly nonlinear problems.In Chapter 2, we proved that, by introducing a non-zero auxiliary parameterc0,the convergence-domain of the power series of a real function (1+ z)−1 can befantastically enlarged to the whole real axis except the singular pointz= −1 only.In this Chapter, we furtherprovethis idea in general. Note that the traditional Taylorseries of a complex functionf (z), i.e.

f (z0)++∞

∑k=1

f (k)(z0)

k!(z− z0)

k,

converges only within a circle∣

∣

∣

∣

z− z0

ζ − z0

∣

∣

∣

∣

< 1,

whereζ is a unique pole off (z). In the z-plane, the convergence-domain of thetraditional Taylor series is within a circle with the radiusr = |ζ − z0|. However,according to Theorem 5.5, by means of introducing a so-called convergence-controlparameterc0, the generalized Taylor series

f (z0)+ limm→+∞

m

∑k=1

f (k)(z0)

k!(z− z0)

k Tm,k(c0,A,B),

can converge tof (z) in thewhole z-plane except a half-line

η =z− z0

ζ − z0> 1.


The comparison of the convergence-domains of the traditional and the generalizedTaylor series is as shown in Fig.5.7, which clearly illustrates that the convergence-domain of a series can be indeed fantastically enlarged in general by introducinga non-zero auxiliary parameterc0, called the convergence-control parameter. Thiswell explains why the so-called convergence-control parameterc0 can guaranteethe convergence of homotopy-series in the frame of the HAM ingeneral. Moreimportantly, it provides a corner-stone for the concept of convergence-control of theHAM in theory.

z0

ζ

Convergence-domain oftranditional Taylor series

z0

ζ

Convergence-domain ofgeneralized Taylor series

Fig. 5.7 Comparison of the convergence-domains of the traditional and the generalized Taylorseries of a complex functionf (z) with unique singular point atz= ζ .

Such kind of generalized Taylor series is further used to define the so-calledhomotopy-transform for a given sequence, which is unnecessary to be a power se-ries. It is well known that the Euler transform is widely applied to accelerate asequence or even to make a divergent sequence convergent. However, we provethat the famous Euler transform is only a special case of the homotopy transform.Besides, we illustrate that the homotopy-transform can derive in the frame of theHAM by means of using the simplest deformation-functionsA(q) = B(q) = q. Thisexplains from the another view point why the HAM can guarantee the convergenceof series solution. In addition, this fact also shows the generality and great potentialof the HAM.

Note that, unlike the Taylor series for a given function and Euler transformfor a given sequence, the HAM is for nonlinear differential equations in general.Therefore, it is the HAM that introduces the convergence-control parameterc0 anddeformation-functions into the nonlinear differential equations in general so as toguarantee the convergence of series solutions of nonlineardifferential equations.Note that, the HAM provides us extremely large freedom not only to choose theconvergence-control parameterc0 and deformation-functionsA(q) and B(q), butalso to choose the auxiliary linear operatorL and the initial approximation. There-


fore, the HAM is much more general than the Euler transform, and thus should bevalid for more complicated nonlinear problems in science and engineering.

In summary, the mathematical proofs given in this chapter provide us a corner-stone for the most important concept ofconvergence-controlin the frame of theHAM, and well explain why the HAM is generally valid for so many highly nonlin-ear problems in science and engineering.

Acknowledgements Sincere thanks to Dr. Graham Little (Dept. of Mathematics, University ofManchester, UK) for his enlightening suggestions and discussions.


References

1. Adams, E.P., Hippisley, C.R.L.: Smithsonian Mathematical Formulae Tables and Table ofElliptic Functions. Smithsonian Institute, Washington (1922)

2. Agnew, R.P.: Euler Transformations. Journal of Mathematics.66, 313-338 (1944)3. Li, Y.J., Nohara, B.T., Liao, S.J.: Series solutions of coupled Van der Pol equation

by means of homotopy analysis method. J. Mathematical Physics 51, 063517 (2010).doi:10.1063/1.3445770

4. Liao, S.J.: The proposed Homotopy Analysis Technique forthe Solution of Nonlinear Prob-lems. PhD dissertation, Shanghai Jiao Tong University (1992)


6. Liao, S.J.: An explicit, totally analytic approximationof Blasius viscous flow problems. Int.J. Nonlin. Mech.34, 759 – 778 (1999)













Chapter 6Some methods based on the HAM

Abstract In this chapter, some analytic and semi-analytic techniques based on thehomotopy analysis method (HAM) are briefly described, including the so-called“homotopy perturbation method”, the optimal homotopy asymptotic method, thespectral homotopy analysis method, the generalized boundary element method, andthe generalized scaled boundary finite element method. The relationships betweenthese methods with the HAM are also revealed.

6.1 A brief history of the homotopy analysis method

To reveal the relationship between the homotopy analysis method (HAM) [16, 17,21–25] with other analytic approximation methods, we first briefly describe the ba-sic ideas of the HAM and its history of development and modification.

The early HAM was first described by Shijun Liao [16] in his PhDdissertationin 1992. For a given nonlinear differential equation

N [u(x)] = 0, x∈ Ω ,

whereN is a nonlinear operator andu(x) is a unknown function, Liao [16] used theconcept of homotopy [12] in topology [33] to construct aone-parameterfamily ofequations in the embedding parameterq∈ [0,1], called the zeroth-order deformationequation


whereL is an auxiliary linear operator andu0(x) is an initial guess. In theory, theconcept of homotopy [12] in topology [33] provides us extremely large freedom tochoose the auxiliary linear operatorL and the initial guessu0(x). At q= 0 andq=1, we haveφ(x;0) = u0(x) andφ(x;1) = u(x), respectively. So, as the embeddingparameterq ∈ [0,1] increases from 0 to 1, the solutionφ(x;q) of the zeroth-orderdeformation equation (6.1) varies (or deforms) from the initial guessu0(x) to the

227

228 6 Some methods based on the HAM

exact solutionu(x) of the original nonlinear differential equationN [u(x)] = 0. Suchkind of continuous variation is called deformation in topology, that is the reason why(6.1) is called the zeroth-order deformation equation. Sinceφ(x;q) is also dependentupon the embedding parameterq∈ [0,1], we can expand it into the Maclaurin serieswith respect toq:

φ(x;q) = u0(x)++∞

∑n=1

un(x) qn, (6.2)

called the homotopy-Maclaurin series. Assuming that, the auxiliary linear operatorL and the initial guessu0(x) are so properly chosen that the above homotopy-Maclaurin series converges atq= 1, we have the so-called homotopy-series solution

u(x) = u0(x)++∞

∑n=1

un(x), (6.3)

which satisfies the original equationN [u(x)] = 0, as proved by Liao [21, 22] ingeneral.

The governing equation ofun(x) is completely determined by the zeroth-orderdeformation equation (6.1). Differentiating (6.1)n times with respect to the embed-ding parameterq, then dividing byn! and finally settingq= 0, we have the so-calledhigh-order deformation equation

L [un(x)− χn un−1(x)] =−Dn−1N [φ(x;q)] , (6.4)

whereχ1 = 0 andχk = 1 whenk ≥ 2, Dk is the so-calledkth-order homotopy-derivative operator defined by

Dk =1k!

∂ k

∂qk

∣

∣

∣

∣

q=0. (6.5)

Note that the high-order deformation equation (6.4) is always linear with the knownterm on the right-hand side, therefore is easy to solve, as long as we choose theauxiliary linear operatorL properly.

Unfortunately, the early HAM mentioned above can not guarantee the conver-gence of approximation series of nonlinear equations in general. To overcome thisrestriction, Liao [17] in 1997 generalized the concept of homotopy and introducedsuch a non-zero auxiliary parameterc0 to construct atwo-parameterfamily of equa-tions, i.e. the zeroth-order deformation equation

(1−q)L [φ(x;q)−u0(x)] = c0 q N [φ(x;q)], x∈ Ω , q∈ [0,1]. (6.6)

In this way, the homotopy-series solution (6.3) is not only dependent upon the phys-ical variablex but also the auxiliary parameterc0. It has been proved (for details,please refer to Chapter 5) that the auxiliary parameterc0 can adjust and controlthe convergence region of homotopy-series solutions, although c0 has no physi-cal meanings at all. In essence, the use of the auxiliary parameterc0 introduces

6.2 Homotopy perturbation method 229

us one more “artificial” degree of freedom, which greatly improves the early HAM:it is the auxiliary parameterc0 which provides us a convenient way to guaranteethe convergence of homotopy-series solution. This is the reason why we callc0

theconvergence-control parameter. Differentiating (6.6)n times with respect to theembedding parameterq, then dividing byn! and finally settingq= 0, we have theso-called high-order deformation equation

L [un(x)− χn un−1(x)] = c0 Dn−1N [φ(x;q)] , (6.7)

Note that (6.1) and (6.4) are special cases of (6.6) and (6.7)whenc0 = −1, respec-tively.

The use of the convergence-control parameterc0 is a milestone in the develop-ment of the HAM. Realizing that more degrees of freedom implylarger possibilityto gain better approximations, Liao [21] in 1999 further introduced more “artifi-cial” degrees of freedom by using the zeroth-order deformation equation in a moregeneral form:

[1−α(q)]L [φ(x;q)−u0(x)] = c0 β (q) N [φ(x;q)], x∈ Ω , q∈ [0,1], (6.8)

whereα(q) andβ (q) are the so-calleddeformation functionssatisfying

α(0) = β (0) = 0,α(1) = β (1) = 1, (6.9)

whose Taylor series

α(q) =+∞

∑m=1

αm qm, β (q) =+∞

∑m=1

βm qm, (6.10)

are convergent for|q| ≤ 1.In fact, the zeroth-order deformation equation (6.8) can befurther generalized, as

shown by Liao [22,23]. Thus, the approximation series givenby the HAM can con-tain many unknown convergence-control parameters, which provide us great possi-bility to guarantee the convergence of homotopy-series solution.

In addition, using these generalized zeroth-order deformation equation, Liao [22]proved that the HAM logically contains other non-perturbation techniques, suchas Lyapunov’s artificial small parameter method [26], Adomian’s decompositionmethod [2,3], theδ -expansion method [13], and so on, and thus is rather general.

6.2 Homotopy perturbation method

In 1998, six years later after Liao [16] proposed the early HAM in his PhD disser-tation, Jihuan He [10,11] published the so-called “homotopy perturbation method”.Like the early HAM, the “homotopy perturbation method” is based on constructinga homotopy equation



which is exactly the same as the zeroth-order deformation equation (6.1). Like theHAM, the solutionφ(x;q) is also expanded into Maclaurin series

φ(x;q) = u0(x)++∞

∑n=1

un(x) qn, (6.12)

and the approximation is gained by settingq= 1, say,

u(x) = u0(x)++∞

∑n=1

un(x). (6.13)

Obviously, (6.12) and (6.13) are exactly the same as (6.2) and (6.3), respectively.The only difference between the “homotopy perturbation method” and the earlyHAM is that the embedding parameterq∈ [0,1] is regarded as a “small parameter”so that the governing equation ofun(x) is gained by substituting the series (6.12)into (6.11) and equating the coefficients of the like-power of q.

However, Hayat and Sajid [9] proved in 2007 that, substituting the Maclaurinseries

N [φ(x;q)] =+∞

∑n=0

DnN [φ(x;q)] qn,

whereDn is defined by (6.5), and the series (6.12) into (6.11), then equating thecoefficients of the like-power ofq, one obtains

L [un(x)− χn un−1(x)] =−Dn−1N [φ(x;q)] (6.14)

for un(x), which is thesameas the high-order deformation equation (6.4) exactly!So, no matter whether or not one regards the embedding parameter q ∈ [0,1] as asmall parameter, one obtains the exactly same approximations as the early HAM.Therefore, Sajid and Hayat [32] pointed out that “nothing isnew in Dr. He’s ap-proach, except the new name the homotopy perturbation method”.

This is easy to understand from the view points of mathematics: the so-called“homotopy perturbation method” is based on (6.11), which isexactly the same asthe zeroth-order deformation equation (6.1) of the early HAM. Therefore, these twoequations have the same solutionφ(x;q). According to the fundamental theorem incalculus, the Maclaurin series of a function is unique. Therefore, as a coefficient ofthe Maclaurin series ofφ(x;q), un(x) is unique. Thus,un(x) must be determined bythe unique equation, say, (6.14) is exactly the same as (6.4).

Unfortunately, like the early HAM, the so-called “homotopyperturbation method”can not guarantee the convergence of approximations, so that it is valid only forweakly nonlinear problems with small physical parameters,as reported by manyresearchers. For example, Abbasbandy [1] compared the modified HAM based on(6.6) and the “homotopy perturbation method” by solving a nonlinear heat transferequation

6.2 Homotopy perturbation method 231

(1+ εu)u′+u= 0, u(0) = 1,

where the prime denotes the differentiation with respect tot, andε ≥ 0 is a physicalparameter. By means of the “homotopy perturbation method” with the auxiliarylinear operatorL u= u′+u and the initial guessu0(t) = exp(−t), one obtains thefollowing approximations

u′(0) =−1+ ε − ε2+ ε3− ε4+ · · · ,

which is convergent only for 0≤ ε < 1. Thus, “the HPM and perturbation methodare valid only for small parameterε”, as concluded by Abbasbandy [1]. This illus-trates that, like perturbation techniques, the “homotopy perturbation method” isnotindependent of small physical parameters in essence. “In fact, for many cases the so-lutions obtained by perturbation method and HPM are identical”, as pointed by Ab-basbandy [1]. However, using the modified HAM based on (6.6) with thesameaux-iliary linear operator and thesameinitial guess but such adifferent1 convergence-control parameterc0 =−(1+ε)−1, Abbasbandy [1] gained the first-order homotopyapproximation ofu′(0) which is accurate forall physical parameter 0≤ ε < +∞.Thus, Abbasbandy [1] came to the conclusion that “HAM provides us with a con-venient way to control the convergence of approximation series, which is a funda-mental qualitative difference in analysis between HAM and other methods”.

Ganji et al. [8] also applied the so-called “homotopy perturbation method” tosolve the following differential equation

ut +ux = 2uxxt, t > 0, −∞ < x<+∞,


u(x,0) = exp(−x),

where the subscript denotes the differentiation. This problem has the closed-formsolution

u(x, t) = exp(−x− t).

Using the auxiliary linear operatorL u= ut and the initial guessu0(x, t) = exp(−x),Ganji et al. [8] reported “an analytic approximation” givenby means of the so-called “homotopy perturbation method”. However, Liang andJeffrey [15] repeatedtheir work and found that the approximations given by the so-called “homotopyperturbation method” are “divergent forall x andt exceptt = 0”. In other words,results given by the so-called “homotopy perturbation method” are divergent in thewhole interval

−∞ < x<+∞, t > 0

exceptt = 0 that however corresponds to the known initial condition! Liang and Jef-frey’s work [15] confirms Abbansbandy’s conclusion [1] thatthe so-called “homo-topy perturbation method” can not guarantee the convergence of the approximation

1 The “homotopy perturbation method” is in fact a special caseof the HAM whenc0 =−1.


and is valid for weakly nonlinear problems only. Besides, Liang and Jeffrey [15]also applied the modified HAM based on (6.6) to solve the same problem: using thesame auxiliary linear operator and the same initial guess, they gained the divergentseries by means of the convergence-control parameterc0 = −1, which is exactlythe same as Ganji’s approximation; however, using a different convergence-controlparameterc0 = +1, they obtained the accurate approximation which converges tothe exact solution exp(−x− t) in thewholeinterval−∞ < x<+∞ andt ≥ 0. Liangand Jeffrey [15] also came to the conclusion that “the HPM is aspecial case of theHAM” when c0 =−1, and that “it is very important to investigate the convergenceof approximation series, otherwise one might get useless results.”

Turkyilmazoglu [36] compared the HAM and the so-called “homotopy pertur-bation method” (HPM) from the view point of convergence, andcame to the con-clusion that “blindly using the HPM yields a non-convergence series to the soughtsolution. In addition to this, HPM is shown not always to generate a continuousfamily of solutions in terms of the homotopy parameter. By the convergence-controlparameter this can however be prevented to occur in the HAM”.

These examples illustrate that the so-called “homotopy perturbation method” isexactly the same as the early HAM. Thus, as a special case of the modified HAMwhenc0 =−1, the so-called “homotopy perturbation method” can not give anythingnew indeed. Besides, they also reveal the importance of the convergence-controlparameterc0 in theory. The use of the convergence-control parameterc0 is a mile-stone of the HAM: it is the convergence-control parameterc0 which provides us aconvenient way to guarantee the convergence of series solution so that the HAMbecomes independent of small/large physical parameters inessence. In fact, it is theconvergence-control parameterc0 which differs the HAM from all other analyticapproximation methods.

6.3 Optimal homotopy asymptotic method

In 2007, Yabushita, Yamashita and Tsuboi [41] first used the minimum of squaredresidual of governing equations to determine optimal convergence-control param-eters in the frame of the HAM. In 2008, Akyildiz and Vajravelu[4] suggested touse the optimal convergence-control parameter determinedby the minimum of thesquared residual of governing equation.

In 2008, Marinca and Herisanu [27, 28] suggested the so-called “optimal homo-topy asymptotic method” based on the homotopy equation

(1−q)L [φ(x;q)−u0(x)] =

(

+∞

∑i=1

ci qi

)

N [φ(x;q)], x∈ Ω , q∈ [0,1], (6.15)

where the optimal value ofci (i = 1,2,3, · · ·) is determined by the minimum ofsquared residual of governing equations. Let

6.3 Optimal homotopy asymptotic method 233

Em =

∫

N

[

m

∑n=0

un(x)

]2

dx

denote the squared residual of governing equationN [u(x)] = 0 at themth-order ofapproximation. Then, one had to solve a set of nonlinear algebraic equations

∂Em

∂ci= 0, 1≤ i ≤ m (6.16)

so as to gain the optimalmth-order approximation.Note that, substituting

α(q) = q, β (q) =1c0

+∞

∑i=1

ci qi , c0 =+∞

∑i=1

ci

into the zeroth-order deformation equation (6.8), one obtains exactly the same equa-tion as (6.15). So, Marinca and Herisanu’s approach [27, 28] is still in the frame ofthe HAM. However, Marinca and Herisanu’s approach [27, 28]is interesting: one

can regard eachci as a convergence-control parameter, as long as+∞∑

i=1ci is convergent

to a bounded value.The optimal approach suggested by Marinca and Herisanu [27, 28] is rigor-

ous in theory. However, the number of unknown convergence-control parame-ters linearly increases as the order of approximation increases, so that it becomestime-consuming in practice to solve the set of corresponding nonlinear algebraicequations related to the high-order optimal approximations, as illustrated by Niuand Wang [31]. In 2010, Liao [25] suggested an optimal HAM with only threeconvergence-control parameters at the most, and illustrated that the optimal HAMwith one or two convergence-control parameters seems to be most efficient compu-tationally.

In Chapter 3, an optimal HAM is proposed based on the zeroth-order deformationequation

(1−q)L [φ(x;q)−u0(x)] =

(

κ

∑i=0

ci qi+1

)

N [φ(x;q)], x∈ Ω , q∈ [0,1], (6.17)

where the optimal value ofci is determined by the minimum of squared residualEm

of themth-order approximation, i.e.

∂Em

∂ci= 0, 1≤ i ≤ minm,κ . (6.18)

The above optimal HAM contains the basic convergence-control parameterc0 whenκ = 1, and becomes Marinca and Herisanu’s approach [27,28] when κ →+∞.


6.4 Spectral homotopy analysis method

In 2010, Motsa, Sibanda and Shateyi [29] suggested the so-called “spectral homo-topy analysis method” (SHAM) by using the Chebyshev pseudospectral method tosolve the linear high-order deformation equations and choosing the auxiliary linearoperatorL in terms of the Chebyshev spectral collocation differentiation matrixdescribed by Don and Solomonoff [7].

For example, to solve the high-order deformation equation (6.7) in a finite inter-val x∈ [a,b] by means of the SHAM [29,30], one approximatesun(x) by means ofthe truncated Chybyshev polynomial

un(x) =M

∑k=0

ak Tk(x),

whereTk(x) is thekth Chebyshev polynomial of the first kind, and the unknowncoefficientak is determined by the discrete collocation points and related boundaryconditions.

In theory, any a continuous function in a bounded interval can be best approxi-mated by Chebyshev polynomial. So, the SHAM provides largerfreedom to choosethe auxiliary linear operatorL and initial guess. The basic idea of the SHAM mightbe expanded to solve nonlinear partial differential equations. Besides, it is easy toemploy the optimal convergence-controlparameter in the frame of the SHAM. Thus,the SHAM has great potential to solve more complicated nonlinear problems in sci-ence and engineering, although further modifications in theory and more applica-tions are needed.

Chebyshev polynomial is a kind of special function. There are many other specialfunctions such as Hermite polynomial, Legendre polynomial, Airy function, Besselfunction, Riemann zeta function, hypergeometric functions and so on. Since theHAM provides us extremely large freedom to choose the auxiliary linear operatorL and the initial guess, it should be possible to develop a “generalized spectralHAM” which can use a proper special function for a given nonlinear problem.

6.5 Generalized boundary element method

In essence, the HAM replaces a nonlinear problem by means of an infinite numberof linear sub-problems, since the high-order deformation equation is always lin-ear and governed by the auxiliary linear operatorL . If the initial guess and theauxiliary linear operatorL are so properly chosen that the analytic solution ofthe high-order deformation equation can be gained, then we obtain the analytichomotopy-approximation exactly, whose convergence is guaranteed by choosingproper value of the convergence-control parameter. However, obviously, the linearhigh-order deformation equation can be solved by means of different types of nu-merical techniques, such as the finite difference method (FDM), the finite element

6.6 Generalized scaled boundary finite element method 235

method (FEM), the finite volume method (FVM), the boundary element method(BEM), and so on. So, in theory, it is very easy to combine the HAM with ad-vanced numerical techniques. Since the numerical techniques are valid for differen-tial equations defined in rather complicated domain, the combination of the HAMwith numerical techniques can greatly enlarge the application fields of the HAM.

For example, based on the HAM, Liao [18–20] proposed the so-called “gener-alized boundary element method”. The traditional BEM is often valid for a lineardifferential equationL0u= 0, whose solution can be expressed by integration of afundamental solution on the boundary. When the traditionalBEM is applied to solvea nonlinear differential equation

L0u+N0u= 0,

whereL0u andN0u denote the linear and nonlinear parts of the governing equa-tion, one often rewritesL0u= −N0u and uses iteration approach by regarding theright-hand side term as the known ones. Unfortunately, thisapproach has strongrestrictions on the linear operatorL0, and thus does not work if the fundamentalsolution ofL is unknown, or if the highest order of derivative ofL0 is lower thanthat of the governing equation, or if the linear operatorL0 does not exist at all, andso on. However, the HAM provides us extremely large freedom to choose the auxil-iary linear operatorL . So, in the frame of the HAM, we can always choose such aproper auxiliary linear operatorL that the linear high-order deformation equationcan be solved by means of the traditional BEM.

Combining the HAM with the traditional BEM in this way, many nonlinear prob-lems can be solved by means of the so-called generalized BEM.For example, bymeans of the generalized BEM, Wu and Liao [40] successfully obtained the conver-gent results of driven cavity viscous flows at Reynolds number up toRe = 10,000,governed by the exact Navier-Stokes equation. Note that, one often obtains con-vergent numerical result of driven cavity flow with onlyRe= 1000 by means oftraditional BEM. This illustrates the great potential of the generalized BEM.

6.6 Generalized scaled boundary finite element method

The scaled boundary finite-element method (SBFEM), developed by Song and Wolf[34], is a novel semi-analytical method to solve linear partial differential equationsin complicated domain. Briefly speaking, in the frame of the SBFEM [34, 37–39],a scaled boundary coordinate system is first introduced, then the weighted resid-ual approximation of finite elements is applied in the circumferential direction, andthe governing partial differential equations are transformed to ordinary differen-tial equations in the radial direction, which can be solved analytically in the radialdirection. Like the BEM, only the boundary of the domain is discretized, but nofundamental solution is required. Thus, this semi-analytical method combines theadvantages of the finite element method and boundary elementmethods.


The traditional SBFEM is widely applied for the problems related to elasto-statics and elasto-dynamics, especially for soil-structure interaction problems in un-bounded domains [34,39]. Recently, the SBFEM has been extended to the fluid flowproblems [35]. By coupling the finite element method and the SBFEM, Dohertyand Deeks [6] captured the nonlinearity of problems in the near field. However, theSBFEM only accurately modeled the linear elastic far field response. Up to now, allproblems solved by the traditional SBFEM are governed by linear partial differentialequations.

In essence, the HAM replaces a nonlinear problem by means of an infinite num-ber of linear sub-problems, since the high-order deformation equations are alwayslinear. Especially, the HAM provides us extremely large freedom to choose the aux-iliary linear operatorL , and besides can guarantee the convergence of approxima-tions by means of proper convergence-control parameterc0. Thus, using the tra-ditional scaled boundary finite element method to solve the linear high-order de-formation equations, Lin and Liao [14] proposed the so-called “generalized scaledboundary finite element method” by means of combing the HAM with the traditionalscaled boundary finite element method. Using a nonlinear heat transfer problem asan example, Lin and Liao [14] illustrated the validity of thegeneralized SBFEM fornonlinear partial differential equations.

Since electronic computers appear, hundreds of numerical techniques and an-alytic methods have been developed independently. However, methods based onthe combination of analytic and numerical techniques are much less. Such kind ofsemi-analytic methods may combine the advantages of the high accuracy of analyticmethods and the flexibility of numerical methods for complicated domains. So, thegeneral SBFEM has great potential in future, although it needs further modificationsin theory and more applications in practice.

References 237

References




4. Akyildiz, F.T., Vajravelu, K. Magnetohydrodynamic flow of a viscoelastic fluid. Phys. Lett.A. 372, 3380 – 3384 (2008)

5. Awrejcewicz, J., Andrianov, I.V., Manevitch, L.I.: Asymptotic Approaches in Nonlinear Dy-namics. Springer-Verlag, Berlin (1998)

6. Doherty, J.P., Deeks, A.J.: Adaptive coupling of the finite-element and scaled boundary finite-element methods for non-linear analysis of unbounded media. Comput. Geotech.32, 436 –444 (2005)

7. Don, W.S., Solomonoff, a.: Accuracy and speed in computing the Chebyshev collocationderivative. SIAM J. Sci. Comput.16, 1253 – 1268 (1995).

8. Ganji, D.D., Tari, H., Jooybari, M.B.: Variational iteration method and homotopy perturbationmethod for nonlinear evaluation equations. Comput. Math. Appl. 54, 1018 – 1027 (2007)

9. Hayat, T., Sajid, M.: On analytic solution for thin film flowof a fourth grade fluid down avertical cylinder. Phys. Lett. A.361, 316–322 (2007)

10. He,J.H. : An approximate solution technique depending upon an artificial parameter. Com-mun. Nonlinear Sci. Numer. Simulat.3, 92 – 97 (1998)

11. He, J.H.: Homotopy perturbation technique. Comput. Method. Appl. M. 178, 257 – 262(1999)

12. Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge(1953)

13. Karmishin, A.V., Zhukov, A.T., Kolosov, V.G.: Methods of Dynamics Calculation and Testingfor Thin-walled Structures (in Russian). Mashinostroyenie, Moscow (1990)

14. Lin, Z.L., Liao, S.J.: The scaled boundary FEM for nonlinear problems. Commun. Nonlinear.Sci. Numer. Simulat.16, 63 – 75 (2011)




18. Liao, S.J.: General boundary element method for nonlinear heat transfer problems governedby hyperbolic heat conduction equation. Computational Mechanics.20, 397 – 406 (1997)

19. Liao, S.J.: Boundary element method for general nonlinear differential operators. EngineeringAnalysis with Boundary Elements.20, 91 – 99 (1997)

20. Liao, S.J.: Numerically solving nonlinear problems by the homotopy analysis method. Com-putational Mechanics.20, 530 – 540 (1997)














33. Sen, S.: Topology and Geometry for Physicists. AcademicPress, Florida (1983)34. Song, C., Wolf, J.P.: The scaled boundary finite-elementmethod - alias consistent infinites-

imal finite-element cell method for elastodynamics. Comput. Meth. Appl. Mech. Eng.147,329 – 355 (1997)

35. Tao, L., Song, H., Chakrabarti, S.: Scaled boundary FEM solution of short-crested wavediffraction by a vertical cylinder. Comput. Meth. Appl. Mech. Eng.197, 232 – 242 (2007)


37. Wolf, J.P., Song, C.: The scaled boundary finite-elementmethod – a primer: derivation. Com-put. Struct.78,191 – 210 (2000)

38. Wolf, J.P., Song, C.: The scaled boundary finite-elementmethod – a fundamental solution-lessboundary-element method. Comput. Meth. Appl. Mech. Eng.190, 5551 – 5568 (2001)

39. Wolf, J.P.: The scaled boundary finite-element method. Wiley, Chichester (2003).40. Wu, Y.Y, Liao, S.J.: Solving high Reynolds-number viscous flows by the general BEM and

domaindecomposition method. Int. J. Numer. Methods in Fluids.47, 185 – 199 (2005)41. Yabushita, K., Yamashita, M., Tsuboi, K.: An analytic solution of projectile motion with the

quadratic resistance law using the homotopy analysis method. J. Phys. A – Math. Theor.40,8403 – 8416 (2007)

Part IIMathematica packageBVPh and its

applications

“ If you shut your door to all errors, truth will be shut out.”

by Rabindranth Tagore (1861 – 1941)

Chapter 7Mathematica packageBVPh

Abstract The BVPh (version 1.0) is a Mathematica package for highly nonlinearboundary-value/eigenvalue problems with singularity and/or multipoint boundaryconditions. It is a combination of the homotopy analysis method (HAM) and thecomputer algebra system Mathematica, and provides us a convenient analytic toolto solve many nonlinear ordinary differential equations (ODEs) and even some non-linear partial differential equations (PDEs). In this chapter, we briefly describe itsscope, the basic mathematical formulas, and the choice of base functions, initialguess and the auxiliary linear operator, and so on, togetherwith a simple usersguide. As open resource, theBVPh 1.0 is given in the appendix of this chapterand free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

7.1 Introduction

By means of high-performance computer and numerical techniques such as Runge-Kutta’s method, it is convenient to gain accurate numericalapproximations of mostof nonlinear initial-value problems (IVPs) except those with chaos [44, 49]. How-ever, it is more difficult to solve boundary-value problems (BVPs), especially whenthere exist high nonlinearity, multiple solutions, singularity and an infinite interval.

The BVP4c is a famous software in MATLAB [25, 62, 63] for multipointboundary-value problems. For more details about theBVP4c, please refer to thewebsite at

http://www.mathworks.com/help/techdoc/ref/bvp4c.html.Many linear ordinary differential equations (ODEs) can be solved by means of theBVP4c. The shooting method [62] is employed inBVP4c, which first transforms aboundary-value problem (BVP) into an initial value problem(IVP) by adding someguessed initial conditions and then correcting them step bystep in such a way thatall original boundary conditions are satisfied. So,BVP4c is a numerical tool inessence. UsingBVP4c, nonlinear problems are often solved by traditional iterationapproaches such as Newton’s iteration. Unfortunately, it is well-known that conver-

241

242 7 Mathematica packageBVPh

gence of iteration cannot be guaranteed by these traditional iteration approaches,especially when the nonlinearity is strong. Besides, by means ofBVP4c, it is gen-erally not easy to find outall multiple solutions of BVPs, especially when thesesolutions are very close, and/or rather sensitive to the guessed initial conditions ofthe shooting method [62]. Furthermore, based-on numericalcomputations, it is dif-ficult for BVP4c to resolve the singularity in governing equations and/or boundaryconditions, such as sin(πz)/zatz= 0, even if sin(πz)/zhas the limitπ asz→ 0. Inaddition,BVP4c regards an infinite interval as a kind of singularity and replaces itby a finite one: this results in additional inaccuracy and uncertainty of solutions.

The Chebfun [17, 70] is a collection of algorithms, and a software systeminMATLAB [25,62,63], developed by Nick Trefethen and ZacharyBattles of OxfordUniversity since 2002 and Toby Driscoll of the University ofDelaware beginningin 2008. For more details about theChebfun , please refer to [17,70] and the web-site athttp://www2.maths.ox.ac.uk/chebfun/. As mentioned by theChebfun team,theChebfun is a powerful “numerical” tool based on Chebyshev expansions, fastFourier transform, barycentric interpolation and so on. But, unlike BVP4c, solu-tions of differential equations given byChebfun are expressed by a sum of somesmooth base functions such as Chebyshev polynomials so thatit can beexactlydif-ferentiatedanytimes atanya given point! This is in essence different fromBVP4c,although both ofChebfun andBVP4c are based on MATLAB. Especially, usingsuch kind of base functions, it is natural and much easier to resolve singularities inequations and/or boundary conditions, and besides to solvedifferential equations inan infinite interval exactly. So, “computing numerically with functions instead ofnumbers” [70] is indeed a wonderful idea!

Although linear differential equations can be solved conveniently byChebfun ,only a few examples for nonlinear differential equations are given. This is mainly be-cause, likeBVP4c, theChebfun uses Newton’s iteration to solve non-linear prob-lems, but it is well-known that convergence of Newton’s iteration is strongly depen-dent upon initial guesses and thus isnot guaranteed. Besides,Chebfun searchesfor multiple solutions of a nonlinear ODE by using differentguess approximations.However, it is not very clear how to gain these different guess approximations. So,it seems difficult to solve highly nonlinear differential equations with multiple solu-tions by means of theChebfun (version 4.0).

Inspirited by the general validity of the homotopy analysismethod (HAM)[28, 33–43, 45, 46, 86] for highly nonlinear problems in so many different fields[1–8, 10–16, 19–24, 26, 27, 29–32, 47, 48, 50–52, 54–61, 64–69, 71–85, 87–95] andby the ability of “computing with functions instead of numbers” [70] provided bycomputer algebra system such as Mathematica [9] and Maple, the author devel-oped a Mathematica packageBVPh (version 1.0) for highly nonlinear differentialequations with multiple solutions and singularities. Our aim is to develop a kind ofanalytictool for as manynonlinear BVPsas possiblesuch that multiple solutions ofhighly nonlinear BVPs can be conveniently found out, and that the infinite intervaland singularities of governing equations and/or boundary conditions can be easilyresolved.


As shown in Part I of this book, the HAM has some obvious advantages overother traditional analytic approximation methods. First,based on the homotopy intopology, the HAM is independent of small/large physical parameters, and thus isvalid even if a nonlinear problem does not contain these perturbed quantities at all.Besides, the so-called homotopy-control parameterc0 introduced by Liao [34] in1997 provides a convenient way to guarantee the convergenceof series solution,so that, different from other analytic approximation methods, the HAM is validfor highly nonlinear problems. Furthermore, the HAM provides us extremely largefreedom to choose initial guess and the auxiliary linear operatorL so that multi-ple solutions of nonlinear problems can be easily found out.In addition, using theidea ofcomputing with functions instead of numbers[70], the infinite interval andsingularities of governing equation and/or boundary conditions can be resolved ina easy and natural way by means of computer algebra system such as Mathemat-ica, Maple and so on. Therefore, in the frame of the HAM, it is possible to developsuch a Mathematica package for nonlinear boundary value problems, which has thefollowing characteristics:

• Guarantee of convergence: the convergence of series solution is guaranteed bychoosing a proper convergence-control parameterc0 in the frame of the HAM;

• Multiple solutions: multiple solutions of nonlinear problems are found out bymeans of the freedom of the HAM on the choice of different guess approxima-tions and different auxiliary linear operators;

• Singularity : singularities of governing equations and/or boundary conditions areresolved analytically by computing with functions of the computer algebra sys-tem Mathematica, instead of numbers;

• Infinite interval : equations are solved in an infinite interval exactly by means ofchoosing proper base functions defined in the infinite interval.

The Mathematica packageBVPh (version 1.0) is given in the appendix of thischapter and free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm. Here, webriefly describe its scope, the basic mathematical formulas, and the choice of basefunctions, initial guess, the auxiliary linear operatorL , the auxiliary function andthe convergence-control parameterc0, together with a simple users guide. Twelveexamples are used to show its validity for some types of nonlinear ODEs and PDEs,with the corresponding files of input data forBVPh 1.0 given in the appendix ofchapters in Part II, which are free available at the same website mentioned above.


7.1.1 Scope

TheBVPh1.0 provides us an analytic tool to solve some nonlinear ordinary differ-ential equations (ODEs) and partial differential equations (PDEs), mainly related tothe following problems:

1. a nonlinear boundary-value equationF [z,u] = 0 in a finite intervalz∈ [0,a],2. a nonlinear boundary-value equationF [z,u] = 0 in an infinite intervalz∈ [b,∞),3. a nonlinear eigenvalue equationF [z,u,λ ] = 0 in a finite intervalz∈ [0,a],4. a nonlinear PDE related to non-similarity and/or unsteady boundary-layer flows,

whereF denotes a nonlinear differential operator,u(z) is an unknown function,λis an unknown eigenvalue,a> 0 andb≥ 0 are known constants, respectively. Theboundary conditions are linear, which may be defined at multipoints including thetwo endpoints.

Some examples are given in the following chapters of Part II to show the validityof theBVPh1.0 for above mentioned problems. In Chapter 8, we illustrate how tofind out multiple solutions of nonlinear ODEs by means of different initial guessesand base functions. Five examples are used in Chapter 9 to illustrate how to useBVPh 1.0 to solve highly nonlinear eigenvalue problems with singularity, and/orvarying (real or complex) coefficients and/or multipoint boundary conditions. InChapter 10, we illustrate how to solve nonlinear ODEs in an infinite interval bymeans of theBVPh1.0, whose solutions decay either exponentially or algebraicallyat infinity. In Chapter 11 and Chapter 12, we illustrate that the BVPh 1.0 can beeven applied to solve some nonlinear PDEs, such as non-similarity and/or unsteadyboundary-layer flows. All of these example show the validityof theBVPh 1.0 forsome types of highly nonlinear ODEs and PDEs with multiple solutions and singu-larity.

Note that theBVPh1.0 is only the beginning of our attempt. Modified versionsfor more types of nonlinear ODEs and PDEs with more complicated and strongersingularities will be released in future. It is indeed very difficult to develop a generalMathematica package valid for as many nonlinear ODEs and PDEs as possible. Forthis reason, our codes are free available online as open resource so that researchersin different countries and different generations can do their contributions to finishthis hard work together, since science belongs to the whole human-being.

7.1.2 Brief mathematical formulas

The BVPh 1.0 is based on the HAM. Here, the mathematical formulas are brieflydescribed.


7.1.2.1 Boundary-value problems in a finite interval

Consider nonlinear boundary-value problems governed by a nonlinearnth-orderODE in a finite interval

F [z,u] = 0, z∈ [0,a], (7.1)

subject to then linear boundary conditions

Bk[z,u] = 0, 1≤ k≤ n, (7.2)

whereF is a nth-order nonlinear differential operator,Bk is a linear differentialoperator,u(z) is a smooth function,a > 0 andγk are constants, respectively. Theboundary conditions may be defined at multipoints includingthe two endpoints.Assume that at least one solution exists, and that all solutions are smooth.

Let q ∈ [0,1] denote the embedding parameter,u0(z) an initial guess ofu(z),respectively. In the frame of the HAM, we construct such a continuous variation (ordeformation)φ(z;q) that, asq increases from 0 to 1,φ(z;q) varies continuously fromthe initial guessu0(z) to the solutionu(z) of (7.1) and (7.2). Such kind of continuousvariation is governed by the so-called zeroth-order deformation equation

(1−q)L [φ(z;q)−u0(z)] = c0 q H(z) F [z,φ(z;q)] , z∈ [0,a], q∈ [0,1], (7.3)


Bk[z,φ(z;q)] = γk, 1≤ k≤ n, (7.4)

whereL is an auxiliary linear operator,c0 is the so-called convergence-controlparameter,H(z) is an auxiliary function, respectively.

As show in Part I, the HAM provides us extremely large freedomto choose theauxiliary linear operatorL , the convergence-control parameterc0 and the auxiliaryfunctionH(η). Assume that all of them are properly chosen so that the Homotopy-Maclaurin series

φ(z;q) = u0(z)++∞

∑m=1

um(z) qm (7.5)

absolutely converges atq= 1, where

um(z) = Dm[φ(z;q)] =1m!

∂ mφ(z;q)∂qm

∣

∣

∣

∣

q=0.

Here,Dm is called themth-order homotopy-derivative operator. Then, we have theso-called homotopy-series solution

u(z) = u0(z)++∞

∑m=1

um(z), (7.6)


whereum(z) is governed by the so-calledmth-order deformation equation

L [um(z)− χm um−1(z)] = c0 H(z) δm−1(z), z∈ [0,a], (7.7)


Bk[z,um(z)] = 0, 1≤ k≤ n, (7.8)

where

χm =

0, m≤ 1,1, m> 1,

(7.9)

and

δk(z) = DkF [z,φ(z;q)] =

1k!

∂ kF [z,φ(z;q)]∂qk

∣

∣

∣

∣

q=0(7.10)

can be easily obtained by means of the Theorems proved in Chapter 4.Note that themth-order deformation equation (7.7) is linear, subject to then lin-

ear boundary conditions (7.8), and therefore is easy to solve, especially by computeralgebra system like Mathematica. Especially, the convergence-control parameterc0

provides us a convenient way to guarantee the convergence ofthe homotopy-series(7.6), whose optimal value is determined by the minimum of the squared residual ofthe governing equation (7.1) at high enough order approximation of u(z). For moredetails, please refer to Chapter 8.

Note that, for boundary-value problems in a finite intervalz∈ [0,a], we shouldsetTypeEQ = 1 for theBVPh1.0.

7.1.2.2 Boundary-value problems in an infinite interval

Consider boundary-value problems governed by a nonlinearnth-order ODE in aninfinite interval

F [z,u] = 0, z∈ [b,+∞), (7.11)


Bk[z,u] = 0, 1≤ k≤ n, (7.12)

whereF is a nth-order nonlinear differential operator,Bk is a linear differentialoperator,u(z) is a smooth function,b ≥ 0 andγk are constants, respectively. Theboundary conditions may be defined at multipoints includingthe two endpoints.Assume that at least one solution exists, and that all solutions are smooth.

All related formulas are the same as those given above in§ 7.1.2.1, except thatthe finite intervalz∈ [0,a] is replacing by the infinite onez∈ [b,+∞). However,completely different initial guessu0(z) and auxiliary linear operatorL are used forthese two types of boundary-value problems, because their solutions are expressedby completely different base functions. For more details, please refer to Chapter 10.


When theBVPh1.0 is used to solve boundary-value problems in an infinite in-tervalz∈ [b,+∞), we should setTypeEQ = 1 andzR = infinity .

7.1.2.3 Eigenvalue problems in a finite interval

Consider eigenvalue problems governed by a nonlinearnth-order ODE in a finiteinterval

F [z,u,λ ] = 0, z∈ [0,a], (7.13)


Bk[z,u] = 0, 1≤ k≤ n, (7.14)

whereF is a nth-order nonlinear differential operator,Bk is a linear differentialoperator,u(z) is a smooth eigenfunction,λ is a unknown eigenvalue,a> 0 andγk

are constants, respectively. The boundary conditions may be defined at multipointsincluding the two endpoints. Assume that at least one eigenfunction and one eigen-value exist, and that all eigenfunctions are smooth.

Eigenvalue problems often have an infinite number of eigenfunctions and eigen-values. To distinguish different eigenfunctions and eigenvalues, we should add oneadditional boundary condition

B0[z,u] = γ0, (7.15)

whereB0 is a linear differential operator andγ0 is a constant.Let q∈ [0,1] denote the embedding parameter,u0(z) an initial guess of the eigen-

functionu(z), λ0 an initial guess of the eigenvalueλ , respectively. In the frame of theHAM, we first construct such two continuous variations (or deformations)φ(z;q)andΛ(q) that, asq increases from 0 to 1,φ(z;q) varies continuously from the initialguessu0(z) to the eigenfunctionu(z) of (7.13) and (7.14), so doesΛ(q) from theinitial guessλ0 to the eigenvalueλ , respectively. Such two continuous variations aregoverned by the so-called zeroth-order deformation equation

(1−q)L [φ(z;q)−u0(z)] = c0 q H(z) F [z,φ(z;q),Λ(q)] , z∈ [0,a], (7.16)

subject to then boundary conditions

Bk[z,φ(z;q)] = γk, 1≤ k≤ n, (7.17)

and the additional boundary condition

B0[z,φ(z;q)] = γ0, (7.18)

whereL is the auxiliary linear operator,c0 is the so-called convergence-controlparameter,H(z) is the auxiliary function, respectively.


As shown in Part II, the HAM provides us extremely large freedom to choose theauxiliary linear operatorL , the convergence-control parameterc0 and the auxiliaryfunctionH(η). Assume that all of them are properly chosen so that the Homotopy-Maclaurin series

φ(z;q) = u0(z)++∞

∑m=1

um(z) qm, Λ(q) = λ0++∞

∑m=1

λm qm, (7.19)

absolutely converge atq= 1. Then, we have the so-called homotopy-series solution

u(z) = u0(z)++∞

∑m=1

um(z), λ = λ0++∞

∑m=1

λm, (7.20)

where the unknownum(z) is governed by themth-order deformation equation

L [um(z)− χm um−1(z)] = c0 H(z) δm−1(z), z∈ [0,a], (7.21)


Bk[z,um(z)] = 0, 1≤ k≤ n, (7.22)

where

δk(z) = DkF [z,φ(z;q),Λ(q)] =

1k!

∂ kF [z,φ(z;q),Λ(q)]∂qk

∣

∣

∣

∣

q=0(7.23)

can be easily obtained by means of the Theorems proved in Chapter 4. Here,Dk

is the so-calledkth-order homotopy-derivative operator. Note thatδm−1(z) in (7.21)contains

λ0, λ1, λ2, · · · ,λm−1.

In addition, the unknownλm−1 is determined by the additional boundary-condition

B0[z,um(z)] = 0. (7.24)

For more details, please refer to Chapter 9.In fact, theBVPh1.0 is valid for more generalized boundary conditions

Bk[z,u,λ ] = 0, 1≤ k≤ n, (7.25)

which may contain the unknown eigenvalueλ .Note that, when theBVPh 1.0 is used to solve eigenvalue problems in a finite

intervalz∈ [0,a], we should setTypeEQ = 2.


7.1.3 Choice of the base functions and initial guess

In the frame of the HAM, we should first of all choose a set of base functions

e0(z),e1(z),e2(z), · · · ,

which is good enough to approximate the unknown solutionsu(z) of a nonlinearboundary-value problem. In other words, we should choose such a kind of basefunctions thatu(z) can be expressed in the form

u(z) =+∞

∑m=0

am em(z), (7.26)

wheream is a coefficient. The above expression is called thesolution-expressionofu(z), which plays an important role in the frame of the HAM for the choice of theauxiliary linear operatorL , the auxiliary functionH(z) and initial guessu0(z).

For boundary-value/eigenvalue problems in a finite interval z∈ [0,a], its solutioncan be expressed by different base functions, such as a powerseries

u(z) =+∞

∑m=0

Am zm, (7.27)

or a Chebyshev series [18,53]

u(z) =+∞

∑m=0

Bm Tm(z), (7.28)

whereTm(z) is the mth Chebyshev polynomial of the first kind,Am and Bm arecoefficients, respectively. Besides, it is well-known thata smooth functionu(z) in afinite intervalz∈ [0,a] can be expressed by Fourier series

u(z) =+∞

∑m=0

[

Am cos(mπz

a

)

+ Bm sin(mπz

a

)]

, (7.29)

whereAm, Bm are coefficients. In addition,u(z) can be expressed more efficiently bymeans of the so-called hybrid-base, i.e. a kind of combination of polynomials andtrigonometric functions, as described in§ 7.2.3. Thus, the polynomials, trigono-metric functions and their combination supply a complete set of base functions toapproximate a smooth solutionu(z) in a finite intervalz∈ [0,1].

In addition, properties of a solution, which can be obtainedfrequently beforesolving a given BVP, are valuable to give a more accurate solution-expression. Forexample, if the solution of a BVP in a finite intervalz∈ [0,a] is odd, we have thecorresponding solution-expression


u(z) =+∞

∑m=0

A2m+1 z2m+1 (7.30)

in power series, and

u(z) =+∞

∑m=0

Bm sin(mπz

a

)

(7.31)

in Fourier series, respectively. Similarly, for an even solution of a BVP inz∈ [0,a],we have the solution-expression

u(z) =+∞

∑m=0

A2m z2m (7.32)

in power series, and

u(z) =+∞

∑m=0

Am cos(mπz

a

)

(7.33)

in Fourier series, respectively.For boundary-value/eigenvalue problems in an infinite interval z ∈ [b,+∞),

whereb≥ 0 is a bounded constant,u(z) can be expressed by

u(z) =+∞

∑k=0

+∞

∑m=0

αk,m zk exp(−mγ z) (7.34)

for exponentially decay solutions (at infinity), or

u(z) =+∞

∑m=0

βm

(1+ γ z)m (7.35)

for algebraically decaying solutions (at infinity), whereγ > 0 is a parameter andαk,m,βm are coefficients.

The initial guessu0(z) should obey the solution-expression (7.26) and besidessatisfy all boundary conditions, if possible. Thus, the initial guessu0(z) is often inthe form

u0(z) =K

∑m=1

bm em(z), K ≥ n, (7.36)

wheren is the number of all boundary conditions,bm is a coefficient,em(z) denotesbase functions,K ≥ n is a positive integer, respectively. In case ofK = n, the ini-tial guess is uniformly determined by then boundary conditions. However, whenµ = K −n> 0, such kind of initial guessu0(z) provides usµ additional degree offreedom. This kind of unknown parameters are calledthe multiple-solution-controlparameters, which provide us a convenient way to search for multiple solutionsand/or to guarantee the convergence of homotopy-series, asillustrated in Chapter 8and Chapter 9


Note that, for the boundary-value/eigenvalue problems mentioned above, thechoice of base functions is mainly determined by the interval and some propertiesof solution, not by governing equations in details. So, a smooth function in afiniteintervalz∈ [0,a] should be approximated by (7.27), (7.28) and (7.29), but a smoothfunction in aninfinite intervalz∈ [b,+∞), whereb> 0, should be approximated byeither (7.34) for exponentially decaying solutions or (7.35) for algebraically decay-ing solutions, respectively. Given a nonlinear boundary-value problem, it is gener-ally not difficult to derive some properties of solution by analyzing governing equa-tions and/or boundary conditions before solving it, such asasymptotic propertiesof solution at infinity, its odd and even property, symmetry and so on. All of theseproperties of solution provide us valuable information to choose a good enough basefunctions.

It should be emphasized that, unlike perturbation techniques which regardssmall/large physical parameters as the starting point, we regard base functions ofsolution as the starting-point of the HAM. This is mainly because base function is akey for analytic approximation of solution: it is as important asnumbersfor numer-ical approximations.Computing with functions instead of numbersimplies that thebase functions are most important forBVPh1.0.

Note that, using base functions instead of numbers and by means of a computeralgebra system, many singular terms such as sin(πz)/z asz→ 0 can be easily re-solved, because both of the numerator sin(πz) and the denominatorz of sin(πz)/zasz→ 0 are now regarded as afunctioninstead of thenumber1 0, and besides thelimit

limz→0

sin(πz)z

= π

is easily obtained by means of a computer algebra system likeMathematica. In ad-dition, some types of base functions are well defined in an infinite interval and thuscan be conveniently used to approximate solutions of a nonlinear boundary-valueproblem in the infinite interval exactly. In this way, many types of singularities canbe easily resolved forBVPh1.0 by means of the ability ofcomputing with functionsinstead of numbersprovided by the computer algebra system Mathematica.

Sinceall smooth functions in a finite intervalz ∈ [0,a] can be expressed byChebyshev series (7.28) and/or Fourier series (7.29), quite different boundary-valueequations in a finite intervalz∈ [0,a] may have the same solution-expression in theframe of the HAM. This suggests the possibility to develop a Mathematica packagefor nonlinear boundary-value problems in general.

1 Note that 0/0 has no meanings for numerical computations, and this results in singularities tomany numerical tools such asBVP4c.


7.1.4 Choice of the auxiliary linear operator

The choice of the auxiliary linear operatorL is mainly determined by base-functions of solutionu(z). In principle, an auxiliary linear operatorL should bechosen properly in such a way that

• all solution-expressions must be satisfied,• all high-order deformation equations have unique solutions, and can be easily

solved by computer algebra system like Mathematica,• the convergence of all homotopy-series can be guaranteed bymeans of choosing

proper convergence-control parameters.

For nth-order boundary-value/eigenvalue equations in afinite intervalz∈ [0,a],we choose the auxiliary linear operator

L u=dnu(z)

dzn (7.37)

whenu(z) is expressed by power series (7.27) or Chebyshev series (7.28). In thiscase, we should chooseTypeL = 1 for BVPh1.0.

However, whenu(z) is expressed by Fourier series (7.29 ) or by the so-calledhybrid-base functions described in§ 7.2.3, we often choose the following auxiliarylinear operator

L u = u′′+ω21 u, whenn= 2,

L u = u′′′+ω21 u′, whenn= 3,

L u = u′′′′+(ω21 +ω2

2) u′′+ω21ω2

2 u, whenn= 4,

L u = u′′′′′+(ω21 +ω2

2) u′′′+ω21ω2

2 u′, whenn= 5,...

for anth-order boundary-valueequation in a finite intervalz∈ [0,a], where the primedenotes the differentiation with respect toz, andωi > 0 is a frequency. Dependingon the related boundary conditions,ωi > 0 may be different each other, such as

ωi = i(κπ

a

)

, (7.38)

or the same, such as

ωi =(κπ

a

)

, (7.39)

whereκ ≥ 1 andi ≥ 1 are positive integers. The above auxiliary linear operatorscan be expressed in a general form

L u=

[

m

∏i=1

(

d2

dx2 +ω2i

)

]

u, whenn= 2m, (7.40)


or

L u=

[

m

∏i=1

(

d2

dx2 +ω2i

)

]

u′, whenn= 2m+1. (7.41)

Note that all of them have the property

L [cos(ωi z)] = L [sin(ωi z)] = 0,

whereωi > 0 is the frequency mentioned above. The auxiliary linear operator (7.40)or (7.41) with different values ofκ in (7.38) or (7.39) provides us a convenient wayto search for multiple solutions of many nonlinear boundary-value/eigenvalue equa-tions, as illustrated in§ 9.3.1,§ 9.3.2 and§ 9.3.3. Note that such kind of auxiliarylinear operator corresponds toTypeL = 2 for BVPh1.0.

Note that all examples in Chapter 8 and Chapter 9 for the boundary-value oreigenvalue problems in a finite intervalz∈ [0,a] are solved by means of either (7.37)or (7.40), (7.41). So, for a nonlinear boundary-value or eigenvalue problem in a fi-nite intervalz∈ [0,a], it is strongly suggested to attempt these two kinds of auxiliarylinear operators first.

Fornth-order boundary-value problems in aninfinite intervalz∈ [b,+∞) with anexponentially decaying solution, whereb≥ 0 is a bounded constant, we often (butnot always) choose such an auxiliary linear operatorL in the form

L u=dnudzn +

n−1

∑m=0

amdmudzm , (7.42)

thatL [exp(mγ z)] = 0, 1+n′−n≤ m≤ n′ (7.43)

holds, wheren′ is the number of the boundary conditions at infinity,γ > 0 is aparameter to be chosen later, ˇam is a coefficient, respectively. Then unknown co-efficientsam in (7.42) are uniquely determined by then linear algebraic equationsgiven by (7.43).

Fornth-order boundary-value problems in an infinite intervalz∈ [b,+∞) with analgebraically decaying solutionu∼ zβ asz→+∞, whereb> 0 andβ are boundedconstants, we often (butnot always) choose such an auxiliary linear operatorL inthe form

L u= zn dnudzn +

n−1

∑m=0

bm zm dmudzm (7.44)

thatL

[

zβ+m]

= 0, 1−n≤ m≤ 0 (7.45)

holds, wherebm is a unknown constant. Then unknown coefficients in (7.44)are uniquely determined by then linear algebraic equations given by (7.45). Forboundary-value problems in the infinite intervalz∈ [0,+∞), we should first usesuch a transform

ξ = 1+ γ z


so thatξ ∈ [1,+∞), whereγ > 0 is a parameter, and then use the auxiliary linear op-erator suggested above. In this case, we can also regardγ as a kind of convergence-control parameter, as shown in§ 10.3. Note that, for boundary-value problems in aninfinite intervalz∈ [b,+∞), we must setzR=infinity for BVPh1.0.

In the frame of the HAM and by means of the ability of “computing with func-tions instead of numbers” provided by computer algebra system like Mathemat-ica, an infinite interval[b,+∞) has essentially no difference from a finite intervalz∈ [0,a]: only different base functions, different auxiliary linear operators and dif-ferent initial guess are used. By means ofBVPh1.0, the two endpoints of an infiniteinterval z∈ [b,+∞) are regarded as the same without fundamental difference: allboundary condition at the two endpoints are regarded as a kind of the limit as eitherz→ b or z→ +∞, respectively. In this way, the infinite interval and many typesof singularities in governing equations and/or boundary conditions of BVPs can beresolved easily.

Since the HAM provides us extremely large freedom to choose the auxiliarylinear operatorL , we can choose an auxiliary operatorL in other forms, whennecessary. In this case, the auxiliary linear operator mustbe explicitly defined in theinput data file for theBVPh1.0.

Note that, for the boundary-value problems, the choice of auxiliary linear oper-ators is mainly determined by base functions. As mentioned above, the choice ofthe base functions is mainly determined by the interval and asymptotic properties ofsolution. So, the choice of the auxiliary linear operator ismainly determined by theinterval and asymptotic properties of solution.

7.1.5 Choice of the auxiliary function

In essence, the choice of the auxiliary functionH(z) may be often regarded as a partof choosing the auxiliary linear operatorL , since the auxiliary linear operatorL

and the auxiliary functionH(z) in a zeroth-order deformation equation such as (7.3)can be often combined as one, i.e.

L u=L uH(z)

.

So, like the choice of the auxiliary linear operatorL , the choice of an auxiliaryfunctionH(z) is also mainly determined by the interval and the propertiesof solu-tion. In principle, the auxiliary functionH(z) should be chosen in such a way

• that the solution-expression must be obeyed,• that solutions of high-order deformation equations exist and are unique,• that the convergence of homotopy-series solutions is guaranteed by means of

proper convergence-control parameters.

Mostly, we can simply choose the auxiliary functionH(z) = 1, especially forBVPs in a finite intervalz∈ [0,a]. For BVPs in an infinite interval[b,+∞), we may


sometimes gain exponentially decaying solutions in the form

u(z) =+∞

∑m=0

am exp(−mγz)

by means of the auxiliary linear operatorH(z) = exp(κγz), or algebraically decay-ing solution in the form

u(z) =+∞

∑m=µ

bm

zm

by means ofH(z) = zκ , whereγ > 0 andµ ≥ 0 are parameters, andκ is a parameterdetermined by the asymptotic properties ofu(z) at infinity and the so-called “rule ofcoefficient ergodicity” suggested by the author in his first book (see§2.3.4 of [39]),i.e. each coefficientam andbm of above solution-expressions may be modified asm→+∞.

7.1.6 Choice of the convergence-control parameterc0

Different from all other analytic approximation techniques, the HAM provides us aconvenient way to guarantee the convergence of series solution by means of intro-ducing the so-called convergence-control parameterc0 in a zeroth-order deforma-tion equation such as (7.3) and (7.16). In fact, it is the convergence-control parame-ter c0 that essentially differs the HAM from all other analytic techniques, as shownin Part I.

As shown in§ 3, at themth-order homotopy-approximation, the optimal conver-gence control parameter is determined by the minimum of the squared residualEm

of governing equation, corresponding to

dEm

dc0= 0, (7.46)

where

Em =

∫

ΩF

[

z,m

∑k=0

uk(z)

]2

dz (7.47)

is a squared residual for BVPs in either a finite intervalΩ : z∈ [0,a] or an infiniteintervalΩ : [b,+∞), and

Em =

∫

ΩF

[

z,m

∑k=0

uk(z),m−1

∑k=0

λm

]2

dz (7.48)

is a squared residual for eigenvalue problems in a finite interval z∈ [0,a], respec-tively. Here,F [z,u] = 0 andF [z,u,λ ] = 0 denote governing equations of BVPs


and eigenvalue problems, respectively. For the sake of computation efficiency, theseintegrals are numerically calculated by means of some discrete points (we set thedefaultNintegral = 50 .

In most cases, the squared residualEm is dependent upon the convergence-controlparameterc0 only. However, to search for multiple solutions in the frameof theHAM, we often use the so-called multiple-solution-controlparameterσ in an initialguessu0(z), which provides us one more degree of freedom to gain multiple solu-tions. In this case, the optimal convergence-control parameterc0 and the optimalmultiple-solution-control parameterσ are determined by

∂Em

∂c0= 0,

∂Em

∂σ= 0. (7.49)

For examples, please refer to§ 8 and§ 9.

7.2 Approximation of solutions and iteration

Although the high-order deformation equations (7.7) and (7.21) are always linear,they are still not easy to solve in general, because the right-hand side termδm−1(z)may be rather complicated. For example, when the auxiliary linear operator (7.40)or (7.41) is used, the linear differential equation

L u= zi[

cos

(

jκπza

)

+ sin

(

jκπza

)]

(7.50)

has a short closed-form solution for arbitrary integersi ≥ 0 and j ≥ 1. However,the termδ0(z) = F [z,u0(z)] or δ0(z) = F [z,u0(z),λ0] maybe be very complicated,since the nonlinear differential operatorF is rather general.

For instance, in case ofF [z,u,λ ] = u′′+λ sin(u) and we use the auxiliary linearoperator (7.40) and the initial guessu0 = cos(κπz/a), we had to solve a lineardifferential equation in the form

u′′+(κπz

a

)2u= sin

[

cos(κπz

a

)]

,

which has no closed-form solution, since the right-hand side term must be expandedinto an infinite series

sin[

cos(κπz

a

)]

= cos(κπz

a

)

− 13!

cos3(κπz

a

)

+ · · ·

In this case, the length ofum(z) is infinite so that it is hard to gain high-order ap-proximations. To avoid this, we must approximate the termδm−1(z) by means of areasonable number of properly chosen base-functions so that themth-order defor-mation equation (7.7) or (7.21) can be solved efficiently with high-enough accuracy.

7.2 Approximation of solutions and iteration 257

More importantly, as illustrated in Chapter 2, iteration can greatly accelerate theconvergence of homotopy-series. However, the necessary condition of using itera-tion approach in the frame of the HAM is to approximate solutions of high-orderdeformation equations by a reasonable number of base-functions in a high-enoughaccuracy, i.e.

um(z)≈Nt

∑m=0

am em(z),

whereem(z) denotes the corresponding base-function,Nt denotes the number oftruncated terms,am is a coefficient, respectively. As mentioned above, the termδm−1(z) may be rather complicated. So, to gainum(z) in the above form, we mustapproximateδm−1(z) in the form

δm−1(z)≈Nt

∑m=0

bm em(z),

where the coefficientbm is uniquely determined byδm−1(z) and the base-functionem(z).

As illustrated in Chapter 2, the HAM provides us great freedom to choose theinitial guessu0(z). Sinceum(z) has a fixed length, it is rather convenient to employthe iteration approach by using theMth-order approximation as a new initial guessu∗0(z), i.e.

u0(z)+M

∑m=1

um(z)→ u∗0(z). (7.51)

The above expression provides us the so-calledMth-order iteration formula. In thisway, the convergence of the homotopy-series can be greatly accelerated, as illus-trated in Chapter 8 and Chapter 9.

The iteration approach of theBVPh1.0 is currently possible only for nonlinearboundary-value/eigenvalue problems in a finite intervalz∈ [0,a]. So, we considerhere the approximation of a smooth functionf (z) in a finite interval∈ [0,a] only.

7.2.1 Polynomials

It is well-known that a smooth solutionu(z) in a finite intervalz∈ [0,a] can be wellapproximated by a polynomial

u(z)≈Nt

∑k=0

ak zk, (7.52)

whereak is a coefficient,Nt is the number of truncated terms. Besides, we have theso-called best approximation by


u(z)≈ b0

2+

Nt

∑k=1

bk Tk(z) (7.53)

whereTk(z) is thekth Chebyshev polynomial of the first kind,Nt denotes the numberof Chebyshev polynomials, and

bk =2π

∫ π

0u[a

2(1+ cosθ )

]

cos(kθ )dθ .

When polynomial is used to solve the boundary-value/eigenvalue problems in afinite intervalz∈ [0,a] by means of theBVPh1.0, we should set

TypeBase = 0 , TypeL = 1with ApproxQ = 0 for the polynomial (7.52), andApproxQ = 1 for the Cheby-shev polynomial (7.53), respectively.

7.2.2 Trigonometric functions

It is well-known that the Fourier series

a0

2+

+∞

∑n=1

[

bn sin(nπz

a

)

+ancos(nπz

a

)]

of a continuous functionf (z) in z∈ (−a,a) converges tof (z) in the intervalz∈(−a,a), where

an =1a

∫ a

af (t)cos

(nπta

)

dt, bn =1a

∫ a

af (t)sin

(nπta

)

dt.

For a continuous functionf (z) in [0,a], we can definef (−z) = f (z) in z∈ (0,a)and the corresponding even Fourier series reads

f (z) =a0

2+

+∞

∑n=1

an cos(nπz

a

)

. (7.54)

Alternatively, definingf (z) =− f (−z) in z∈ (0,a), we have the odd Fourier series

f (z) =+∞

∑n=1

bn sin(nπz

a

)

. (7.55)

In practice, we have the corresponding approximations

f (z) =a0

2+

Nt

∑n=1

an cos(nπz

a

)

(7.56)

or


f (z) =Nt

∑n=1

bn sin(nπz

a

)

, (7.57)

whereNt denotes the number of truncated terms.When the above-mentioned trigonometric functions are usedto solve boundary-

value/eigenvalue problems in a finite intervalz∈ [0,a] by means of theBVPh1.0,we should set

ApproxQ = 1 , TypeL = 2with TypeBase = 1 for the odd expression (7.57) andTypeBase = 2 for theeven expression (7.57), respectively.

7.2.3 Hybrid-base functions

Note that the first-order derivative of the even Fourier series (7.54) equals to zero atthe two endpointsx= 0 andx= a, but the original functionf (z) may have arbitraryvalues off ′(0) and f ′(a). So, in case off ′(0) 6= 0 and/orf ′(a) 6= 0, one had to usemany terms of the even Fourier series (7.54) so as to obtain anaccurate approxima-tion near the two boundary points. To overcome this disadvantage, we first expressf (z) by such a combination

f (z) ≈Y(z)+w(z), (7.58)

where

Y(z) =

f ′(0) z− [ f ′(0)+ f ′(a)]2a

z2

cos(πz

a

)

(7.59)

and then approximatew(z) = f (z)−Y(z) by the even Fourier series

w(z)∼ a0

2+

NT

∑n=1

ancos(nπz

a

)

, (7.60)

with the Fourier coefficient

an =2a

∫ a

0[ f (t)−Y(t)]cos

(nπta

)

dt.

Here,NT is the number of truncated terms. Note thatY(z) satisfies

Y′(0) = f ′(0), Y′(a) = f ′(a)

so thatw′(0) = w′(a) = 0. Therefore, we often need a few terms of the even Fourierseries to accurately approximatew(z). Note also that both of the trigonometric andpolynomial functions are used in (7.58) to approximatef (z). It is found that, bymeans of such kind of approximations based on hybrid-base functions, one oftenneeds much less terms to approximate a given smooth functionf (z) in [0,a] than


the traditional Fourier series. For example, the 15-term hybrid-base approximationof f (z) = 1/(1+ z) in z∈ [0,π ] is much better than its 50-term traditional Fourierapproximation, especially nearx= 0, as shown in Fig. 7.1.

Fig. 7.1 Comparison of dif-ferent approximations of(1+ z)1 in z∈ [0,π ]. Solidline: (1+ z)1; Symbols:hybrid-base approximationwith 15 terms; Dashed line:traditional Fourier approxi-mation with 50 terms.

z

1/(

1+

z)

0 0.05 0.1 0.15 0.2

0.85

0.9

0.95

1

Alternatively, for a continuous functionf (z) in [0,a], we can use

Y(z) = f (0)+[ f (a)− f (0)]

az (7.61)

or

Y(z) =[ f (0)+ f (a)]

2+

[ f (0)− f (a)]2

cos(πz

a

)

, (7.62)

and approximatew(z) by the odd Fourier series

w(z) ∼NT

∑n=1

bnsin(nπz

a

)

, (7.63)

where

bn =2a

∫ a

0[ f (t)−Y(t)]sin

(nπta

)

dt.

It is suggested to use the hybrid-base approximation (7.58)with (7.59) for aneven functionf (z), and (7.58) with (7.61) or (7.62) for an odd functionf (z), re-spectively. If f (z) is neither an odd nor even function, both of them work.

By means of theBVPh 1.0, the above-mentioned hybrid-base approximationmethod is possible to solve themth-order deformation equation (7.7) and (7.21).We first approximateδi(z) by means of the above-mentioned hybrid-base approxi-mation method with a reasonable number of truncated terms, for instance,

δi(z)≈Yi(z)+wi(z), (7.64)

where


Yi(z) =

δ ′i (0) z− [δ ′

i (0)+ δ ′i (a)]

2az2

cos(πz

a

)

(7.65)

and

wi(z)∼a0

2+

NT

∑n=1

ancos(nπz

a

)

, (7.66)

with the Fourier coefficient given by

an =2a

∫ a

0[δi(t)−Yi(t)]cos

(nπta

)

dt.

Alternatively (especially when the solution is odd), we canalso use

Yi(z)≈ δi(0)+[δi(a)− δi(0)]

az, (7.67)

or

Yi(z)≈[δi(0)+ δi(a)]

2+

[δi(0)− δi(a)]2

cos(πz

a

)

, (7.68)

where

wi(z) ∼NT

∑n=1

bnsin(nπz

a

)

, (7.69)

and

bn =2a

∫ a

0[δi(t)−Yi(t)]sin

(nπta

)

dt.

In this way, as long asNt is large enough, the original high-order deformationequation (7.7) and (7.21) can be replaced by

L [um(z)− χm um−1(z)]≈ c0 H(z) [Ym−1(z)+wm−1(z)] (7.70)

with high accuracy, where we choose the auxiliary linear operator (7.40) or (7.41).Then, the above linear equation can be divided into a finite number of linear differ-ential equations in the form of (7.50), which are easy to solve by means of the com-puter algebra system Mathematica [9]. More importantly, givenNT , i.e. the numberof truncated terms,um(z) has always a fixed length, which does not increase evenwhen the approximation order is very high. Therefore, by means of this hybrid-baseapproximation method, the high-order deformation equation (7.7) and (7.21) can besolved rather efficiently, no matter how complicated the nonlinear operatorF is.

When the above-mentioned hybrid-base approximation is used, we have evenlarger freedom to choose the initial guessu0(z). For example, for a 2nd-orderboundary-value/eigenvalue problem in a finite intervalz∈ [0,a], we may choosesuch an initial guess in the form

u0(z) = B0+B1 cos(κπz

a

)

+B2 sin(κπz

a

)

, (7.71)

or


u0(z) =(

B0+B1 z+B2 z2) cos(κπz

a

)

, (7.72)

whereB0,B1 andB2 are determined by linear boundary conditions.When the above-mentionedhybrid-base approximation is used to solve boundary-

value/eigenvalue problems in a finite intervalz∈ [0,a] by means of theBVPh1.0,we should set

ApproxQ = 1 , TypeL = 2with TypeBase = 1 for the odd expression (7.63) andTypeBase = 2 for theeven expression (7.60), respectively.

7.3 A simple users guide of theBVPh 1.0

7.3.1 Key modules

BVPh The moduleBVPh[k ,m ] gives thekth to mth-order homotopy approxi-mations of a nonlinear boundary-value problem (whenTypeEQ = 1) or a non-linear eigenvalue problem (whenTypeEQ = 2), as defined above. It is the ba-sic module. For example,BVPh[1,10] gives the 1st to 10th-order homotopy-approximations. Thereafter,BVPh[11,15] further gives the 11th to 15th-orderhomotopy-approximations.

iter The moduleiter[k ,m ,r ] provides us homotopy-approximations ofthe kth to mth iteration by means of therth-order iteration formula (7.51). Itcalls the basic moduleBVPh. For example,iter[1,10,3] gives homotopy-approximations of the 1st to 10th iteration by the 3rd-orderiteration formula. Fur-thermore,iter[11,20,3] gives the homotopy-approximations of the 11th to20th iterations. For eigenvalue problems, the initial guess of eigenvalue is deter-mined by an algebraic equation. Thus, if there are more than one initial guessesof eigenvalue, it is required to choose one among them by inputting an integer,such as 1 or 2, corresponding to the 1st or the 2nd initial guess of the eigen-value, respectively. If the convergence-control parameter c0 is unknown at thebeginning of iteration, curves of squared residual of governing equation at theup-to 3rd-order approximations versusc0 are given at the 1st iteration, in orderto choose a proper value ofc0. This value ofc0 will be renewed afterNupdatetimes iterations. The iteration stops when the squared residual of governing equa-tion is less than a critical valueErrReq , whose default is 10−20.

GetErr The moduleGetErr[k ] gives the squared residual of the govern-ing equation at thekth-order homotopy-approximation gained by the moduleBVPh, or at thekth-iteration homotopy-approximation obtained by the mod-ule iter . Note that,error[k] provides the residual of governing equa-tion at thekth-order homotopy-approximation gained byBVPh, or at thekth-iteration homotopy-approximation obtained byiter , andErr[k] gives the

7.3 A simple users guide of theBVPh1.0 263

averaged squared residual of the governing equation at thekth-order homotopy-approximation gained byBVPh, andERR[k] gives the averaged squared resid-ual of the governing equation at thekth-iteration homotopy-approximation ob-tained byiter , respectively.

GetMin1D The moduleGetMin1D[f ,x ,a ,b ,Npoint ] searches forthe local minimums of a real functionf (x) in the intervalx ∈ [a,b] by meansof dividing the interval[a,b] into Npoint equal parts. IfNpoint is largeenough, it gives all local minimums with the corresponding positionx. In general,Npoint = 20 is suggested. This module is often used to search for the optimalconvergence-controlparameterc0, or multiple solutions of a nonlinear boundary-value problem. It calls the moduleGetMin1D0[f ,x ,a ,b ,Npoint ] .

GetMin2D Using GetMin2D[f ,x ,a ,b ,y ,c ,d ,Npoint ] , we cansearch for the local minimums of a real functionf (x,y) in the intervala≤ x≤ b,c ≤ y ≤ d by dividing it into Npoint × Npoint equal parts. IfNpoint islarge enough, it gives all local minimums with the corresponding position(x,y).In general,Npoint = 20 is suggested. This module is often used to search formultiple solutions of a nonlinear boundary-value problem.It calls the moduleGetMin2D0[f ,x ,a ,b ,c ,d ,Npoint ] .

hp The modulehp[f ,m ,n ] gives the[m,n] homotopy-Pade approximationof the homotopy-approximationf , wheref[0],f[1],f[2] denotes the ze-roth, first and 2nd-order homotopy-approximation off . For details about thehomotopy-Pade approximation, please refer to Chapter 2.

7.3.2 Control parameters

TypeEQ A control parameter for the type of governing equation:TypeEQ = 1corresponds to a nonlinear boundary-value equation,TypeEQ = 2 correspondsto a nonlinear eigenvalue problem, respectively.

TypeL A control parameter for the type of base functions:TypeL = 1 corre-sponds to polynomial (7.52) or Chebyshev polynomial (7.53), andTypeL = 2corresponds to a trigonometric approximation or a hybrid-base approximationdescribed in§7.2.3, respectively. It is valid only for boundary-value/eigenvalueproblems in a finite intervalz∈ [0,a], wherea> 0 is a constant.

ApproxQ A control parameter for approximation of solutions. WhenApproxQ= 1, the right-hand side term of all high-order deformation equations is ap-proximated by Chebyshev polynomial (7.53), or the hybrid-base approximationdescribed in§ 7.2.3. WhenApproxQ = 0 , there is no such kind of approxi-mation. When the moduleiter is employed,ApproxQ = 1 is automatically


assigned. For theBVPh(version 1.0), this parameter is valid only for boundary-value/eigenvalue problems in a finite intervalz∈ [0,a], wherea> 0 is a constant.

TypeBase A control parameter for the type of base functions to approximatesolutions:TypeBase = 0 corresponds to the Chebyshev polynomial (7.53),TypeBase = 1 corresponds to the hybrid-base (7.58) with the odd Fourierapproximation (7.63),TypeBase = 2 corresponds to the hybrid-base (7.58)with the even Fourier approximation (7.60), respectively.This parameter is validonly whenApproxQ = 1 for boundary-value/eigenvalue problems in a finiteintervalz∈ [0,a].

Ntruncated A control parameter to determine the positive integerNt > 0, cor-responding to the number of truncated terms in (7.53), (7.60) and (7.63). Thelarger the numberNt , the better the approximations, but the more CPU times. Itis valid only whenApproxQ = 1 for boundary-value/eigenvalue problems ina finite intervalz∈ [0,a]. The default is 10.

NtermMax A positive integer used in the moduletruncated , which ignoresall polynomial terms whose order is higher thanNtermMax . The default is 90.

ErrReq A critical value of squared residual of governing equation to stop theiteration of the moduleiter . The default is 10−10.

NgetErr A positive integer used in the moduleBVPh. The squared residual ofgoverning equation is calculated when the order of approximation divided byNgetErr is an integer. The default is 2.

Nupdate A critical value of the times of iteration to update the convergence-control parameterc0. The default is 10.

Nintegral Number of discrete points with equal space, which are used tonu-merically calculate the integral of a function. It is used inthe moduleint . Thedefault is 50.

ComplexQ A control parameter for complex variables.ComplexQ = 1 cor-responds to the existence of complex variables in governingequations and/orboundary conditions.ComplexQ = 0 corresponds to the nonexistence of suchkind of complex variables. The default is 0.

c0L A real number to determine the interval of the convergence-control param-eter c0 for plotting curves of the squared residual of the governingequationversusc0 in the moduleiter . The default is -2, corresponding to the interval−2≤ c0 ≤ 0. The value ofc0L can be positive, such asc0L = 1 , correspond-ing to the interval 0≤ c0 ≤ 1.


7.3.3 Input

f[z ,u ,lambda ] The governing equation, corresponding toF [z,u] = 0 fornonlinear boundary-value problems in either a finite interval z∈ [0,a] or an infi-nite intervalz∈ [b,+∞), or corresponding toF [z,u,λ ] = 0 for nonlinear eigen-value problems in a finite intervalz∈ [0,a], wherea > 0,b ≥ 0 are boundedconstants. Note that,lambda denotes the eigenvalue for nonlinear eigenvalueproblems, but has no meanings at all for nonlinear boundary-value problems.

BC[k ,z ,u ,lambda ] Thekth boundary condition, corresponding to eitherBk[z,u] = 0 for nonlinear boundary-value problems, orBk[z,u,λ ] = 0 for non-linear eigenvalue problems, respectively, where 0≤ k ≤ n. Note that,lambdadenotes the eigenvalue for nonlinear eigenvalue problems,but has no meaningsat all for nonlinear boundary-value problems.

u[0] The initial guessu0(z).

L[f ] The auxiliary linear operator. For boundary-value/eigenvalue problemsin a finite intervalz∈ [0,a], the auxiliary linear operator (7.37) is automaticallychosen whenTypeL=1 , and the auxiliary linear operator (7.40) or (7.41) is usedotherwise. For boundary-value problems in an infinite intervalz∈ [b,+∞), whereb≥ 0 is a bounded constant, one may choose either (7.42) for exponentially de-caying solutions or (7.44) for algebraically decaying solutions, respectively. Inany cases, the auxiliary linear operator must be clearly defined and properly cho-sen.

H[z ] The auxiliary linear operator. The default isH[z ]:=1 .

OrderEQ The order of the boundary-value equationF [z,u] = 0 or the eigen-value equationF [z,u,λ ] = 0 .

zL The left end-point of the intervalz∈ [a,b], corresponding toz= a. For exam-ple, zL=1 corresponds toa = 1. For boundary-value/eigenvalue problems in afinite intervalz∈ [0,a], zL = 0 is automatically assigned. The default is 0.

zR The right end-point of the intervalz∈ [a,b], corresponding toz= b. For ex-ample,zR=1 corresponds tob= 1. For boundary-value problems in an infiniteinterval,zR = infinity must be used forBVPh(version 1.0).


7.3.4 Output

U[k] Thekth-order homotopy-approximation ofu(z) given by the basic moduleBVPh.

V[k] The kth-iteration homotopy-approximation ofu(z) given by the iterationmoduleiter .

Uz[k] Thekth-order homotopy-approximation ofu′(z) given by the basic mod-uleBVPh.

Vz[k] Thekth-iteration homotopy-approximation ofu′(z) given by the iterationmoduleiter .

Lambda[k] Thekth-order homotopy-approximation of the eigenvalueλ givenby the basic moduleBVPh.

LAMBDA[k] The kth-iteration homotopy-approximation of the eigenvalueλgiven by the iteration moduleiter .

error[k] The residual of governing equation given by either thekth-orderhomotopy-approximation (obtained by the basic moduleBVPh) or the kth-iteration homotopy-approximation (obtained by the iteration moduleiter ).

Err[k] The averaged squared residual of governing equation given by thekth-order homotopy-approximation (obtained by the basic moduleBVPh).

ERR[k] The averaged squared residual of governing equation given by thekth-iteration homotopy-approximation (obtained by the iteration moduleiter ).

7.3.5 Global variables

All control parameters and output variables mentioned above are global. Excepttheses, the following variables and parameters are also global.

z The independent variablez.

u[k] The solutionuk(z) of thekth-order deformation equation.

lambda[k] A constant variable, corresponding toλk.


delta[k] A function dependent uponz, corresponding to the right-hand sidetermδk(z) in the high-order deformation equation.

L The auxiliary linear operatorL .

Linv The inverse operator ofL , corresponding toL −1.

nIter The number of iteration, used in the moduleiter .

sNum A positive integer, which determines which initial guessλ0 of eigenvalueis chosen when there exist multiple solutions ofλ0.


Mathematica packageBVPh (version 1.0)

TheBVPh (version 1.0) is a HAM-based Mathematica package for some types ofnth-order nonlinear boundary-value/eigenvalue problems,developed by Shijun Liaoof the Shanghai Jiao Tong University beginning in 2010. It isan open resource andfree available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

Copyright StatementCopyright c©2011, The University of Shanghai Jiao Tong University, and the

BVPhDevelopers. All rights reserved.Redistribution and use in source and binary forms, with or without modification,

are permitted provided that the following conditions are met:

• Redistributions of source code must retain the above copyright notice, this list ofconditions and the following disclaimer.

• Redistributions in binary form must reproduce the above copyright notice, thislist of conditions and the following disclaimer in the documentation and/or othermaterials provided with the distribution.

• Redistributions and uses in source and binary forms for profit purpose, with orwithout modification, are not allowed without written agreement from theBVPhdevelopers.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBU-

TORS ”AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT

LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR

A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT

HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,

SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES HOWEVER CAUSED AND ON

ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT

ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF

THE POSSIBILITY OF SUCH DAMAGE.

Appendix 7.1 Mathematica packageBVPh (version 1.0) 269

Mathematica packageBVPh(version 1.0)by Shijun LIAO

Shanghai Jiao Tong UniversityAugust 2010


( *************************************************** )( * Default of control parameters * )( *************************************************** )Ntruncated = 10;Nupdate = 10;NtermMax = 90;Nintegral = 50;ErrReq = 10ˆ(-20);NgetErr = 2;ComplexQ = 0;kappa = 1;PRN = 1;c0L = -2;zL = 0;c0 = .;

( *************************************************** )( * Default auxiliary function * )( *************************************************** )H[z_] := 1;

U[0] = u[0];( *************************************************** )( * Define Linv * )( * Find solution of linear equation: L[u ] = f * )( *************************************************** )Linv[f_] := Module[temp,w,temp = DSolve[ L[w[z]] == f, w[z], z];temp[[1,1,2]] /. C[_]->0 // TrigReduce];

( *************************************************** )( * Property of the inverse operator of L * )( * Linv[f_+g_] := Linv[f] + Linv[g] * )( *************************************************** )Linv[p_Plus] := Map[Linv,p];Linv[c_ * f_] := c * Linv[f] /; FreeQ[c,z];

( *************************************************** )( * Define Getdelta[k] * )( * Get delta[k] automatically based on * )( * the definition of f[z,u,lambda] * )( *************************************************** )


Getdelta[k_]:=Module[temp,phi,LAMBDA,w,lamb,q,eq,m,n,coeff,Coeff,

eq = f[z,phi[z,q],LAMBDA[q]];temp[0] = D[eq,q,k]/k! // Expand;temp[1] = temp[0]/.D[phi[z,q],z,m_,q,n_]

->n! * diff[w[n],z,m];temp[2] = temp[1]/.D[LAMBDA[q],q,n_]

->n! * lamb[n];temp[3] = temp[2]/.phi[z,q]->w[0],

LAMBDA[q]->lamb[0];temp[4] = temp[3]/.diff[w[m_],z,0]->w[m];temp[5] = temp[4]/.w->u,diff->D;delta0[k] = temp[5] /. lamb->lambda;If[ApproxQ == 0,

temp[-1] = delta0[k]//Expand//TrigReduce];

If[ApproxQ == 1 && TypeL == 1,If[TypeEQ == 1,temp[-1]=ChebyApproxA[delta0[k],z,0,zR,Ntruncated],For[m=0,m<=k,m++,

coeff[m]=Coefficient[temp[5],lamb[m]]];

coeff[-1] = temp[5]/.lamb[_]->0;For[n=-1,n<=k,n++,Coeff[n]=ChebyApproxA[coeff[n],z,0,zR,Ntruncated];

];lamb[-1] = 1;temp[-1] = Sum[Coeff[n] * lamb[n],n,-1,k]

/. lamb->lambda;];];If[ApproxQ == 1 && TypeL == 2,If[TypeEQ == 1,temp[-1] = TrigApprox[delta0[k],z,zR,

Ntruncated,TypeBase],For[m=0,m<=k,m++,

coeff[m]=Coefficient[temp[5],lamb[m]]];

coeff[-1] = temp[5]/.lamb[_]->0;For[n=-1,n<=k,n++,

Coeff[n] = TrigApprox[coeff[n],z,zR,Ntruncated,TypeBase];

];lamb[-1] = 1;temp[-1] = Sum[Coeff[n] * lamb[n],n,-1,k]

/. lamb->lambda;];];delta[k] = temp[-1]//Expand//GetDigit;];


( *************************************************** )( * Define GetBC[k] * )( * Get boundary conditions automatically * )( * based on the definition BC[n,z,u,lambda] * )( *************************************************** )GetBC[i_,k_] := Module[temp,j,phi,q,LAMB,phi = Sum[u[j] * qˆj,j,0,k];LAMB = Sum[lambda[j] * qˆj,j,0,k];temp[0] = BC[i,z,phi,LAMB];temp[1] = Series[temp[0],q,0,k]//Normal;temp[2] = Coefficient[temp[1],qˆk];temp[2]//Expand//GetDigit];

( *************************************************** )( * Define functions chi[m] * )( *************************************************** )chi[m_] := If[m <= 1, 0, 1];

( *************************************************** )( * Define truncated[zˆn_.] * )( *************************************************** )truncated[a_] := a /; FreeQ[a,z];truncated[a_ * f_] := a * truncated[f] /; FreeQ[a,z];truncated[p_Plus] := Map[truncated,p];truncated[zˆn_.] := If[n > NtermMax, 0, zˆn];

( *************************************************** )( * Define GetReal[] and GetDigit[] * )( *************************************************** )Naccu = 100;GetReal[c_] := N[IntegerPart[c * 10ˆNaccu]

/10ˆNaccu, Naccu] /; NumberQ[c];GetDigit[c_] := N[IntegerPart[c * 10ˆNaccu]

/10ˆNaccu, Naccu] /; NumberQ[c];Default[GetDigit, 1] = 0;GetDigit[p_Plus] := Map[GetDigit, p];GetDigit[c_. * f_] := GetReal[c] * f /; NumberQ[c];GetDigit[c_.] := 0 /; NumberQ[c]

&& Abs[c] < 10ˆ(-Naccu+1);

( *************************************************** )( * Define int[f,x,x0,x1,Nintegral] * )( * Integration of f in the interval [x0,x1] * )( * by Nintegral points * )( *************************************************** )int[f_, x_, x0_, x1_, Nintegral_]:= Module[temp,dx,s,t,i,M,M = Nintegral;dx = N[x1-x0,100]/M;temp[0] = Series[f,x,0,2]//Normal;temp[0] = temp[0] /. xˆ_. ->0 ;s = temp[0];For[i=1,i<=M,i=i+1,


t = x0 + i * dx;temp[i] = f /.x->t;s = s + temp[i]//Expand;];

temp[M+1] = (temp[0]+temp[M])/2;(s - temp[M+1]) * dx//Expand];

( *************************************************** )( * Define ChebyApproxA[f,x,a,b,M] * )( * Approximate a function by Chebyshev polynomial * )( *************************************************** )ChebyApproxA[f_,x_,a_,b_,Ntruncated_]:= Module[temp,n,z,t,F,A,temp[0] = f/. x-> a + (b-a) * (z+1)/2;F = temp[0] /. z -> Cos[t];For[n=0, n<=Ntruncated, n++,

A[n] = 2 * int[F * Cos[n * t],t,0,Pi,100]/Pi];

For[n = 0, n<=Ntruncated, n++,temp[0] = A[n]//Expand;A[n] = GetDigit[temp[0]];];

temp[1] = A[0]/2 + Sum[A[n] * ChebyshevT[n,z],n,1,Ntruncated];

temp[1] /. z-> -1 + 2 * (x-a)/(b-a)//Expand];

( *************************************************** )( * Define TrigApprox[f,x,Ntruncated,TypeBase] * )( * Approximate f[x] in [0,xR] : * )( * TypeBase = 0 : Fourier series * )( * TypeBase = 1 : hybrid-base * )( * Ntruncated : number of truncated terms * )( *************************************************** )TrigApprox[f_,x_,xR_,Ntruncated_,TypeBase_]:= Module[temp,a,b,c,y,i,j,t,If[TypeBase == 0, temp[0] = f/.x->t];If[TypeBase == 1,

temp[0] = Series[f,x,0,2]//Normal;temp[1] = temp[0] /. xˆ_. -> 0;temp[2] = f /. x->xR;temp[3] = temp[1]+(temp[2]-temp[1])/xR * x//Expand;y = GetDigit[temp[3]];temp[0] = f - y /. x->t;];

If[TypeBase == 2,temp[0] = D[f,x]//Expand;temp[1] = Series[temp[0],x,0,2]//Normal;temp[1] = temp[1] /. xˆ_. -> 0;temp[2] = temp[0] /. x->xR;a = temp[1];b = -(temp[1] + temp[2])/2/xR;temp[3] = (a * x+b * xˆ2) * Cos[Pi * x/xR]//Expand;


y = GetDigit[temp[3]];temp[0] = f - y /. x-> t ;];

If[TypeBase == 0 || TypeBase == 2,For[i = 0, i <= Ntruncated, i++,

c[i] = 2 * int[temp[0] * Cos[i * t * Pi/xR],t,0,xR,Nintegral]/xR;

];temp[4] = c[0]/2

+ Sum[c[j] * Cos[j * x* Pi/xR],j,1,Ntruncated];];

If[TypeBase == 1,For[i = 1, i <= Ntruncated, i++,

c[i] = 2 * int[temp[0] * Sin[i * t * Pi/xR],t,0,xR,Nintegral]/xR;

];temp[4] = Sum[c[j] * Sin[j * x* Pi/xR],

j,1,Ntruncated];];

If[TypeBase == 0,temp[5] = GetDigit[temp[4]],temp[5] = GetDigit[temp[4] + y]];

temp[5]];

( *************************************************** )( * Define GetMin1D[f,x,a,b,Npoint] * )( *************************************************** )GetMin1D[f_,x_,x0_,x1_,Npoint_] := Module[temp,fmin,Xmin,xmin0,xmin1,dx,Num,i,j,X,s,Num = 0;dx = (x1-x0)/Npoint //GetDigit;For[i = 0, i <= Npoint, i++,

X[i] = x0 + i * dx //GetDigit;temp[i] = f /. x -> X[i];];

For[i = 1, i <= Npoint-1, i++,If[temp[i] < temp[i-1] && temp[i] < temp[i+1],

Num = Num + 1;fmin[Num] = temp[i];Xmin[Num] = X[i];];

];For[i = 1, i <= Num, i++,

j = 0;xmin0 = Xmin[i];Label[100];j = j + 1;temp[0] = xmin0 - dx/5ˆ(j-1) //GetDigit;temp[1] = xmin0 + dx/5ˆ(j-1) //GetDigit;s = GetMin1D0[f,x,temp[0],temp[1],10];xmin1 = s[[2]];If[Abs[xmin1 - xmin0] < 10ˆ(-20),


Xmin[i] = xmin1;fmin[i] = f /. x-> xmin1;Goto[200];];

xmin0 = xmin1;Goto[100];Label[200];Print[" Minimum = ",fmin[i]//N, " at ",

x, " = ",N[Xmin[i],20]];];

];

GetMin1D0[f_,x_,x0_,x1_,Npoint_]:= Module[temp, fmin, Xmin, dx, i,z,F,dx = (x1-x0)/Npoint // GetDigit;fmin = 10ˆ100;Xmin = 0;For[i = 1, i <= Npoint, i++,

z = x0 + i * dx // GetDigit;F = f /. x->z // GetDigit;If[F < fmin,

fmin = F;Xmin = z;];

];fmin,Xmin];

( *************************************************** )( * Define GetMin2D[f,x,x0,x1,y,y0,y1,Npoint] * )( *************************************************** )GetMin2D[f_,x_,x0_,x1_,y_,y0_,y1_,Npoint_]:=Module[temp,fmin,Xmin,Ymin,xmin0,xmin1,

ymin0,ymin1,dx,dy,Num,i,j,X,Y,s,Num = 0;dx = (x1-x0)/Npoint //GetDigit;dy = (y1-y0)/Npoint //GetDigit;For[i = 0, i <= Npoint, i++,

X[i] = x0 + i * dx //GetDigit;For[j = 0, j <= Npoint, j++,

Y[j] = y0 + j * dy //GetDigit;temp[i,j] = f /. x -> X[i], y->Y[j];];

];For[i = 1, i <= Npoint-1, i++,

For[j = 1, j <= Npoint-1, j++,If[temp[i,j] < temp[i-1,j]

&& temp[i,j] < temp[i+1,j]&& temp[i,j] < temp[i,j-1]&& temp[i,j] < temp[i,j+1],Num = Num + 1;fmin[Num] = temp[i,j];Xmin[Num] = X[i];Ymin[Num] = Y[j];


];];

];For[i = 1, i <= Num, i++,

j = 0;xmin0 = Xmin[i];ymin0 = Ymin[i];Label[100];j = j + 1;X[0] = xmin0 - dx/5ˆ(j-1) //GetDigit;X[1] = xmin0 + dx/5ˆ(j-1) //GetDigit;Y[0] = ymin0 - dy/5ˆ(j-1) //GetDigit;Y[1] = ymin0 + dy/5ˆ(j-1) //GetDigit;s = GetMin2D0[f,x,X[0],X[1],y,Y[0],Y[1],10];xmin1 = s[[2]];ymin1 = s[[3]];If[Abs[xmin1 - xmin0] < 10ˆ(-20)

&& Abs[ymin1 - ymin0] < 10ˆ(-20),Xmin[i] = xmin1;Ymin[i] = ymin1;fmin[i] = f /. x-> xmin1, y->ymin1;Goto[200];];

xmin0 = xmin1;ymin0 = ymin1;Goto[100];Label[200];Print[" Minimum = ",fmin[i]//N];Print[" at ", x, " = ",N[Xmin[i],20]];Print[" ", y, " = ",N[Ymin[i],20]];];

];

GetMin2D0[f_,x_,x0_,x1_,y_,y0_,y1_,Npoint_]:= Module[temp,fmin,Xmin,Ymin,dx,dy,i,X,Y,F,dx = (x1-x0)/Npoint // GetDigit;dy = (y1-y0)/Npoint // GetDigit;fmin = 10ˆ10;Xmin = 10ˆ10;Ymin = 10ˆ10;For[i = 1, i <= Npoint, i++,

X = x0 + i * dx // GetDigit;For[j = 1, j <= Npoint, j++,

Y = y0 + j * dy // GetDigit;F = f /. x->X, y->Y // GetDigit;If[F < fmin,

fmin = F;Xmin = X;Ymin = Y];

];];

fmin,Xmin,Ymin];


( *************************************************** )( * Define GetErr[m] * )( * This module gives averaged squared residual * )( *************************************************** )GetErr[k_] := Module[temp,error[k] = f[z,U[k],Lambda[k-1]];If[ComplexQ == 0, temp[0] = error[k]ˆ2 ] ;If[ComplexQ == 1,

temp[1] = Re[error[k]];temp[2] = Im[error[k]];temp[0] = temp[1]ˆ2 + temp[2]ˆ2;

];If[!NumberQ[zRintegral],

If[FreeQ[zR,infinity] && zR < 100,zRintegral = zR,Print["Squared residual is integrated

in the interval [",zL,",b]"];Print[" The value of b = ? "];zRintegral = Input[];];

];temp[1] = Abs[zRintegral-zL];Err[k] = int[temp[0],z,zL,zRintegral,Nintegral]

/temp[1];];

( *************************************************** )( * Define hp[f_,m_,n_] * )( *************************************************** )hp[f_,m_,n_]:=Block[k,i,df,res,q,df[0] = f[0];For[k=1,k<=m+n,k++,df[k]=f[k]-f[k-1]//Expand];res = df[0] + Sum[df[i] * qˆi,i,1,m+n];Pade[res,q,0,m,n]/.q->1];

( *************************************************** )( * Main Code * )( * HAM approach without iteration * )( *************************************************** )BVPh[begin_,end_]:=Block[uSpecial,CC,temp,ss,w,i,j,EQ,Unknown,phi,LAM,

sLeng,sMin,sMinI,RHS,base,a,time[0] = SessionTime[];For[k=begin,k<=end,k=k+1,

If[PRN == 1, Print["k = ",k]];If[k == 1 && NumberQ[c0], c0 = GetDigit[c0]];If[k == 1, Clear[lambda]];

( * Get special solution * )Getdelta[k-1];RHS = H[z] * delta[k-1]//Expand;temp[1] = Linv[RHS];


uSpecial = c0 * temp[1] + chi[k] * u[k-1]//Expand;

( * Get lambda[k-1] and u[k] * )temp[1] = DSolve[L[w[z]]==0,w[z],z];temp[2] = temp[1][[1,1,2]];Unknown = ;For[i = 1, i <= OrderEQ, i++,

base[i] = Coefficient[temp[2],C[i]];If[FreeQ[base[i],Exp[n_. * z]],

Unknown = Union[Unknown,CC[i,k]],If[( Abs[base[i]] /. z -> 1. ) < 1,

Unknown = Union[Unknown,CC[i,k]],CC[i,k] = 0;base[i] = 0;];

];];

ss = Sum[base[i] * CC[i,k],i,1,OrderEQ];u[k] = uSpecial + ss // Expand;For[i = 1, i <= OrderEQ, i++,

eq[i] = GetBC[i,k]//Expand//GetDigit];

EQ = ;For[i=1, i <= OrderEQ, i++,

If[FreeQ[BC[i,z,w[z],a],infinity],EQ = Union[EQ,eq[i] == 0];];

];

If[TypeEQ != 1,eq[0] = GetBC[0,k]//Expand//GetDigit;EQ = Union[EQ,eq[0] == 0];Unknown = Union[Unknown,lambda[k-1]];];

ss = Solve[EQ,Unknown];sLeng = Length[ss];If[sLeng == 1, sNum = 1];If[k == 1 && !NumberQ[sNum],

For[j = 1, j <= sLeng, j++,temp[1] = lambda[0] /. ss[[j]];Print[j," th solution of lambda[0] = ",

temp[1]//N];];

Print["which solution is chosen ? "];sNum = Max[1,Input[]];];

If[k == 1 && TypeEQ != 1 && NumberQ[sNum]&& NumberQ[LAMBDA[nIter-1]],

sMin = 10ˆ10;For[j = 1, j <= sLeng, j++,

temp[1] = lambda[0] /. ss[[j]];


temp[2] = Abs[temp[1]-LAMBDA[nIter-1]];If[ temp[2] < sMin,

sNum = j;sMin = temp[2];];

];];

For[i = 1, i <= OrderEQ, i++,temp[1] = CC[i,k] /. ss[[sNum]]//Expand;CC[i,k] = temp[1]//GetDigit;];

u[k] = u[k]//Expand//GetDigit;U[k] = Expand[U[k-1] + u[k]];Uz[k] = D[U[k],z];

If[TypeEQ != 1,temp[1] = lambda[k-1] /. ss[[sNum]]//Expand;lambda[k-1] = temp[1]//GetDigit;Lambda[k-1] = Sum[lambda[i],i,0,k-1]

//Expand//GetDigit;If[PRN == 1,

Print[k-1,"th-order approx. of eigenvalue= ",Lambda[k-1]//N ];

];];

( * Print results * )If[PRN == 1 && NumberQ[c0] && TypeEQ == 1,

output[z,U,k]];

If[IntegerQ[k/NgetErr] && PRN == 1,GetErr[k];If[NumberQ[Err[k]],

Print["Squared residual = ",Err[k]//N];];

If[Err[k] < ErrReq,Print["Congratulation: the required

squared residual is satisfied !"];Goto[end]];

];

If[PRN == 1,time[k] = SessionTime[];temp[0] = time[k]-time[0];Print["Used CPU times = ",temp[0],

" (seconds) "];];

];If[PRN == 1, Print[" Successful !"]];Label[end];];


( *************************************************** )( * Main Code * )( * HAM approach with iteration * )( * begin : number of starting iteration * )( * end : number of stopping iteration * )( * OrderIter : order of iteration formula * )( *************************************************** )iter[begin_, end_, OrderIter_]:=Block[temp,time,PRN = 0;If[TypeL != 1, ApproxQ = 1];For[nIter=Max[1,begin],nIter<=end,nIter=nIter+1,

If[nIter == 1,sNum = .;If[NumberQ[c0], c0 = GetDigit[c0]];Time[0] = SessionTime[];V[0] = u[0]//Expand//GetDigit;Vz[0] = D[V[0],z];];

If[nIter > 1,u[0] = V[nIter-1];U[0] = u[0];];

Print[ nIter, " th iteration:"];

If[NumberQ[c0], BVPh[1,OrderIter]];If[!NumberQ[c0],

BVPh[1,Max[3,OrderIter]];GetErr[1];GetErr[2];GetErr[3];Print[" Squared Residual of Governing EQ:"];Print[" Red line : 1st-order approx."];Print[" Green line : 2nd-order approx."];Print[" Blue line : 3rd-order approx."];LogPlot[Abs[Err[1]],Abs[Err[2]],Abs[Err[3]],

c0,c0L,0,PlotStyle->RGBColor[1,0,0],RGBColor[0,1,0],RGBColor[0,0,1]];

Print[" Adjust the interval of c0 by choosinga better end-point "];

Print[" New end-point on left (c0 < 0)or right (c0 > 0) = ?(Input 0 to skip it) "];

c0L = Input[];If[c0L != 0,LogPlot[Abs[Err[1]],Abs[Err[2]],Abs[Err[3]],

c0,c0L,0,PlotStyle->RGBColor[1,0,0],RGBColor[0,1,0],RGBColor[0,0,1]];

];Print[" Choose the value of

convergence-control parameter c0"];Print[" c0 = ? (Input 0 to stop the code)"];temp[0] = Input[];c0 = GetDigit[temp[0]];Print[" c0 = ",c0//N];


If[c0==0,c0 =.;Print["Stop!"];Goto[stop]];];

temp[0] = U[OrderIter]//Expand;If[PolynomialQ[temp[0],z],

temp[0] = truncated[temp[0]]];

V[nIter] = GetDigit[temp[0]];Vz[nIter] = D[V[nIter],z];If[TypeEQ == 1,

output[z,V,nIter],LAMBDA[nIter]=Lambda[OrderIter-1]//Expand;Print[" Eigenvalue = ",LAMBDA[nIter]//N ];];

GetErr[OrderIter];ERR[nIter] = Err[OrderIter];Print[" Squared Residual = ",ERR[nIter]//N];If[ERR[nIter] < ErrReq,

Print["STOP:Required accuracy satisfied!"];Goto[stop];];

If[nIter > 1 && ERR[nIter] > ERR[nIter-1]&& Err[nIter] < 10ˆ(-7),Print["Squared residual does NOT decrease!"];Goto[stop];];

Time[nIter] = SessionTime[];Temp[-1] = Time[nIter]-Time[0];Print[" Used CPU times = ", Temp[-1],

" (seconds) "];If[IntegerQ[nIter/Nupdate]&&Abs[c0]< 1/2,c0 =.];];

Print[" End of iteration ! "];Label[stop];PRN = 1;];

( *************************************************** )( * Print equation, boundary conditions * )( * and control parameters * )( *************************************************** )PrintInput[s_] := Module[,Print["-------------------------------------------" ];Print["The values of control parameters: "];Print[" Nupdate = ", Nupdate];Print[" Ntruncated = ", Ntruncated];Print[" NtermMax = ", NtermMax];Print[" Nintegral = ", Nintegral];Print[" ErrReq = ", ErrReq//N];Print[" NgetErr = ", NgetErr];Print[" ComplexQ = ", ComplexQ];Print[" PRN = ", PRN];Print[" c0L = ", c0L];


Print[" zL = ", zL];Print["-------------------------------------------" ];Print[" Governing Equation : ",

f[z,s,lambda], " = 0 "];Print[" Interval of z : ",

zL, " < z < ", zR];If[TypeEQ != 1,

Print[" ",0,"th Boundary Condition : ",BC[0, z, s, lambda]," = 0"];

];For[i=1, i<= OrderEQ, i++,

Print[" ",i,"th Boundary Condition : ",BC[i, z, s, lambda]," = 0"];

];Print["-------------------------------------------" ];If[ApproxQ == 0,

Print[" Delta[k] is NOT approximatedby other base functions ! "],

If[TypeL == 1,Print[" Base function : Chebyshev "],If[TypeBase == 1,

Print[" Base function : hybrid-basewith sine and polynomial"],

Print[" Base function : hybrid-basewith cosine and polynomial "]

];];

];Print[" Auxiliary linear operator L[u] = ",L[s]];If[TypeL != 1,

Print[" kappa = ",kappa]];

Print[" Initial guess u[0] = ",u[0]];Print[" Auxiliary function H[z] = ",H[z]];If[NumberQ[c0],

Print[" Convergence-control parameter c0 = ",c0]];

Print["-------------------------------------------" ];];


References

1. Abbas, Z., Wang, Y., Hayat, T., Oberlack, M.: Hydromagnetic flow in a viscoelstic fluid dueto the oscillatory stretching surface. Int. J. Nonlin. Mech. 43, 783 – 793 (2008)


3. Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hiro-taSatsuma coupled KdV equation. Phys. Lett. A.361, 478 – 483 (2007)

4. Abbasbandy, S., Parkes, E.J.: Solitary smooth hump solutions of the Camassa-Holm equationby means of the homotopy analysis method. Chaos Soliton. Fract. 36, 581 – 591 (2008)

5. Abbasbandy, S.: Solitary wave equations to the Kuramoto-Sivashinsky equation by means ofthe homotopy analysis method. Nonlinear Dynam.52, 35 – 40 (2008)


7. Abbasbandy, S., Parkes, E.J.: Solitary-wave solutions of the DegasperisProcesi equation bymeans of the homotopy analysis method. Int. J. Comp. Math.87, 2303 – 2313 (2010)

8. Abbasbandy, S., Shivanian, E.: Predictor homotopy analysis method and its application tosome nonlinear problems. Commun. Nonlinear Sci. Numer. Simulat.16, 2456 – 2468 (2011)


10. Akyildiz, F.T., Vajravelu, K. Magnetohydrodynamic flowof a viscoelastic fluid. Phys. Lett.A. 372, 3380 – 3384 (2008)

11. Akyildiz, F.T., Vajravelu, K., Mohapatra, R.N., Sweet,E., Van Gorder, R.A.: Implicit differ-ential equation arising in the steady flow of a Sisko fluid. Applied Mathematics and Compu-tation.210, 189 – 196 (2009)

12. Alizadeh-Pahlavan, A., Borjian-Boroujeni, S.: On the analytical solution of viscous fluid flowpast a flat plate. Physics Letters A.372, 3678 – 3682 (2008)

13. Alizadeh-Pahlavan, A., Aliakbar, V., Vakili-Farahani, F., Sadeghy, K.: MHD flows of UCMfluids above porous stretching sheets using two-auxiliary-parameter homotopy analysismethod. Commun. Nonlinear Sci. Numer. Simulat.14, 473 – 488 (2009)

14. Allan, F.M., Syam, M.I.: On the analytic solutions of thenonhomogeneous Blasius problem.J. Comp. Appl. Math.182, 362 – 371 (2005)

15. Allan, F.M.: Derivation of the Adomian decomposition method using the homotopy analysismethod. Appl. Math. Comput.190, 6 – 14 (2007)

16. Allan, F.M.: Construction of analytic solution to chaotic dynamical systems using the Homo-topy analysis method. Chaos, Solitons and Fractals.39, 1744 – 1752 (2009)

17. Battles, Z., Trefethen, L.N.: An extension of Matlab to continuous functions and operators.SIAM J. Sci. Comp.25, 1743 – 1770 (2004)

18. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. DOVER Publications, Inc. New York(2000)

19. Cai, .W.H.: Nonlinear Dynamics of Thermal-Hydraulic Networks. PhD dissertation, Univer-sity of Notre Dame (2006)

20. Cheng, J.: Application of The Homotopy Analysis Method in Nonlinear Mechanics and Fi-nance. PhD dissertation, Shanghai Jiao Tong University (2008)

21. Gao, L.M.: Analysis of the Propagation of Surface Acoustic Waves in Functionally GradedMaterial Plate. PhD dissertation, Tong Ji University (2007)

22. Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down avertical cylinder. Phys. Lett. A.361, 316–322 (2007)

23. Jiao, X.Y.: Approximate Similarity Reduction and Approximate Homotopy Similarity Reduc-tion of Several Nonlinear Problems. PhD dissertation, Shanghai Jiao Tong University (2009)

24. Jiao, X.Y., Gao, Y., Lou, S.Y.: Approximate homotopy symmetry method – Homotopy seriessolutions to the sixth-order Boussinesq equation. Sciencein China (G).52, 1169 – 1178(2009)

References 283

25. Kierzenka, J., Shampine, L.F.: A BVP solver based on residual control and the MATLABPSE. ACM TOMS.27, 299 – 316 (2001).

26. Kumari, M., Nath, G.: Unsteady MHD mixed convection flow over an impulsively stretchedpermeable vertical surface in a quiescent fluid. Int. J. Non-Linear Mech.45, 310 – 319 (2010)

27. Kumari, M., Pop, I., Nath, G.: Transient MHD stagnation flow of a non-Newtonian fluid dueto impulsive motion from rest. Int. J. Non-Linear Mech.45, 463 – 473 (2010)


29. Liang, S.X.: Symbolic Methods for Analyzing Polynomialand Differential Systems. PhDdissertation, University of Western Ontario (2010)


31. Liang, S.X., Jeffrey, D.J.: An efficient analytical approach for solving fourth order boundaryvalue problems. Computer Physics Communications.180, 2034 – 2040 (2009)

32. Liang, S.X., Jeffrey, D.J.:Approximate solutions to a parameterized sixth order boundaryvalue problem. Computers and Mathematics with Applications.59, 247 – 253 (2010)












44. Liao, S.J.: On the reliability of computed chaotic solutions of non-linear differential equa-tions. Tellus.61A, 550 – 564 (2009)



47. Liu, Y.P.: Study on Analytic and Approximate Solution ofDifferential equations by SymbolicComputation. PhD Dissertation, East China Normal University (2008)

48. Liu, Y.P., Li, Z.B.: The homotopy analysis method for approximating the solution of themodified Korteweg-de Vries equation. Chaos Soliton. Fract.39, 1 – 8 (2009)

49. Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmos.Sci.20, 130 – 141 (1963).50. Mahapatra, T. R., Nandy, S.K., Gupta, A.S.: Analytical solution of magnetohydrodynamic

stagnation-point flow of a power-law fluid towards a stretching surface. Applied Mathematicsand Computation.215, 1696 – 1710 (2009)




53. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall/CRC Press, BocaRaton (2003)





58. Pandey, R.K. , Singh, O.P., Baranwal, V.K.: An analytic algorithm for the space – time frac-tional advection – dispersion equation. Computer Physics Communications.182, 1134 – 1144(2011)

59. Pirbodaghi, T., Ahmadian, M.T., Fesanghary, M.: On the homotopy analysis method for non-linear vibration of beams. Mechanics Research Communications.36, 143 – 148 (2009)

60. Sajid, M.: Similar and Non-Similar Analytic Solutions for Steady Flows of Differential TypeFluids. PhD dissertation, Quaid-I-Azam University (2006)


62. Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge Uni-versity Press. Cambridge (2003)

63. Shampine, L.F., Reichelt, M.W., Kierzenka, J.: Solvingboundary value prob-lems for ordinary differential equations in MATLAB with BVP4c. Available athttp://www.mathworks.com/bvptutorial.

64. Shidfar, A., Babaei, A., Molabahrami, A.: Solving the inverse problem of identifying an un-known source term in a parabolic equation. Computers and Mathematics with Applications.60, 1209 – 1213 (2010)

65. Shidfar, A., Molabahrami, A.: A weighted algorithm based on the homotopy analysis method- application to inverse heat conduction problems. Commun.Nonlinear Sci. Numer. Simulat.15, 2908 – 2915 (2010)

66. Siddheshwar, P.G.: A series solution for the Ginzburg-Landau equation with a time-periodic coefficient. Applied Mathematics.3, 542 – 554 ( 2010). Online available athttp://www.SciRP.org/journal/am

67. Singh, O.P., Pandey, R.K., Singh, V.K.: An analytic algorithm of LaneEmden type equationsarising in astrophysics using modified Homotopy analysis method. Computer Physics Com-munications.180, 1116 – 1124 (2009)

68. Song, H., Tao, L.: Homotopy analysis of 1D unsteady, nonlinear groundwater flow throughporous media. J. Coastal Res.50, 292 – 295 (2007)

69. Tao, L., Song, H., Chakrabarti, S.: Nonlinear progressive waves in water of finite depth – ananalytic approximation. Coastal Engineering.54, 825 – 834 (2007)


71. Turkyilmazoglu, M.: Purely analytic solutions of the compressible boundary layer flow dueto a porous rotating disk with heat transfer. Physics of Fluids.21, 106104 (2009)

72. Turkyilmazoglu, M.: A note on the homotopy analysis method. Appl. Math. Lett.23, 1226 –1230 (2010)

73. Turkyilmazoglu, M.: Series solution of nonlinear two-point singularly perturbed boundarylayer problems. Computers and Mathematics with Applications.60, 2109 – 2114 (2010)

74. Turkyilmazoglu, M.: An optimal analytic approximate solution for the limit cycle of Duffing- van der Pol equation. ASME J. Appl. Mech.78, 021005 (2011).

References 285

75. Turkyilmazoglu, M.: Numerical and analytical solutions for the flow and heat transfer near theequator of an MHD boundary layer over a porous rotating sphere. Int. J. Thermal Sciences.50 831 – 842 (2011)

76. Turkyilmazoglu, M.: An analytic shooting-like approach for the solution of nonlinear bound-ary value problems. Math. Comp. Modelling.53, 1748 – 1755 (2011)


78. Van Gorder, R.A., Vajravelu, K.: Analytic and numericalsolutions to the Lane-Emden equa-tion. Phys. Lett. A.372, 6060 – 6065 (2008)

79. Van Gorder, R.A., Vajravelu, K.: On the selection of auxiliary functions, operators, and con-vergence control parameters in the application of the Homotopy Analysis Method to nonlin-ear differential equations - A general approach. Commun. Nonlinear Sci. Numer. Simulat.14,4078 – 4089 (2009)

80. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Analytical solutions of a coupled nonlinear sys-tem arising in a flow between stretching disks. Applied Mathematics and Computation.216,1513 – 1523 (2010)

81. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Nano boundary layers over stretching surfaces.Commun. Nonlinear Sci. Numer. Simulat.15, 1494 – 1500 (2010)

82. Van Gorder, R.A., Vajravelu, K.: Convective heat transfer in a conducting fluid over a perme-able stretching surface with suction and internal heat generation/absorption. Applied Mathe-matics and Computation.217, 5810 – 5821 (2011)

83. Wu, Y.Y.: Analytic Solutions for Nonlinear Long Wave Propagation. PhD dissertation, Uni-versity of Hawaii (2009)

84. Wu, Y., Cheung, K.F.: Explicit solution to the exact Riemann problems and application innonlinear shallow water equations. Int. J. Numer. Meth. Fl.57, 1649 – 1668 (2008)

85. Wu, Y.Y., Cheung, K.F.: Homotopy solution for nonlineardifferential equations in wave prop-agation problems. Wave Motion.46, 1 – 14 (2009)



88. Zand, M.M., Ahmadian, M.T., Rashidian, B.: Semi-analytic solutions to nonlinear vibrationsof microbeams under suddenly applied voltages. J. Sound andVibration. 325, 382 – 396(2009)

89. Zand, M.M., Ahmadian, M.T: Application of homotopy analysis method in studying dynamicpull-in instability of microsystems. Mechanics Research Communications.36, 851 – 858(2009)

90. Zhao, J., Wong, H.Y.: A closed-form solution to Americanoptions under general diffusions(2008). Available at SSRN: http://ssrn.com/abstract=1158223.

91. Zhu, J.: Linear and Non-linear Dynamical Analysis of Beams and Cables and Their Combi-nations. PhD dissertation, Zhe Jiang University (2008)



94. Zou, L.: A Study of Some Nonlinear Water Wave Problems Using Homotopy AnalysisMethod. PhD dissertation, Dalian University of Technology(2008)

95. Zou, L., Zong, Z., Wang, Z., He, L.: Solving the discrete KdV equation with homotopy anal-ysis method. Phys. Lett. A.370, 287 – 294 (2007)

Chapter 8Nonlinear boundary-value problems withmultiple solutions

Abstract In this chapter, using three different types of nonlinear boundary-valueequations with multiple solutions, we verify the validity of the Mathematica packageBVPh(version 1.0) fornth-order nonlinear boundary-value equationsF [z,u] = 0 ina finite interval 0≤ z≤ a, subject to then linear boundary conditionsBk[z,u] = γk

(1≤ k≤ n), whereF is anth-order nonlinear differential operator,Bk is anth-orderlinear operator,γk is a constant, respectively. Especially, the so-called multiple-solution-control parameter is introduced into initial guess in order to search formultiple solutions. We illustrate that, using theBVPh 1.0 as a tool, multiple so-lutions of some nonlinear boundary-value equations can be found out by means ofsuch kind of multiple-solution-control parameter.

8.1 Introduction

In this chapter, we illustrate the general validity of the Mathematica packageBVPh(version 1.0) for nonlinear boundary-value problems with multiple solutions, gov-erned by anth-order nonlinear differential equation in a finite interval

F [z,u] = 0, z∈ [0,a], (8.1)

subject to then linear multipoint boundary conditions

Bk [z,u] = γk (8.2)

at the two endpoints and some separated points in the interval z∈ (0,a), whereF

is a nonlinearnth-order differential operator,Bk is a linear differential operator,z isan independent variable,u(z) is the unknown function,γk is a constant, respectively.Assume that this kind of nonlinear boundary-value problemshas at least one smoothsolution.

A nonlinear boundary-value equation has often multiple solutions, which are dif-ficult to find out by numerical techniques in general, especially when they are close

287

288 8 Nonlinear boundary-value problems with multiple solutions

each other and/or numerically unstable. Based on the homotopy analysis method(HAM) [5–19,22], some analytic approaches were suggested to gain multiple solu-tions of nonlinear boundary-value problems. In 2005, Liao [14] applied the HAMto successfully obtain two branches of solutions of the boundary-layer equation

F ′′′+12

FF ′′−βF ′2 = 0, F(0) = 0, F ′(0) = 1, F ′(+∞) = 0,

where−1< β <+∞ is a constant, by means of introducing an additional unknownquantityδ =F(+∞). Solving a nonlinear algebraic equation related to this unknownquantityδ , Liao [14] found one new branch of solutions whenβ > 1, which hadbeennever reported by other analytic methods and even neglected by numericalmethods, mainly because the difference between the values of F ′′(0) of the twobranches of solutions is so small that it is hard to distinguish them. This illustratesthe great potential of the HAM for nonlinear problems with multiple solutions. Inaddition, it verifies that multiple solutions of some nonlinear problems can be foundout by introducing an unknown parameter properly.

In 2009, Abbasbandy et al. [1] applied the HAM to gain the multiple solutionsof a 2nd-order reaction-diffusion equation by solving a nonlinear algebraic equationabout the convergence-control parameter. In 2010, Xu et al.[22] used the HAMto obtain multiple solutions of viscous boundary-layer flows between two movingparallel walls by means of a shooting-like technique.

In this Chapter, we illustrate that, using theBVPh1.0, multiple solutions of non-linear boundary-value equations can be found out by means ofintroducing a so-calledmultiple-solution-control parameterin initial guess. Three examples are usedto illustrate the validity of this strategy, and their inputdata files forBVPh1.0 aregiven in the appendix of this chapter and free available at

http://numericaltank.sjtu.edu.cn/BVPh.htm.

8.2 Brief mathematical formulas

Let u0(z) denote an initial guess of the solutionu(z) of (8.1) and (8.2),q ∈ [0,1]an embedding parameter, respectively. In the frame of the HAM, we should firstconstruct such a continuous deformation (homotopy)φ(z;q) that, asq increasesfrom 0 to 1,φ(z;q) varies from the initial guessu0(z) to the solutionu(z) of (8.1)and (8.2). Such a kind of continuous deformation is constructed by the so-calledzeroth-order deformation equation

(1−q)L [φ(z;q)−u0(z)] = c0 q F [z,φ(z;q)] , (8.3)


Bk [z,φ(z;q)] = γk, (8.4)

8.2 Brief mathematical formulas 289

whereL is an auxiliary linear operator with the propertyL [0] = 0, whose high-est order of derivative is exactlyn, andc0 6= 0 is a convergence-control parameter,respectively. Obviously, it holds

φ(z;0) = u0(z), φ(z;1) = u(z), (8.5)

at q = 0 and q = 1, respectively. So, if the auxiliary linear operatorL andthe convergence-control parameterc0 are so properly chosen that the Maclaurin–homotopy series

φ(z;q) = u0(z)++∞

∑m=1

um(z) qm (8.6)

converges toφ(z;1), we have the homotopy-series solution

u(z) = u0(z)++∞

∑m=1

um(z), (8.7)

whereum(z) is governed by themth-order deformation equation

L [um(z)− χm um−1(z)] = c0 δm−1(z), (8.8)


Bk [z,um] = 0, (8.9)

whereδk(z) = Dk F [z,φ(z;q)]

and

Dk [φ ] =1k!

∂ kφ∂qk

∣

∣

∣

∣

q=0, χk =

0, k≤ 1,1, k> 1.

Here,Dk is thekth-order homotopy-derivative operator. Equation (8.8) isgiven bymeans of Theorem 4.15. For details, please refer to Chapter 4. Note thatδk(z) onlydepends upon the nonlinear differential operatorF [z,u], and can be easily deducedby the theorems proved in Chapter 4. In fact, for nonlinear ordinary differentialequations (ODEs), it is unnecessary for users to deduce the termδk(z) by means oftheBVPh1.0.

At the mth-order homotopy-approximation

u(z) ≈ u0(z)+m

∑k=1

uk(z), (8.10)

the discrete squared residualEm of the governing equation (8.1) is defined by

Em =1

Np+1

Np

∑k=0

F [zk, u(zk)]2 , (8.11)


where

zi = k

(

aNp

)

denotes the coordinate of the(Np +1) separate points. The optimal convergence-control parameterc0 is determined by the minimum of the squared residualEm.

In the frame of the HAM, we have extremely large freedom to choose the auxil-iary linear operatorL and the initial guessu0(z). It is well-known that the contin-uous functionf (z) in a finite intervalz∈ [0,a] can be well approximated by eitherChebyshev series or Fourier series. If the power or Chebyshev polynomial is usedto approximateu(z), we often choose the auxiliary linear operator

L u=dnudzn , (8.12)

wheren is the order of the highest-derivative of (8.1). If the hybrid-base approxima-tion method described in§ 7.2.3 is used to approximate the solutionu(z), we choosethe following auxiliary operator

L u = u′′+ω21 u, whenn= 2,

L u = u′′′+ω21 u′, whenn= 3,

L u = u′′′′+(ω21 +ω2

2) u′′+ω21ω2

2 u, whenn= 4,

L u = u′′′′′+(ω21 +ω2

2) u′′′+ω21ω2

2 u′, whenn= 5,...

whereωk > 0 is a frequency of the corresponding base function cos(ωk z) andsin(ωk z), i.e.

L [cos(ωk z)] = L [sin(ωk z)] = 0.

The above auxiliary linear operators can be expressed in a general form

L u=

[

m

∏i=1

(

d2

dx2 +ω2i

)

]

u, whenn= 2m, (8.13)

or

L u=

[

m

∏i=1

(

d2

dx2 +ω2i

)

]

u′, whenn= 2m+1, (8.14)

where the frequencyωi may be different each other, such as

ωi = i(κπ

a

)

, (8.15)

or the same, like

ωi =(κπ

a

)

, (8.16)

depending on the boundary conditions (8.2). Here,κ ≥ 1 is a positive integer.

8.3 Examples 291

Note that the initial guessu0(z) should satisfy then linear boundary-conditions(8.2) at least, and should be expressed by either polynomialor trigonometric func-tions, or the hybrid-base functions.

For the choice of the auxiliary linear operatorL and the initial guessu0(z),please refer to Chapter 7.

8.3 Examples

Many nonlinear boundary-value equations have multiple solutions, which are dif-ficult to obtain by numerical techniques in general. In this section, three non-linear boundary-value problems with multiple solutions are used to illustrate thevalidity of the BVPh 1.0, which is given in Appendix 7.1 and free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm. The input data files of all these exam-ples forBVPh1.0 are given in the appendix of this chapter and free available at thesame website.

8.3.1 Nonlinear diffusion-reaction model

First of all, let us consider a nonlinear diffusion-reaction model in porous catalysts,governed by

u u′′− γ2 = 0, u′(0) = 0, u(1) = 1, (8.17)

whereγ is a given parameter. The above equation has the implicit closed-form so-lution

z=u(0)i γ

√

π2

Erf

[

i

√

lnu(z)u(0)

]

, (8.18)

wherei =√−1, Erf(z) denotes the error function, andu(0) is determined by

1=u(0)i γ

√

π2

Erf

[

i

√

ln1

u(0)

]

, (8.19)

respectively.This example is a special case of (8.1) when

F [z,u] = u u′′− γ2.

In this case, themth-order homotopy-approximation is given by (8.10), whereuk(z)is governed by thekth-order deformation equation

L [uk(z)− χk uk−1(z)] = c0 δk−1(z), (8.20)



u′k(0) = 0, uk(1) = 0, (8.21)

where

δn(z) =n

∑j=0

u j u′′n− j − (1− χn+1) γ2 (8.22)

is given by Theorem 4.1.Equation (8.17) has two solutions when 0≤ γ ≤ 0.775152, as pointed out by

Magyari [20]. Using the HAM with power polynomial as base function, Abbas-bandy et al. [1] gained the two solutions by solving a nonlinear algebraic equa-tion about the convergence-control parameterc0. In this chapter, we suggest a newoptimal HAM approach to find out multiple solutions by introducing a unknownparameter in initial guess, calledthe multiple-solution-control parameter.

Without loss of generality, let us first consider the case ofγ = 3/5. The nonlinearalgebraic equation (8.19) has two solutionsu(0) = 0.101046 andu(0) = 0.779034whenγ = 3/5, corresponding to the two exact solutions in the implicit form (8.18),respectively.

First of all, let us apply the hybrid-base approximation method mentioned in§ 7.2.3 to solve this nonlinear boundary value problem with multiple solutions. Ac-cording to (8.13), we choose the auxiliary linear operator

L u= u′′+π2 u. (8.23)

Note that, ifu(z) satisfies (8.17), thenu(−z) = u(z) is also its solution. So,u(z) isan even function. This information is useful for us to choosethe initial guessu0(z).So, we choose the initial guess

u0(z) =12(σ +1)+

12(σ −1) cos(π z), (8.24)

whereσ = u0(0) is an unknown parameter. Note thatu0(z) is dependent upon theunknown parameterσ and thus different values ofσ give different initial guesses.The unknown parameterσ is introduced here to search for the multiple solutions,as shown later.

Using the auxiliary linear operator (8.23) and the initial guess (8.24), we gain the10th-order homotopy-approximation by means of theBVPh1.0. Note that, the ap-proximation contains two unknown parameters: the convergence-control parameterc0 and the unknown parameterσ in (8.24). However, givenc0, the squared resid-ual Em is only dependent uponσ . For example, in case ofc0 = −1, the curves ofEm versusσ are as shown in Fig. 8.1, which clearly indicates that, in a large range−0.8< σ < 0.5, the squared residualEm decreases as the order of approximationincreases, and the optimal value ofσ (for c0 = −1) is about -0.2, corresponding tothe fastest convergent series. This is indeed true. It is found that, whenc0 =−1, the3rd-order iteration approach gives convergent resultu(0) = 0.101046 by means of

8.3 Examples 293

Fig. 8.1 Squared residualEm

of (8.17) versusσ = u0(0)by means of the auxiliarylinear operator (8.23) andthe initial guess (8.24) whenγ = 3/5 andc0 =−1. Dashedline: 5th-order approximation;Dash-dotted line: 6th-orderapproximation; Solid line:8th-order approximation.

σ

Em

-1 -0.5 0 0.5 110-6

10-5

10-4

10-3

10-2

10-1

100

γ = 3/5, c0 = -1


of (8.17) versusσ = u0(0) bymeans of the auxiliary linearoperator (8.23) and the initialguess (8.24) whenγ = 3/5andc0 = −1. Dashed line:10th-order approximation;Dash-dotted line: 15th-orderapproximation; Solid line:20th-order approximation.

σ

Em

0.722 0.724 0.726 0.728 0.73 0.732 0.73410-4

10-3

10-2

γ = 3/5, c0 = -1

Fig. 8.3 Approximations oftwo branches of solutionsu(z) of (8.17) whenγ = 3/5by means of the auxiliarylinear operator (8.23) andthe initial guess (8.24). Solidline: 5th-order homotopy-approximation of the low-branch solution given byc0 =−1 andσ = −1/5; Dashedline: 10th-order homotopy-approximation of the up-branch solution given byc0 = −0.71848810 andσ =0.72834766; Symbols: exactsolution.

z

u

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1 γ = 3/5


Table 8.1 The approximation ofu(0) at themth iteration by means of the auxiliary linear operator(8.23) and the initial guess (8.24) whenNt = 20,γ = 3/5, c0 =−1 with different values ofσ .

m, number of iteration σ =−1/2 σ =−1/5 σ = 1/2

5 0.113536 0.101924 0.13834810 0.100775 0.101008 0.09929215 0.101057 0.101046 0.10103220 0.101046 0.101045 0.10104125 0.101046 0.101046 0.10104530 0.101046 0.101046 0.101046

Table 8.2 Squared residualEm of (8.17) at themth iteration by means of the auxiliary linearoperator (8.23) and the initial guess (8.24) whenNt = 20, γ = 3/5, c0 = −1 with different valuesof σ .

m, number of iteration σ =−1/2 σ =−1/5 σ = 1/5

5 4.6×10−4 3.5×10−6 3.2×10−3

10 4.6×10−7 1.5×10−8 1.8×10−5

15 1.5×10−8 2.1×10−9 1.5×10−7

20 2.3×10−9 1.9×10−9 6.1×10−9

25 1.9×10−9 1.8×10−9 2.0×10−9

30 1.9×10−9 1.8×10−9 1.9×10−9

σ = −1/2,−1/5,+1/2, and the iteration approach by means ofσ = −1/5 corre-sponds to the fastest convergence among them, as shown in Table 8.1 and Table 8.2,respectively. Besides, the corresponding 5th-order approximation agrees well withthe low-branch exact solution, as shown in Fig. 8.3. Thus, when c0 = −1, thereexists such a intervalΩ1 ⊃ [−1/2,+1/2] that for anyσ ∈ Ω1, the correspondinghomotopy-series converges to the low-branch solution of (8.17). Note that one cannot gain accurate approximation of the low-branch solutionif σ 6⊂ Ω1. In this mean-ing, like c0, the unknown parameterσ in the initial guess (8.24) can be regarded asa kind of convergence-control parameter, too.

According to Fig. 8.1, there exists another regionΩ2 ⊂ (0.5,1) such that thesquared residualEm decreases as the order of approximation increases. However,whenc0 =−1, this interval is rather small, i.e.Ω2 ⊂ (0.7276,0.7292), as shown inFig. 8.2. It is found that such a kind of interval becomes smaller and smaller as theorder of approximation increases, so that it is time-consuming to determine its exactboundary. Besides, even using the optimal valueσ∗ = 0.7284 (whenc0 =−1) at the20th-order of approximation, we still can not gain convergent series solution. Usingother values ofc0, such asc0 = −1/4, such kind of intervalΩ2 is still very small.So, whenc0 = −1, the corresponding intervalΩ2 is so small that it is very difficultto gain the accurate enough approximation of the up-branch solution of (8.17) bymeans of iteration approach.

To gain an accurate enough approximation of the up-branch solution of (8.17),we regard both ofc0 andσ as the unknown parameters and search for the mini-

8.3 Examples 295

Table 8.3 Optimal parametersc∗0 andσ ∗ of themth-order approximation of the up-branch solutionof (8.17) by means of the auxiliary linear operator (8.23) and the initial guess (8.24) whenγ = 3/5.

m, order of approx. c∗0 σ ∗0 Em u(0)

4 -0.71489261 0.72844802 2.59×10−5 0.7789795 -0.71891704 0.72832456 1.51×10−6 0.7790886 -0.71353321 0.72835306 3.32×10−7 0.7790228 -0.71350498 0.72834803 3.20×10−8 0.77903210 -0.71848810 0.72834766 2.68×10−8 0.779034

mum of the squared residual of the governing equation at a given order of approx-imation. By means of the Mathematica commandMinimize under a restrictionσ > 1/2 (otherwise, we always find the low-branch solution), the minimum of thesquared residual and the corresponding optimal parametersc∗0 andσ∗ are easily ob-tained, as shown in Table 8.3. Alternatively, we can use the moduleGetMin2Dof the packageBVPh 1.0 to search for the local minimum of the squared resid-ual. At the 10th-order approximation, we have the optimal convergence-control pa-rametersc∗0 =−0.71848810 andσ∗ = 0.72834766, corresponding to the minimumof the squared residualE10 = 2.68×10−8. Note that the corresponding 10th-orderhomotopy-approximation ofu(z) agrees well with the up-branch exact solution, asshown in Fig. 8.3, and besides it gives the accurate approximationu(0) = 0.779034that is equal to the exact value, as show in Table 8.3.

In this way, we successfully gain the two solutions of (8.17)by means of intro-ducing the unknown parameterσ in the initial guess (8.24). We call such kind ofunknown parameterthe multiple-solution-control parameter. Like the convergence-control parameterc0 which provides us a convenient way to guarantee the conver-gence of solution series, the so-called multiple-solution-control parameter providesus a convent way to search for multiples solutions of nonlinear differential equa-tions, as shown later.

Secondly, since the HAM provides us extremely large freedomto choose theauxiliary linear operator and the initial guess, we can alsochoose the auxiliary linearoperator

L u= u′′, (8.25)

and the initial guessu0(z) = σ +(1−σ) z2 (8.26)

so as to approximate the solutions of (8.17) by means of powerpolynomial, whereσ = u0(0) is a unknown parameter, which is used here as a multiple-solution-controlparameter.

Using theBVPh 1.0, it takes a few seconds CPU time to gain the correspond-ing 5th-order homotopy-approximation. By means of the Mathematica commandMinimize , the optimal convergence-control parameterc∗0 = −1.16236 and theoptimal multiple-solution-control parameterσ∗ = 0.76500 are found, correspond-ing to the minimum of the squared residual of the governing equation (at the 5th-


Table 8.4 Squared residualEm of (8.17) andu(0) at themth-order approximation whenγ = 3/5by means of the auxiliary linear operator (8.25) and the initial guess (8.26) withc0 = −1.16236andσ = 0.765.

m, order of approximation Squared residualEm u(0)

4 1.5×10−9 0.7790028 2.4×10−13 0.77903412 1.1×10−17 0.77903416 8.2×10−23 0.77903420 3.4×10−25 0.779034

order homotopy-approximation) 6.3× 10−12. By means ofc0 = −1.16236 andσ = 0.76500, the corresponding homotopy-series converges rather quickly to theup-branch solution: even the 8th-order homotopy-approximation gives the exactvalueu(0) = 0.779034, as shown in Table 8.4. Settingc0 = −1, the curves of thesquared residualEm at up-to 5th-order of approximation versusσ are as shown inFig 8.4. It indicates that, whenc0 =−1, there exists an intervalΩ2 such that, for anyσ ∈ Ω2, Em decreases as the order of approximation increases. It is found that thehomotopy-series converges by means ofc0 =−1 and 0.6≤ σ ≤ 1, therefore it holdsΩ2 ⊃ [0.6,1] whenc0 = −1. It is interesting that, by means of 3rd-order iterationapproach withanya multiple-solution-control parameterσ ∈ [0.6,1], we also gainaccurate approximation of the up-branch solution of (8.17).


of (8.17) versusσ = u0(0)in the intervalσ ∈ [0,1] bymeans of the auxiliary linearoperator (8.25) and the initialguess (8.26) whenγ = 3/5andc0 = −1. Dashed line:3rd-order approximation;Dash-dotted line: 4th-orderapproximation; Solid line:5th-order approximation.

σ

Em

0 0.2 0.4 0.6 0.8 110-11

10-9

10-7

10-5

10-3

10-1 γ = 3/5

According to Fig. 8.4, there exists an intervalΩ1 near σ = 0 such that thesquared residualEm of the governing equation decreases as the order of approxi-mation increases. Unfortunately, this interval is rather small, as shown more clearlyin Fig. 8.5, so that it is time-consuming to determine the boundary of the intervalΩ1 exactly. To avoid it, we use the Mathematica commandMinimize with therestrictionσ < 1/10 (otherwise, we always find the up-branch solution) to gainthe optimal homotopy-approximation. It is found that, the minimum of the squared

8.3 Examples 297


of (8.17) versusσ = u0(0) inthe local intervalσ ∈ [0,0.15]by means of the auxiliarylinear operator (8.25) andthe initial guess (8.26) whenγ = 3/5 andc0 =−1. Dashedline: 3rd-order approximation;Dash-dotted line: 4th-orderapproximation; Solid line:5th-order approximation.

σ

Em

0 0.05 0.1 0.1510-3

10-2

10-1

γ = 3/5

Table 8.5 Optimal parametersc∗0 andσ ∗ for themth-order approximation of the low-branch so-lution of (8.17) by means of the auxiliary linear operator (8.25) and the initial guess (8.26) whenγ = 3/5.

m, order of approx. c∗0 σ ∗0 Em u(0)

5 -1.52954 0.05137 6.3×10−4 0.10237910 -1.6299959583 0.05123009640906 2.3×10−5 0.10116515 -1.6648354484 0.05122982392509 1.1×10−6 0.10106020 -1.6833563121 0.05122982327004 5.2×10−8 0.10104725 -1.6951993251 0.05122982326872 2.7×10−9 0.101046

residual at the 25th-order homotopy-approximation arrives 2.7× 10−9 by meansof the optimal convergence-control parameterc∗0 = −1.6951993251 and the opti-mal multiple-solution-control parameterσ∗ = 0.05122982326872, which give anaccurate approximationu(0) = 0.101046 of the low-branch solution, as shown inTable 8.5. So, using the auxiliary linear operator (8.25) and the initial guess (8.26),we also gain the two solutions of (8.17) by means of the so-called multiple-solution-control parameterσ .

All of these illustrate that the so-calledmultiple-solution-control parameterσintroduced in the initial guess (8.24) and (8.26) indeed provides us a convenient wayto search for multiple solutions of nonlinear boundary value problems. For (8.17),given a reasonable convergence-control parameterc0, there exist two intervalsΩ1

and Ω2 such that, as long asσ ∈ Ω1 or σ ∈ Ω2, the corresponding homotopy-series converges to either the low-branch or the up-branch solution, respectively. Itis interesting that the intervalsΩ1 andΩ2 depend upon not only the convergence-control parameterc0 but also the auxiliary linear operator and the initial guess. Bymeans of the auxiliary linear operator (8.23) and the initial guess (8.24), the intervalΩ1 is much larger thanΩ2 so that it is easier to gain the low-branch solution than theup-branch ones. However, using the auxiliary linear operator (8.25) and the initialguess (8.26), the intervalΩ2 is much larger thanΩ1 so that it is easier to gain theup-branch solution than the low-branch ones. Since the HAM provides us extremely


large freedom to choose the auxiliary linear operator and the initial guess, we can usethe auxiliary linear operator (8.23) and the initial guess (8.24) to gain the low-branchof solution, and the auxiliary linear operator (8.25) and the initial guess (8.26) toobtain the up-branch solution! Using this strategy, we gainthe two branches1 ofsolutions for the possible values ofγ. This also illustrates that, using theBVPh1.0,we can search for multiple solutions of nonlinear boundary-value equations (8.1) bymeans of different types of auxiliary linear operators and different types of initialguesses that contains an unknown parameter, called the multiple-solution-controlparameter.

8.3.2 A three-point nonlinear boundary-value problem

Let us further consider a three-point nonlinear boundary-value problem

u′′′′ = βz(1+u2), u(0) = u′(1) = u′′(1) = 0, u′′(0)−u′′ (α) = 0, (8.27)

whereα ∈ (0,1) andβ are given constants. Graef, Qian and Yang [3,4] proved thatthe above equation has at least two positive solutions whenα = 1/5 andβ = 10.

This problem is a special case of (8.1) when

F [z,u] = u′′′′−βz(1+u2).

Similarly, themth-order homotopy-approximation is given by (8.10), whereuk(z) isgoverned by thekth-order deformation equation

L [uk(z)− χk uk−1(z)] = c0 δk−1(z) (8.28)


uk(0) = 0, u′k(1) = 0, u′′k(1) = 0, u′′k(0)−u′′k(α) = 0, (8.29)

where

δi(z) = u′′′′i − (1− χi+1) β z−β zi

∑j=0

u j ui− j (8.30)

is gained by Theorem 4.1.Note that the governing equation (8.27) contains the termz. So, the solutionu(z)

in the finite intervalz∈ [0,1] can be expressed by power polynomial. Thus, wechoose the auxiliary linear operator

L u= u′′′′ (8.31)

1 More truncated terms are necessary in order to gain accuratelow-branch solution for smallγ ,because there exists the singularity asγ → 0.

8.3 Examples 299

and the initial guess in the form

u0(z) = a0+4

∑i=1

ak zk,

where ak is a unknown constant. Letσ = u(1) denote the unknown multiple-solution-control parameter, which provides us the additional condition

u(1) = σ . (8.32)

Then, the five unknown constantsak (0≤ k ≤ 4) of the initial guessu0(z) are de-termined by the four boundary conditions in (8.27) and the additional boundarycondition (8.32). Thus, we have the initial guess

u0(z) =σ

(2α −3)

[

2(3α −4)z+6(1−α)z2+2αz3− z4] . (8.33)

Note that the above initial guess is dependent upon the position z= α, where thereexists the multipoint boundary-conditionu′′(0) = u′′(α).

The multipoint boundary conditions can be easily resolved by means of computeralgebra system like Mathematica: all boundary conditions are satisfied in a similarway. Letu∗k(z) denote a special solution of (8.28). According to (8.31), its generalsolution reads

uk(z) = u∗k(z)+C0+C1 z+C2 z2+C3 z3,

where the four coefficientsC0,C1,C2 and C3 are determined by the four linearboundary conditions (8.29). This is mainly because computer algebra system likeMathematica provide us the ability to “compute with functions instead of num-bers” [21].


of (8.27) versusσ in the inter-val σ ∈ [0,2.5] by means ofthe auxiliary linear operator(8.31) and the initial guess(8.33) whenα = 1/5, β = 10andc0 = −1. Dashed line:3rd-order approximation;Solid line: 5th-order approxi-mation.

σ

Em

0 0.5 1 1.5 2 2.510-9

10-7

10-5

10-3

10-1

101

103

α = 0.2, β = 10

Let us first consider the case ofα = 1/5 and β = 10. Regarding both ofthe convergence-control parameterc0 and the multiple-solution-control parame-


Table 8.6 Squared residualEm of (8.27) andu(1) at themth-order approximation whenα = 1/5andβ =10 by means of the auxiliary linear operator (8.31) and the initial guess (8.33) withc0 =−1andσ = 0.62.

m, order of approximation Em u(1)

1 1.7×10−8 0.6272992 1.2×10−10 0.6273133 8.7×10−13 0.6273155 4.4×10−17 0.62731510 8.2×10−28 0.627315

ter σ as unknowns, the squared residualEm of the governing equation at themth-order homotopy-approximation is dependent uponc0 andσ . It is found that,at the 5th-order homotopy-approximation, the minimum of the squared residualreads 1.6× 10−20, corresponding to the optimal convergence-control parameterc∗0 =−1.00661 and the optimal multiple-solution-control parameter σ∗ = 0.62018.The curves ofEm versusσ whenc0 = −1 are as shown in Fig. 8.6, which indicatesthat there exists an intervalΩ1 such that, for anyσ ∈Ω1, the squared residualEm de-creases asm increases. It is found that thesameconvergent series solution can be ob-tained by means ofc0 =−1 andσ = 0,0.5 and 1.1, so thatΩ1 ⊃ [0,1.1]. Especially,usingσ = 0.62 andc0 =−1, we gain the accurate result withu(1) = 0.627315 in afew seconds of CPU time by means of the 3rd-order iteration approach, as shown inTable 8.6.

According to Fig. 8.6, there should exist a solution nearσ = 2. Let Ω2 de-note such an interval that, for anyσ ∈ Ω2 and c0 = −1, we gain a convergenthomotopy-series of the 2nd solution. However, it is found that the intervalΩ2 isso small that it is time-consuming to determine it exactly. To avoid it, we searchfor the 2nd solution by means of the minimum of the squared residual at thehigh enough order of approximation. Using the Mathematica commandMinimizewith the restrictionσ > 2 (otherwise we always gain the 1st solution), it is foundthat, at the 5th-order homotopy-approximation, the minimum of the squared resid-ual reads 1.7× 10−15, corresponding to the optimal convergence-control parame-terc∗0 =−1.0202828199 and the optimal multiple-solution-control parameterσ∗ =2.2062238124. Furthermore, at the 10th-order of approximation, the minimum ofthe squared residual reads 7.4×10−31, corresponding to the optimal convergence-control parameterc∗0 = −1.0166130822 and the optimal multiple-solution-controlparameterσ∗ = 2.206223812318637. It is found that, atz = 1, the 5th to 10thhomotopy-approximations tend to the same valueu(1) = 2.2411770066, as shownin Table 8.7. Therefore, we indeed obtain two different positive solutions of thethree-point nonlinear boundary-value equation (8.27) when α = 1/5 andβ = 10, asshown in Fig. 8.7 and listed in Table 8.8.

Similarly, we can find out the multiple solutions of (8.27) for different values ofα andβ by means of theBVPh 1.0. For example, whenα = 1/5, it is found that(8.27) has two positive solutions in the intervalβ ∈ (0,12.05], as listed in Table 8.9.

8.3 Examples 301

Fig. 8.7 Approximationsof two solutionsu(z) of(8.27) whenα = 1/5 andβ = 10 by means of theauxiliary linear operator(8.31) and the initial guess(8.33). Solid line: homotopy-approximation of the 1st so-lution given by 10th iterationwith c0 = −1 andσ = 0.62;Dashed line: 10th-orderhomotopy-approximationof the 2nd solution givenby the optimal parametersc0 = −1.0166130822 andσ = 2.206223812318637.

z

u

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

α = 0.2, β = 10

Fig. 8.8 u(1) of two positivesolutionsu(z) of (8.27) whenα = 1/5 with different valuesof β by means of the auxiliarylinear operator (8.31) and theinitial guess (8.33). Solid line:low-branch solution; Dashedline: up-branch solution.

β

u(1

)

2 4 6 8 10 12

10-2

10-1

100

101

102

α = 0.2

Fig. 8.9 u(1) of two positivesolutionsu(z) of (8.27) whenβ = 10 with different valuesof α by means of the auxiliarylinear operator (8.31) and theinitial guess (8.33). Solid line:low-branch solution; Dashedline: up-branch solution.

α

u(1

)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

β = 10


Note that, asβ tends to zero,u(1) of the up-branch solution increases quickly, butu(1) of the low-branch solution tends to zero, as shown in Fig. 8.8.

Similarly, whenβ = 10, we find out the multiple solutions of (8.27) for differentvalues ofα in the boundary condition

u′′(0)−u′′ (α) = 0.

It is found that there exist at least two positive solutions in the intervalα ∈ (0,1], asshow in Fig. 8.9. The corresponding values ofu(1) versusα are listed in Table 8.10.They show the influence of the above boundary condition at a separate pointz= αto the solutions of nonlinear boundary-value equation (8.27).

Note that the (8.27) can be first transferred into a nonlineareigenvalue equationand then be solved by theBVPh1.0, as shown in Example 9.3.4.

This example illustrates that, using theBVPh 1.0, we can gain multiple solu-tions of nonlinear multipoint boundary-value problems by introducing the so-calledmultiple-solution-control parameter in initial guess.

Table 8.7 Optimal convergence-control parametersc∗0 and the optimal multiple-solution-controlparameterσ ∗ at themth-order approximation of the 2nd solution of (8.27) whenα = 1/5 andβ = 10 by means of the auxiliary linear operator (8.31) and the initial guess (8.33).

m c∗0 σ ∗0 Em u(1)

3 -1.0156751650 2.206225834958724 3.2×10−9 2.24118520315 -1.0202828199 2.206223812447131 1.7×10−15 2.24117700766 -1.0186293138 2.206223812311513 6.1×10−19 2.24117700767 -1.0147412057 2.206223812318337 2.2×10−21 2.24117700768 -1.0216642167 2.206223812318337 3.8×10−25 2.24117700769 -1.0198947262 2.206223812318337 2.1×10−28 2.241177007610 -1.0166130822 2.206223812318337 7.4×10−31 2.2411770076

Table 8.8 Convergent homotopy-approximation of the two positive solutions of (8.27) whenα =1/5 andβ = 10 by means of the auxiliary linear operator (8.31) and the initial guess (8.33).

z 1st solution 2nd solution

0 0 00.1 0.144853 0.5083350.2 0.269514 0.9481590.3 0.374042 1.3195590.4 0.458715 1.6230390.5 0.524156 1.8599520.6 0.571473 2.0331310.7 0.602414 2.1476510.8 0.619519 2.2116680.9 0.626286 2.2372531.0 0.627315 2.241177

8.3 Examples 303

Table 8.9 The value ofu(1) of the two positive solutions of (8.27) whenα = 1/5 with differentvalues ofβ by means of the auxiliary linear operator (8.31) and the initial guess (8.33).

β 1st solution 2nd solution

0.125 0.0061113 230.0726560.25 0.012224 115.0270190.5 0.024455 57.4948851 0.048974 28.7101352 0.098462 14.2799683 0.149017 9.4353544 0.201271 6.9856956 0.314218 4.4745958 0.447555 3.1414319 0.528490 2.66030510 0.627315 2.24117711 0.762509 1.84378012 1.054690 1.33297312.05 1.10240 1.275268

Table 8.10 The value ofu(1) of the two positive solutions of (8.27) whenβ = 10 with differentvalues ofα by means of the auxiliary linear operator (8.31) and the initial guess (8.33).

α 1st solution 2nd solution

0.01 0.648176 2.2137740.05 0.646932 2.2150930.1 0.643033 2.2195120.2 0.627315 2.2411770.3 0.601033 2.2892740.4 0.564806 2.3775410.5 0.520086 2.5221820.6 0.468852 2.7444480.7 0.413081 3.0776740.8 0.354329 3.5831590.9 0.293562 4.3918600.95 0.262562 4.9991961 0.231157 5.848855

8.3.3 Channel flows with multiple solutions

The steady laminar flow of a viscous incompressible fluid in a two-dimensionalporous channel with uniform injection or suction [22] is described by a nonlinearboundary-value differential equation

F ′′′′+α(

z F′′′+3F ′′)+(

F F ′′′−F ′ F ′′)= 0, 0≤ z≤ 1, (8.34)


F(0) = 0, F ′′(0) = 0, F(1) = R, F ′(1) = 0, (8.35)


where the prime denotes the differentiation with respect toz, R is the cross-flowReynolds number,α is a physical constant related to the wall expansion ratio, re-spectively. Here,F(z) is related to the stream function

ψ(x,y) =(νx

d

)

F(z), z= y/d,

whered is the half of the distance between the two parallel walls. For details, pleaserefer to Berman [2] and Xu et al. [22]. Writingu(z) = F(z)/R, the above equationbecomes

u′′′′+α(

z u′′′+3u′′)

+R(

u u′′′−u′ u′′)

= 0, 0≤ z≤ 1, (8.36)


u(0) = 0, u′′(0) = 0, u(1) = 1, u′(1) = 0. (8.37)

In the frame of the HAM, Xu et al. [22] developed an analytic approach to findout the multiple solutions of this problem, which is in principle based on the shoot-ing method. Here, we illustrate that, using theBVPh1.0, we can find out its multi-ple solutions by introducing the so-called multiple-solution-control parameter in theinitial guess.

This problem is a special case of (8.1) when

F [z,u] = F ′′′′+α(

z F′′′+3F ′′)+(

F F ′′′−F ′ F ′′) .

Themth-order homotopy-approximation is given by (8.10), whereuk(z) is governedby thekth-order deformation equation

L [uk(z)− χk uk−1(z)] = c0 δk−1(z) (8.38)


uk(0) = 0, u′′k(0) = 0, uk(1) = 0, u′k(1) = 0, (8.39)

where

δi(z) = u′′′′i +α(

z u′′′i +3u′′i)

+Ri

∑j=0

(

u j u′′′i− j −u′j u′′i− j

)

(8.40)

is gained by Theorem 4.1.Analyzing the governing equation (8.36) and the boundary conditions (8.37) ,

it is easy to find that the solution is an odd function. Besides, noticing that (8.36)contains the termz, it is convenient to expressu(z) by power polynomials. So, wechoose the auxiliary linear operator

L u= u′′′′ (8.41)

8.3 Examples 305

and the initial guess in the form

u0(z) = a1 z+a2 z3+a3 z5,

wherea1,a2,a3 are constants. Note that this initial guess automatically satisfies theboundary conditions atz= 0. Thus, there are onlytwo boundary conditions left atz= 1 for the threeunknown constantsa1,a2 anda3 so that one of them is unde-termined. This provides us one additional degree of freedomin the initial guess,which is useful for us to find out the multiple solutions, as shown later. To searchfor the multiple solutions, we introduce the so-called multiple-solution-control pa-rameterσ = u′0(0). Then, the three unknown constants are uniquely determinedbyσ = u′0(0) and the two boundary conditions (8.37) atz= 1. Thus, we have the initialguess

u0(z) = σ z+12(5−4σ) z3− 1

2(3−2σ) z5. (8.42)

Fig. 8.10 Squared residualEm of (8.36) versusσ whenR= −11 andα = 3/2 bymeans of the auxiliary linearoperator (8.41) and the initialguess (8.42) withc0 = −1.Solid line:E10; Dashed line:E6.

σ

Em

-2 -1 0 1 2 310-6

10-4

10-2

100

102

104

106

108

R = -11, α = 3/2, c0 = -1

Without loss of generality, let us consider the caseR=−11 andα = 3/2. Regard-ing both of the convergence-control parameterc0 and the multiple-solution-controlparameterσ as unknowns, the squared residualEm of the governing equation at themth-order approximation is dependent upon both ofc0 andσ . Obviously, the smallerthe value ofEm, the better the correspondingmth-order homotopy approximation.So, the optimal approximation is given by the minimum ofEm. Besides, any a verysmall value ofEm corresponds to an accurate approximation ofu(z). To gain someinformation about the multiple solutions of (8.36) in case of R=−11 andα = 3/2,we first plot the curves ofE6 andE10 versusσ whenc0 =−1, as shown in Fig. 8.10.Note that there are three local minimums, which suggest thatthere might exist threesolutions in the intervalσ < 0, 0< σ < 1 andσ > 1, respectively. To confirmthis guess, we use the Mathematica commandMinimize first with the restrictionσ < 0 to search for the solution of (8.36). The minimum of the squared residualdecreases to 1.8×10−7 and 6.9×10−17 at the 10th and 20th-order approximation,


Table 8.11 u′(0) and the optimal parametersc∗0 andσ ∗ at themth-order approximation of the 1stsolution of (8.36) whenR=−11 andα = 3/2 by means of the auxiliary linear operator (8.41) andthe initial guess (8.42).

m c∗0 σ ∗0 Em u′(0)

10 -1.15185 -1.081208 1.8×10−7 -1.0237512 -1.16501 -1.081396 2.3×10−9 -1.0237714 -1.25436 -1.081389 2.5×10−10 -1.0237716 -1.17126 -1.081372 3.6×10−13 -1.0237718 -1.24314 -1.08137198 4.6×10−14 -1.0237720 -1.17586 -1.08137151 6.9×10−17 -1.02377

Table 8.12 u′(0) and the optimal parametersc∗0 andσ ∗ at themth-order approximation of the 2ndsolution of (8.36) whenR=−11 andα = 3/2 by means of the auxiliary linear operator (8.41) andthe initial guess (8.42).

m c∗0 σ ∗0 Em u′(0)

10 -0.87023 0.293224 4.9×10−4 0.1710312 -0.96316 0.292173 2.2×10−5 0.1690014 -0.90210 0.292288 1.1×10−6 0.1692616 -0.91428 0.292329 5.6×10−8 0.1693718 -0.94502 0.292321 3.0×10−9 0.1693520 -0.89768 0.292322 1.8×10−10 0.16935

respectively, and the corresponding value ofu′(0) converges to -1.02377, as shownin Table 8.11. So, there indeed exists a solution in the interval σ < 0. It is found that,whenc0 = −1, there exists such an intervalΩ1 ⊃ [−2,−0.7] that, for anyσ ∈ Ω1,we gain the same homotopy-approximation with the same valueu′(0) =−1.02377by means of theBVPh1.0 without iteration. Especially, usingσ =−1 andc0 =−1,which are close to their optimal values, we gain the convergent solution in a fewseconds of CPU time by a laptop computer.

Similarly, using the Mathematica commandMinimize with the restriction 0<σ < 1, we find out the 2nd solution of (8.36) whenR=−11 andα = 3/2 by meansof the BVPh 1.0. The minimum of the squared residual decreases to 1.8× 10−10

at the 10th-order approximation, and the optimal homotopy-approximation givenby the corresponding optimal convergence-control parameter c∗0 and the optimalmultiple-solution-control parameterσ∗ gives a convergent valueu′(0) = 0.16935,as shown in Table 8.12. LetΩ2 denote such an interval that, for anyσ ∈ Ω2, thecorresponding homotopy-series converges to the 2nd solution. However, it is foundthat the intervalΩ2 is so small that it is time-consuming to find its boundary exactly.So, it is more convenient to gain the 2nd solution by means of theBVPh1.0withoutiteration.

To gain the 3rd solution, we use the Mathematica commandMinimize with therestrictionσ > 1 to search for the minimum of the squared residual. It is found thatthe minimum of the squared residualEm indeed decreases as the order of approx-

8.3 Examples 307

Table 8.13 Squared residualEm of (8.36) andu′(0) of themth-iteration approximation whenR=11 andα = 3/2 by means of the auxiliary linear operator (8.41) and the initial guess (8.42) withc0 =−1/2 andσ = 2, corresponding to the 3rd solution.

m, time of iteration Em u′(0)

1 6.9×103 2.310185 1.8×102 2.6736610 4.5 2.7502115 9.8×10−2 2.7597720 1.9×10−3 2.7609425 3.2×10−5 2.7610830 5.1×10−7 2.7611035 7.7×10−9 2.7611140 1.0×10−10 2.76111

Fig. 8.11 Multiple solutionsof (8.36) whenR= −11 andα = 3/2 gained by means ofthe auxiliary linear operator(8.41) and the initial guess(8.42). Solid line: the 1stsolution; Dashed line: the 2ndsolution; Dash-dotted line: the3rd solution.

z

u’(

z)

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3 R = -11, α = 3/2

imation increases, but rather slowly. Even at the 20th-order of approximation, theminimum of the squared residual reads 1.6 by means of the optimal convergence-control parameterc∗0 =−0.75716 and the optimal multiple-solution-control param-eterσ∗ = 1.960920. To accelerate the convergence, the 3rd-order iteration approachis used. It is found that, whenc0 =−1/2, there exists such an intervalΩ3 ⊃ [0.5,3]that, for anyσ ∈ Ω3, we obtain the same convergent homotopy-approximation withu′(0) = 2.76111. For example, whenc0 =−1/2 andσ = 2, the squared residual ofthe governing equation decreases quickly, as shown in Table8.13.

In this way, we successfully obtain the three solutions of (8.36), corresponding tou′(0) =−1.02377,u′(0) = 0.16935 andu′(0) = 2.76111, respectively, as shown inFig. 8.11. This confirms once again that, using theBVPh1.0, multiple solutions ofnonlinear boundary-value problems can be found out by introducing the so-calledmultiple-solution-control parameter in initial guess.



In this chapter, using three nonlinear boundary-value equations as examples, weillustrate the validity of theBVPh 1.0 for nonlinear boundary-value equationswith multiple solutions, governed bynth-order nonlinear boundary-value equationF [z,u] = 0 in a finite interval 0≤ z≤ a, subject to then linear boundary conditionsBk[z,u] = γk (1≤ k≤ n).

An unknown parameter, namelymultiple-solution-control parameter, is intro-duced into initial guess, for the first time, so as to search for multiple solutions.We illustrate that, using theBVPh 1.0 as a tool, multiple solutions of somenth-order nonlinear boundary-value problems can be found out bymeans of the so-called multiple-solution-control parameter, whose optimal value is determined bythe minimum of the squared residual of governing equation ata high-enough or-der homotopy-approximation. In essence, the unknown multiple-solution-controlparameter provides us one additional degree of freedom in aninitial guess. Notethat, it is the HAM that provides us great freedom to introduce such a kind of un-known parameter in the initial guessu0(z), whose different values correspond todifferent initial guesses. So, in addition tothe convergence-control parameter c0

that provides us a convenient way to guarantee the convergence of homotopy-series,we introduce in this chapter a new concept,the multiple-solution-control parameter,that provides us a convenient way to search for multiple solutions, although, as men-tioned in Chapter 10, it also has influence on the convergenceof homotopy-series.Our examples illustrate that, using an optimal multiple-solution-control parameterand an optimal convergence-control parameterc0 determined by the minimum ofsquared residual of governing equation, we can find out multiple solutions of somenth-order nonlinear multipoint boundary-value problems ina finite interval[0,a] bymeans of theBVPh1.0.

Note that, different from numerical softwareBVP4, our approach is in principlean analytic ones, so that these two control-parameters are used as unknowns to gainhomotopy-approximations until optimal values are found for them. This is an ad-vantage of analytic approaches over numerical ones. From this point of view, theBVPh1.0 provides us more freedom and flexibility to guarantee theconvergence ofseries solution and to search for multiple solutions of nonlinear problems than thenumerical packageBVP4.

Based on the HAM, theBVPh 1.0 provides us great freedom to use differentbase functions to approximate solutions of anth-order nonlinear boundary-valueequationF [z,u] = 0 in a finite interval 0≤ z≤ a. This is mainly because the HAMprovides us extremely large freedom to choose the auxiliarylinear operatorL andthe initial guessu0(z). In fact, it is due to such kind of freedom of the HAM that wecan introduce the so-called multiple-solution-control parameter in the initial guess.

Finally, it must be emphasized that thenth-order nonlinear boundary-value equa-tion (8.1) and then linear boundary conditions (8.2) are so general that it is quitedifficult to develop a package valid for all of them. Note that, as mentioned in Chap-ter 7, our aim is to develop a package valid for as many (butnot all) nth-order non-linear multipoint boundary-value problems as possible. The Mathematica package


BVPh(version 1.0) provides us a convenient analytic tool to search for multiple so-lutions of nonlinear boundary-value equations, although further modifications (seethe Problems in this chapter) and more applications are needed in future.

Note that theChebfun 4.0 also provides us the ability to “compute with func-tions instead of numbers” [21]. So, it is very interesting toestablish a similar HAM-based package for highly nonlinear multipoint BVPs with singularities by means ofChebfun , an open resource available athttp://www2.maths.ox.ac.uk/chebfun/.


Appendix 8.1 Input data of BVPh for Example 8.3.1

( * Input Mathematica package BVPh version 1.0 * )<<BVPh.txt;

( * Define the physical and control parameters * )TypeEQ = 1;TypeL = 1;TypeBase = 2;ApproxQ = 1;Ntruncated = 20;

( * Define the governing equation * )f[z_,u_,lambda_] := u * D[u,z,2] - gammaˆ2;gamma = 3/5 ;

( * Define Boundary conditions * )zR = 1;OrderEQ = 2;BC[1,z_,u_,lambda_] := Limit[D[u,z], z->0 ];BC[2,z_,u_,lambda_] := u - 1 /. z->zR;

( * Define initial guess * )U[0] = u[0];If[TypeL == 1,

u[0] = sigma + (1-sigma) * zˆ2,u[0] = (sigma+1)/2 + (sigma-1)/2 * Cos[kappa * Pi * z];];

sigma = .;

( * Define output term * )output[z_,u_,k_]:= Print["output = ",

u[k] /. z->0//N];

( * Define the auxiliary linear operator * )omega[1] = Pi/zR;L[f_] := Module[temp,numA,numB,i,If[TypeL == 1,

temp[1] = D[f,z,OrderEQ],numA = IntegerPart[OrderEQ/2];numB = OrderEQ - 2 * numA//Expand;temp[0] = D[f,z,numB];For[i=1, i<=numA, i++,

temp[1] = D[temp[0],z,2]+ (kappa * omega[i])ˆ2 * temp[0];

temp[0] = temp[1];];

];temp[1]//Expand];

Appendix 8.1 Input data ofBVPh for Example 8.3.1 311

( * Print input and control parameters * )PrintInput[u[z]];

( * Gain 3rd-order approximation * )BVPh[1,3];

( * Gain squared residual at 3rd-order approx. * )GetErr[3];

( * Gain optimal values of c0 and sigma * )res=Minimize[Err[3],c0,sigma];Print["Minimum square residual = ", res];

( * Set optimal c0 and sigma * )c0=-1;sigma = 3/4;Print["c0 = ",c0," sigma = ",sigma];

( * Gain 10th-order HAM approximation * )BVPh[1,10];

The Mathematica packageBVPh (version 1.0) and the above input data are freeavailable athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

Note that, for nonlinear boundary-value/eigenvalue problems defined in a finiteinterval,BVPh(version 1.0) has the defaultzL = 0 .




( * Define the physical and control parameters * )TypeEQ = 1;TypeL = 1;TypeBase = 2;ApproxQ = 0;

( * Define the governing equation * )f[z_,u_,lambda_] := D[u,z,4]-beta * z* (1+uˆ2);beta = 10 ;

( * Define Boundary conditions * )zR = 1;OrderEQ = 4;alpha = 1/5 ;BC[1,z_,u_,lambda_] := Limit[u, z->0 ];BC[2,z_,u_,lambda_] := D[u,z] /. z->zR;BC[3,z_,u_,lambda_] := D[u,z,2] /. z->zR;BC[4,z_,u_,lambda_] := Module[temp,

temp[1]=D[u,z,2]/.z->0;temp[2]=D[u,z,2]/.z-> alpha;temp[1]-temp[2] //Expand

];

( * Define initial guess * )U[0] = u[0];u[0] = sigma/(2 * alpha-3) * ((6 * alpha-8) * z

+6* (1-alpha) * zˆ2+2 * alpha * zˆ3-zˆ4);sigma = .;


N[ u[k] /. z->1, 24] ];

( * Define the auxiliary linear operator * )L[f_] := Module[temp,numA,numB,i,If[TypeL == 1,




];temp[1]//Expand];



( * Gain 3rd-order HAM approximation * )BVPh[1,3];

( * Gain squared residual at 3rd-order approx. * )GetErr[3];

( * Gain optimal values of c0 and sigma * )res=Minimize[Err[3],c0,sigma];Print["Minimum square residual = ", res];

( * Set optimal c0 and sigma * )c0=-1;sigma = 62/100;Print["c0 = ",c0," sigma = ",sigma];







( * Define the physical and control parameters * )TypeEQ = 1;TypeL = 1;TypeBase = 2;ApproxQ = 0;NgetErr = 1;

( * Define the governing equation * )f[z_,u_,lambda_]:=D[u,z,4]

+ alpha * ( z * D[u,z,3]+3 * D[u,z,2] )+ R( u * D[u,z,3] - D[u,z] * D[u,z,2] );

alpha = 3/2;R = -11;

( * Define Boundary conditions * )zR = 1;OrderEQ = 4;BC[1,z_,u_,lambda_] := Limit[u, z->0] ;BC[2,z_,u_,lambda_] := Limit[D[u,z,2], z->0 ];BC[3,z_,u_,lambda_] := u - 1 /. z->zR;BC[4,z_,u_,lambda_] := D[u,z] /. z->zR;

( * Define initial guess * )u[0]=sigma * z+(5-4 * sigma)/2 * zˆ3-(3-2 * sigma)/2 * zˆ5;sigma = .;


D[u[k],z]/.z->0//N];

( * Define the auxiliary linear operator * )omega[1] = Pi/zR;omega[2] = Pi/zR;L[f_] := Module[temp,numA,numB,i,If[TypeL == 1,




];temp[1]//Expand];



( * Set c0 and sigma * )c0 =-1/2;sigma = 2;Print[" c0 = ",c0, " sigma = ",sigma];

( * Gain approximation by 3rd-order iteration * )iter[1,40,3];




Problems

8.1. Boundary-value problems with nonlinear boundary conditionsDevelop a HAM-based analytic approach for thenth-order nonlinear boundary-value equation in a finite interval:

F [z,u] = 0, 0≤ z≤ a,

subject to then nonlinear multipoint boundary conditions

Bk[z,u] = γk,

whereBk is a nonlinear differential operator andγk is a constant. Assume that theabove equation has at least one smooth solution. Modify the Mathematica packageBVPh(version 1.0) for this problem.

8.2. Coupled nonlinear boundary-value problemsDevelop a HAM-based analytic approach forn coupled nonlinear boundry-valueequations in a finite intervalz∈ [0,a]:

Fk[z,u] = 0, 1≤ k≤ n,

subject to some linear/nonlinear multipoint boundary conditions, wheren≥ 2. As-sume that there exists at least one smooth solution. Modify the Mathematica packageBVPh(version 1.0) for this kind of problems in general.

8.3. Boundary-value problems in an infinite intervalDevelop a HAM-based analytic approach for thenth-order nonlinear boundary-value equation in an infinite interval

F [z,u] = 0, 0≤ z<+∞,

subject to then multipoint nonlinear boundary conditions

Bk[z,u] = γk,

whereBk is a nonlinear differential operator andγk is a constant. Assume that thereexists at least one smooth solution. Modify the MathematicapackageBVPh(version1.0) for this kind of problems in general.

8.4. Coupled nonlinear boundary-value problems in an infinite intervalDevelop a HAM-based analytic approach forn coupled nonlinear boundary-valueequations in an infinite intervalz∈ [0,+∞):

Fk[z,u] = 0, 1≤ k≤ n,

subject to some linear/nonlinear boundary conditions, where n ≥ 2. Assume thatthere exists at least one smooth solution. Modify the Mathematica packageBVPh(version 1.0) for this kind of problems in general.

References 317

References


2. Berman, A. S.: Laminar flow in channels with porous walls. J. Appl. Phys.24, 1232 – 1235(1953)

3. Graef, J.R., Qian. C. Yang, B.: A three point boundary value problem for nonlinear forth orderdifferential equations. J. Mathematical Analysis and Applications.287, 217 – 233 (2003)

4. Graef, J.R., Qian. C. Yang, B.: Multiple positive solutions of a boundary value prolem forordinary differential equations. Electronic J. of Qualitative Theory of Differential Equations.11, 1 – 13 (2004)
















20. Magyari, E.: Exact abalytic solution of a nonlinear reaction-diffusion model in porous cata-lysts. Chem. Eng. J.143, 167 – 171 (2008)



Chapter 9Nonlinear eigenvalue equations with varyingcoefficients

Abstract Five different types of examples are used to illustrate the validity ofthe Mathematica packageBVPh (version 1.0) for nonlinear eigenvalue equationsF [z,u,λ ] = 0 in a finite interval 0≤ z≤ a, subject to then linear boundary con-ditionsBk[z,u] = γk (1≤ k ≤ n), whereF denotes anth-order nonlinear ordinarydifferential operator,Bk is a linear differential operator,γk is a constant,u(z) andλdenote eigenfunction and eigenvalue, respectively. Theseexamples verify that, us-ing theBVPh1.0, multiple solutions of some highly nonlinear eigenvalue equationswith singularity and/or multipoint boundary conditions can be found by means ofdifferent initial guesses and different types of base functions.

9.1 Introduction

In science and engineering, one often needs to solve nonlinear differential equa-tions about eigenfunction and eigenvalue. Many nonlinear eigenvalue equationshave multiple solutions. However, it is well known that multiple solutions of nonlin-ear boundary-value problems are not easy to gain by means of numerical techniquessuch as the shooting method [31].

In 2005, Liao [20] applied the homotopy analysis method (HAM) [12–26,33] tosuccessfully obtain two branches of solutions of the boundary-layer equation

F ′′′+12

FF ′′−βF ′2 = 0, F(0) = 0, F ′(0) = 1, F ′(+∞) = 0,

where−1< β <+∞ is a constant, by means of introducing an additional unknownquantityδ =F(+∞). Solving a nonlinear algebraic equation related to this unknownquantityδ , Liao [20] found one new branch of solutions whenβ > 1, which hadbeennever reported by other analytic methods and even neglected by numericalmethods, mainly because the difference between the values of F ′′(0) of the twobranches of solutions is so small that it is hard to distinguish them. It verifies that

319

320 9 Nonlinear eigenvalue equations with varying coefficients

multiple solutions of some nonlinear problems can be found out by introducing anunknown parameter properly.

In Chapter 8, we illustrate that, using Mathematica packageBVPh (version 1.0)as a analytic tool, multiple solutions of some highly nonlinear boundary-value prob-lems in a finite intervalz∈ [0,a] can be found by introducing a unknown parameterinto initial guess, called themultiple-solution-control parameter. In this chapter, wefurther illustrate the validity of theBVPh 1.0 for nonlinear eigenvalue problemsgoverned by anth-order nonlinear ordinary differential equation in a finite intervalz∈ [0,a]:

F [z,u,λ ] = 0, z∈ [0,a], (9.1)


Bk [z,u] = γk, 1≤ k≤ n, (9.2)

whereF is anth-order nonlinear ordinary differential operator,Bk is a linear dif-ferential operator,z is an independent variable,u(z) is an eigenfunction in the finiteintervalz∈ [0,a], λ is an unknown eigenvalue,γk anda> 0 are bounded constants,respectively. Note that, whenn ≥ 3, then linear boundary conditions (9.2) can bedefined at separated points in the interval[0,a] (including the two endpoints). Thelinear operatorBk is in the general form

Bk [z,u] =n

∑i=0

ak,n−i(z)diu(z)

dzi , (9.3)

whereak,i(z) is a smooth function ofz. Note that at least one ofak,n−i(z) is non-zero.Assume that the above eigenvalue problem has at least one smooth eigenfunctionand the corresponding eigenvalue.

In general, there exist many different non-zero eigenfunctions and eigenvalues,depending on a term such asu(a), u′(0), and so on. Therefore, we add such anadditional linear boundary condition

B0 [z,u] = γ0, at z= z∗, (9.4)

wherez∗ ∈ [0,a] and γ0 is a constant, to distinguish different eigenfunctions andeigenvalues. Note that the above boundary condition must belinearly independentof then original boundary conditions (9.2).

Some numerical techniques are developed to solve nonlineareigenvalue prob-lems. One of them is the famousBVP4c in MATLAB [31], which is based onthe shooting method [31]. In 2009, by means of trigonometricfunctions as basefunctions, Liao [23] employed the HAM to analytically solvea nonlinear eigen-value problem about a non-uniform beam acted by an axial force. In 2011, usingpower polynomial as base functions, Abbasbandy [1] proposed a HAM-based ana-lytic approach for linear eigenvalue problems, and suggested a way to gain multipleeigenfunctions by using different values of the convergence-control parameterc0

governed by a nonlinear algebraic equation.

9.2 Brief mathematical formula 321

In this chapter, using the hybrid-base approximation method described in§ 7.2.3,we propose a HAM-based analytic approach for the nonlinear eigenvalue equation(9.1) with the multipoint boundary conditions (9.2) in general. Then, we illustratethat such kind of nonlinear eigenvalue problems can be solved by means of theBVPh1.0. The validity and generality of theBVPh1.0 are shown by five differenttypes of eigenvalue equations.

9.2 Brief mathematical formula

Note that both of the eigenfunctionu(z) and eigenvalueλ are unknown. Letq∈ [0,1]denote the homotopy-parameter,u0(z) andλ0 denote the initial guess of the eigen-function u(z) and eigenvalueλ , respectively, whereu0(z) satisfies then originalboundary conditions (9.2) and the additional boundary condition (9.4). In the frameof the HAM, we should first construct two continuous deformations (homotopies)φ(z;q) andΛ(q) such that, asq∈ [0,1] increases from 0 to 1,φ(z;q) varies from theinitial guessu0(z) to the eigenfunctionu(z), andΛ(q) from the initial guessλ0 tothe eigenvalueλ , respectively. Such kind of continuous deformations (homotopies)are constructed by the so-called zeroth-order deformationequation

(1−q)L [φ(z;q)−u0(z)] = c0 q F [z,φ(z;q),Λ(q)] , q∈ [0,1], (9.5)


Bk [z,φ(z;q)] = γk, 1≤ k≤ n, (9.6)

and the additional linear boundary condition

B0 [z,φ(z;q)] = γ0, (9.7)

wherec0 6= 0 is the so-called convergence-control parameter, andL is an auxiliarylinear operator with the propertyL [0] = 0, whose highest order of derivative isn.Since the initial guessu0(z) satisfies all boundary conditions, obviously,

φ(z;0) = u0(z), Λ(0) = λ0

andφ(z;1) = u(z), Λ(1) = λ

are solutions of (9.5) to (9.7) whenq= 0 andq= 1, respectively. If the initial guessu0(z), the auxiliary linear operatorL and the convergence-control parameterc0 areproperly chosen so that the homotopy-Maclaurin series

φ(z;q) = u0(z)++∞

∑m=1

um(z) qm, (9.8)


Λ(q) = λ0++∞

∑m=1

λm qm, (9.9)

are absolutely convergent atq= 1, we have the homotopy-series

u(z) = u0(z)++∞

∑m=1

um(z), (9.10)

λ = λ0++∞

∑m=1

λm, (9.11)

respectively. At theMth-order of approximation, we have the homotopy approxima-tions

u(z)≈M

∑m=0

um(z), λ ≈M

∑m=1

λm−1. (9.12)

According to Theorem 4.15,um(z) andλm−1 are governed by themth-order de-formation equation

L [um(z)− χm um−1(z)] = c0 δm−1(z), (9.13)

whereδk(z) = DkF [z,φ(z;q),Λ(q)]

and

χk =

0, k≤ 1,1, k> 1.

Here,Dk is thekth-order homotopy-derivative operator, defined by

Dk =1k!

∂ k

∂qk

∣

∣

∣

∣

q=0.

Substituting the homotopy-Maclaurin series (9.8) into then boundary conditions(9.6) and (9.7), then equating the like-power ofq, we have the(n+1) linear bound-ary conditions

Bi [z,um(z)] = 0, 0≤ i ≤ n. (9.14)

For details, please refer to Chapter 2 and Chapter 4.Note thatδn(z) in (9.13) only depends upon the nonlinear operatorF . By means

of the basic properties of the homotopy-derivative operator Dk proved in Chapter 4,it is easy to deduce an explicit expression ofδk(z) in most cases. For example, incase of

F [z,u,λ ] = L0[u]+λ f (z,u),

whereL0 is a nth-order linear differential operator andf (z,u) is a smooth func-tion of z andu, using the basic property of the homotopy-derivative operator Dn

described by Theorem 4.1, the linearity property describedby Theorem 4.2 and thecommutativity property described by Theorem 4.3, we have the explicit expression


δk(z) = L0[uk(z)]+k

∑n=0

λk−n Dn f [z,φ(z;q)] (9.15)

with the recurrence formulas

D0 f [z,φ(z;q)] = f (z,u0), (9.16)

Dn f [z,φ(z;q)] =n−1

∑j=0

(

1− jn

)

un− j(z)∂D j f [z,φ(z;q)]

∂u0, (9.17)

given by Theorem 4.10. In this case, we can always gain the explicit expression ofδk(z) for arbitrary linearnth-order differential operatorL0 and arbitrary nonlinearfunction f (z,u). For a general nonlinear operatorF [z,u,λ ], it is easy to gainδk(z)efficiently by means of computer algebraic system like Mathematica [2]. In fact, tosolve nonlinear ODEs in a finite intervalz∈ [0,a] by means of theBVPh1.0, it isunnecessary for us to deduce the expression ofδk(z), since it is given automaticallyby the package. For details, please refer to theBVPh1.0 given in the appendix ofChapter 7, which is free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

Note that both of themth-order deformation equation (9.13) and the(n+ 1)boundary conditions (9.14) are linear. Letu∗m(z) denote a special solution of (9.13).One has

u∗m(z) = χm um−1(z)+ c0 L−1 [δm−1(z)] ,

whereL −1 is the inverse operator of the auxiliary linear operatorL . Note that thespecial solutionu∗m(z) contains the unknownλm−1. The general solutionum(z) of(9.13) reads

um(z) = χm um−1(z)+ c0 L−1 [δm−1(z)]+

n

∑i=1

Ai ϕi(z),

whereAi is a unknown coefficient andϕi(z) is a known function, which satisfies theequation

L [ϕi(z)] = 0, 1≤ i ≤ n.

The unknownλm−1 and then unknown integral coefficientsAi (1 ≤ i ≤ n) are ex-actly determined by the(n+1) linear boundary conditions (9.14). Therefore, giventhe initial guessu0(z), we can gainλ0,u1(z), thenλ1,u2(z), and so on. It is ex-tremely efficient to do such a kind of work by means of computeralgebra systemslike Mathematica.

In the frame of the HAM, we have extremely large freedom to choose the aux-iliary linear operatorL and the base function for the general eigenvalue equation(9.1). Therefore, theBVPh 1.0 provides us great freedom to choose the auxiliarylinear operatorL and the initial guessu0(z), so that different types of nonlineareigenvalue problems can be solved.

It is well-known that a continuous functionu(z) in z∈ [0,a] can be approximatedby Fourier series [7]. Leonhard Euler (15 April 1707 – 18 September 1783) solved asimple but famous eigenvalue problem, governed by the 2nd-order linear differential


equationu′′+λ u= 0, u(0) = 0, u(a) = 0,

whose eigenfunction and eigenvalue read

u= A sin(κπz

a

)

, λ =(κπ

a

)2

with a positive integerκ ≥ 1 and an arbitrary constantA. The above linear eigen-value problem has an infinite number of eigenfunctions and eigenvalues. Note thatit is a special case of (9.1) and (9.2). In order to include such kind of eigenfunctions,we often choose the auxiliary linear operator

L u= u′′+(κ π

a

)2u (9.18)

for the 2nd-order eigenvalue differential equation (9.1),which has the property

L

[

C1cos(κπz

a

)

+C2sin(κπz

a

)]

= 0

for arbitrary constantsC1, C2 and positive integerκ ≥ 1.Similarly, for a 3rd-order eigenvalue differential equation (9.1), we can choose

the auxiliary linear operator

L u= u′′′+(κ π

a

)2u′, (9.19)

which has the property

L

[

C0+C1cos(κπz

a

)

+C2sin(κπz

a

)]

= 0

for arbitrary constantsC0,C1, C2 and positive integerκ ≥ 1. In general, for thenth-order eigenvalue differential equation (9.1), we can choose

L u=

[

m

∏i=1

(

d2

dx2 +ω2i

)

]

u, whenn= 2m, (9.20)

or

L u=

[

m

∏i=1

(

d2

dx2 +ω2i

)

]

u′, whenn= 2m+1. (9.21)

whereωi is a frequency, which may be different each other, such as

ωi = i(κπ

a

)

, (9.22)

or the same, like

ωi =(κπ

a

)

, (9.23)


depending on the boundary conditions (9.2), whereκ ≥ 1 is a positive integer.Note that the differentκ of the above auxiliary linear operators correspond to a

set of different base functions

sin

(

jκπza

)

,cos

(

jκπza

)

∣

∣

∣j = 0,1,2,3, · · ·

,

which provides us a convenient way to search for multiple eigenfunctions andeigenvalues of a 2nd-order nonlinear eigenvalue equation (9.1) in a finite intervalz∈ [0,a], as shown in§ 9.3.1,§ 9.3.2 and§ 9.3.3.

It is well-known that a continuous functionu(z) in a finite intervalz∈ [0,a] canbe well approximated by a power series

u(z) =+∞

∑k=0

ak zk,

or by a Chebyshev series

u(z) =+∞

∑k=0

bk Tk(z),

whereTk(z) is the Chebyshev polynomial of the first kind. So, we sometimes expressan eigenfunction in power or Chebyshev polynomial, as shownin § 9.3.4 and§ 9.3.5.In this case, we simply choose the auxiliary linear operator

L u=dnudzn (9.24)

for anth-order eigenvalue problem in a finite intervalz∈ [0,a].Note thatum(z) andλm−1 contain the so-called convergence-control parameter

c0, which has no physical meanings but provides us a convenientway to guaranteethe convergence of the homotopy-series (9.10) and (9.11), as shown in Chapter 2and proved in Chapter 4. Let

Em(c0) =1a

∫ a

0

F

[

z,m

∑i=0

ui(z),m−1

∑i=0

λi

]2

dz (9.25)

denote the averaged squared residual of the governing equation (9.1) at themth-order of approximation, where the integral is calculated numerically by means ofmany enough discrete points. The optimal valuec∗0 of the convergence-control pa-rameterc0 is determined by

dEm(c0)

dc0= 0.

In most cases, choosing a convergence-control parameter near its optimal valuec∗0,we can obtain quickly convergent homotopy-series solutionof the eigenfunctionu(z) and eigenvalueλ , as illustrated later. In this way, we can solve highly nonlineareigenvalue problems inz∈ [0,a] by means of theBVPh1.0, as illustrated later.


For the choice of the auxiliary linear operatorL and the initial guessu0(z),please also refer to Chapter 7.

9.3 Examples

All examples given below are solved by theBVPh 1.0, which is given in the ap-pendix of Chapter 7 and free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.In addition, the input data files of all these examples for theBVPh1.0 are given inthe appendix of this chapter and free available at the above website.

In the following examples, we set the number of truncated terms Nt = 20, anduse 50 discrete points with equal space in the intervalz∈ [0,a] to numerically calcu-late the related integrals mentioned above. Besides, if notmentioned, the 3rd-orderHAM iteration approach is used, i.e.M = 3 in (7.51).

9.3.1 Non-uniform beam acted by axial load

Let us first consider a non-uniform beam with arbitrary cross-section [5, 8, 10, 11]on two supports under an axial loadP, as shown in Fig. 9.1, whereP is positive forcompressive force and negative for tensile force. Letl andθ denote the length andslope of the beam, andu its deflection, respectively. Letsdenote the arc-coordinateof the natural axis which passes through the centroid of eachcross-section of thebeam in its straight or unbuckled state,I(s) the smallest moment of inertia of thecross-section about a line in its plane through the centroid, E the Young’s modulusof the material, respectively. Assume that all of the principle axes of inertia areparallel so that the beam is not twisted. Mathematically, the problem is governed by

EI(s)dθds

+Pu= 0, u(0) = u(l) = 0,

whereI(s), the moment of inertia, is non-negative. Assume thatI(s) andI ′(s) arebounded and continuous functions, where the prime denotes the differentiation with

Fig. 9.1 Beam acted by anaxial loadP with variablemoment of inertiaI(z)

9.3 Examples 327

respect tos. Differentiating the original governing equation with respect tos andusing the relationship sinθ = du/ds, we have the nonlinear buckling equation

(EIθ ′)′+P sinθ = 0.

Substituting the boundary conditionu(0) = u(l) = 0 into the original governingequation, we have the equivalent boundary conditions in theform

θ ′(0) = θ ′(l) = 0,

which means that the bending moment at two ends of the beam is zero.Define the dimensionless arc-coordinate of the natural axis

z=π sl

and writeI = I0 µ(z),

whereµ(z) is a distribution function ofI , andI0 > 0 is a reference moment of inertia,respectively. For example,I0 can be defined by

I0 =1l

∫ l

0I(s)ds=

1π

∫ π

0I(z)dz,

although this is not absolutely necessary. Then, the problem under consideration isgoverned by

µ(z)θ ′′(z)+ µ ′(z)θ ′(z)+λ sin[θ (z)] = 0, θ ′(0) = θ ′(π) = 0, (9.26)

where the prime denotes the differentiation with respect toz, and

λ =P

EI0

(

lπ

)2

(9.27)

is called the axial load parameter. For givenλ , it is straightforward to gain thecorresponding axial load

P= λ (EI0)(π

l

)2.

So,λ > 0 corresponds to a compressive forceP, andλ < 0 to a tensile one, respec-tively.

Equation (9.26) with varying coefficientsµ(z) andµ ′(z) is only a special case of(9.1), i.e.

F [z,θ ,λ ] = µ(z)θ ′′(z)+ µ ′(z)θ ′(z)+λ sin(θ ).

Obviously, for different non-zero eigenfunctionθ (z), the value ofθ (0) is different.This provides us an additional boundary condition

θ (0) = γ (9.28)


to distinguish different non-zero eigenfunctions.TheMth-order approximations ofu(z) andλ are given by (9.12), whereθm(z)

andλm−1 are determined by the correspondingmth-order deformation equation

L [θm(z)− χm θm−1(z)] = c0 δm−1(z), (9.29)


θ ′m(0) = 0, θ ′

m(π) = 0, θm(0) = 0, (9.30)

where

δk(z) = µ(z)θ ′′k (z)+ µ ′(z)θ ′

k(z)+k

∑i=0

λk−i Di sin[φ(z;q)] (9.31)

is given by Theorem 4.1, andDi sin[φ(z;q)] is gained by a recursion formuladescribed by Theorem 4.8 and Theorem 4.10.

9.3.1.1 Uniform beam

Firs of all, let us consider the case of uniform beam, i.e.µ = 1, corresponding to

F [z,θ ,λ ] = θ ′′+λ sin(θ ). (9.32)

Since it is a 2nd-order differential equation, we use the auxiliary linear operator(9.18), which contains a positive integerκ . To satisfy the two original boundaryconditions in (9.26) and the additional boundary condition(9.28), we choose suchan initial guessθ0(z) that

θ ′0(0) = θ ′

0(π) = 0, θ0(0) = γ.

There are many functions satisfying the above conditions, such as

θ0(z) = γ cos(κ z) (9.33)

andθ0(z) = σ − (σ − γ) cos(κ z), (9.34)

whereσ is a real number, which provides us one additional degree of freedom inthe initial guess. Note that the initial guess (9.33) is a special case of (9.34) whenσ = 0. We callσ the multiple-solution-control parameter, because it provides us aconvenient way to search for multiple eigenfunctions, as shown later.

This nonlinear eigenvalue problem is solved by means of theBVPh 1.0. With-out loss of generality, we first consider the case ofγ = 1 andκ = 1. Using theinitial guess (9.33) withκ = 1, the curves of the squared residual of the governingequation versus the convergence-control parameterc0 at up-to 3rd-order approx-imation are as shown in Fig. 9.2, which indicates that the optimal convergence-

9.3 Examples 329

Fig. 9.2 Squared residual ofgoverning equation versusc0when µ = 1,γ = 1, κ = 1and θ0(z) = cos(z). Solidline: 1st-order approxima-tion; dashed line: 2nd-orderapproximation; Dash-dottedline: 3rd-order approximation.

c0

Em

-2 -1.5 -1 -0.5 010-14

10-12

10-10

10-8

10-6

10-4

µ = 1, γ = 1, κ = 1

u0 = cos(z)

control parameterc∗0 is close to -1. Usingc0 = −1, even the 3rd-order homotopy-approximation without iteration is quite accurate: the corresponding squared resid-ual is only 4.1× 10−14. At the 2nd iteration by means of the 3rd-order iterationformula, the squared residual decreases to 1.3×10−26, which is so small that moreiterations are unnecessary. In this case, the eigenvalueλ converges to 1.137069rather quickly, as shown in Table 9.1.

Table 9.1 First-type eigenvalue and squared residual given by the 3rd-order iteration approachwith c0 =−1 whenγ = 1, κ = 1 andθ0(z) = cos(z)

Number of iterationm λ Squared residualEm

1 1.137069 4.1×10−14

2 1.137069 1.3×10−26

Table 9.2 Second-type eigenvalue and squared residual given by the 3rd-order iteration approachwith c0 =−1 whenγ = 1, κ = 1 andθ0(z) = 3−2cos(z)

Number of iterationm λ Squared residualEm

1 -1.922288 1.5×10−3

2 -1.954534 4.6×10−6

3 -1.951247 3.6×10−8

4 -1.951353 1.2×10−9

5 -1.951371 9.7×10−12

6 -1.951368 2.2×10−14

7 -1.951368 5.0×10−16

8 -1.951368 1.2×10−17

9 -1.951368 2.0×10−18

10 -1.951368 1.9×10−18


Fig. 9.3 Squared residual ofgoverning equation versusc0whenµ = 1,γ = 1, κ = 1 andθ0(z) = 3− 2cos(z). Solidline: 1st-order approxima-tion; dashed line: 2nd-orderapproximation; Dash-dottedline: 3rd-order approximation.

c0

Em

-2 -1.5 -1 -0.5 010-4

10-3

10-2

10-1

100

µ = 1, γ = 1, κ = 1

u0 = 3 - 2 cos(z)

Fig. 9.4 Two different eigen-functions whenµ = 1,γ = 1and κ = 1. Solid line: the1st eigenfunction relatedto λ = 1.137069 given byθ0(z) = cos(z); dashed line:the 2nd eigenfunction relatedto λ = −1.951368 given byθ0(z) = 3−2cos(z).

z

θ

0 0.5 1 1.5 2 2.5 3 3.5-2

-1

0

1

2

3

4

5

6

µ = 1, γ = 1, κ = 1

Noticing that the initial guess (9.33) is a special case of (9.34) whenσ = 0,we attempt the initial guess (9.34) with different values ofσ , such asσ = 2,3,4and so on. In a surprise, it is found that a new type of solutioncan be obtainedwhenσ = 2 andσ = 3 by choosing proper convergence-control parameterc0. Forexample, in case ofσ = 3, the curves of the squared residual versusc0 at up-to the3rd-order of homotopy-approximation are as shown in Fig. 9.3, which indicates thatthe optimal convergence-control parameterc0 is about -1.25. Indeed, we obtain theconvergent approximation of the eigenvalueλ = −1.951368 by means of the 3rd-order HAM iteration approach withc0=−1, and the corresponding squared residualdecreases quickly, as shown in Table 9.2. Note that we introduce in Chapter 8 asimilar parameter in initial guesses so as to find out multiple solutions of nonlinearODEs inz∈ [0,a]. This reveals the reasons why we call itmultiple-solution-controlparameter.

Therefore, in case ofµ = 1, γ = 1 andκ = 1, there exist two different typesof eigenvalues: one positive eigenvalueλ = 1.137069 and one negative eigenvalueλ = −1.951368, corresponding to the two different types of eigenfunctionsθ (z),as shown in Fig. 9.4. They give completely different displacements, as shown in

9.3 Examples 331

Fig. 9.5 Displacement givenby two different eigenfunc-tions whenµ = 1,γ = 1andκ = 1. Solid line: dis-placement correspondingto λ = 1.137069 given byθ0(z) = cos(z); dashed line:displacement correspondingto λ = −1.951368 given byθ0(z) = 3−2cos(z).

x

u

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

µ = 1, γ = 1, κ = 1

AB

Fig. 9.5, wherex andu denote the horizontal and vertical displacement of the beam,given by

x(z) =∫ z

0cosθ (s)ds, u(z) =

∫ z

0sinθ (s)ds,

respectively. For other different values ofγ = θ (0), we can obtain the two differenttypes of eigenfunctions and eigenvalues in a similar way by means of theBVPh1.0.

In the above expressions aboutx(z) andu(z), we regard one end of the beamat z= 0 as a fixed point, i.e.x(0) = 0 andu(0) = 0. Let (xA,0) and (xB,0) de-note the position of the other end of the beam, correspondingto the 1st and 2nd-type eigenfunction, respectively, as shown in Fig. 9.5. Obviously,xA andxB dependon the value ofγ. It is found that, for the 1st-type eigenfunction, asγ increasesfrom 0 to π , xA decreases monotonously fromxA = π . For the 2nd-type eigen-function, asγ decreases fromπ to 0, xB increases monotonously fromxB = −π .Especially, it is found that, for the 1st-type eigenfunction, xA = −1.4×10−6 whenγ = 2.281319= γ∗, with the corresponding eigenvalueλ = 2.183380= λ ∗. For the2nd-type eigenfunction,xB = 7.2× 10−7 whenγ = 0.860274= γ, with the corre-sponding eigenvalueλ =−2.183380= λ . It is interesting that

λ ∗ =−λ , γ∗+ γ ≈ π , (9.35)

and besides, the displacements given by these two differenttype eigenvalues arealmost the same, as shown in Fig. 9.6.

In general case, letλ ∗ denote the eigenvalue of the 1st-type eigenfunction whenθ (0) = γ∗, and λ the eigenvalue of the 2nd-type eigenfunction whenθ (0) = γ,respectively. It is found that, as long as

γ∗+ γ = π ,

it holdsλ ∗ =−λ ,


Fig. 9.6 Displacement givenby two different types ofeigenfunctions whenµ =1 andκ = 1. Solid line:displacement correspondingto the 1st-type eigenfunctionwith the positive eigenvalueλ ∗ = 2.183380 given byγ∗ = 2.281319; Symbols:displacement correspondingto the 2nd-type eigenfunctionwith the negative eigenvalueλ = −2.183380 given byγ = 0.860274.

x

u

-0.75 -0.5 -0.25 0 0.25 0.5 0.750

0.25

0.5

0.75

1

1.25

1.5

µ = 1, κ = 1

Fig. 9.7 Displacement givenby two different types ofeigenfunctions whenµ =1 andκ = 1. Solid line:displacement correspondingto the 1st-type eigenfunctionwith the eigenvalueλ ∗ =1.951368 given byγ∗ = π−1;dashed line: displacementcorresponding to the 2nd-type eigenfunction with theeigenvalueλ = −1.951368given byγ = 1.

x

u

-0.75 -0.5 -0.25 0 0.25 0.5 0.750

0.25

0.5

0.75

1

1.25

1.5 µ = 1, κ = 1

AB

and the displacements corresponding to the two different eigenfunctions are sym-metric, as shown in Fig. 9.7 and Fig. 9.8. So, the relationship (9.35) holds in general.

Fig. 9.8 Displacement givenby two different types ofeigenfunctions whenµ = 1andκ = 1. Solid line: dis-placement correspondingto the 1st-type eigenfunc-tion with the eigenvalueλ ∗ = 3.202901 given byγ∗ = π − 1/2; dashed line:displacement correspondingto the 2nd-type eigenfunc-tion with the eigenvalueλ = −3.202901 given byγ = 1/2.

x

u

-0.75 -0.5 -0.25 0 0.25 0.5 0.750

0.25

0.5

0.75

1

1.25

1.5

µ = 1, κ = 1

A B

9.3 Examples 333

Fig. 9.9 Eigenvalue versusγ = θ (0) when µ = 1 andκ = 1. Solid line: positiveeigenvalue corresponding tothe 1st-type eigenfunction;dashed line: negative eigen-value corresponding to the2nd-type eigenfunction.

γ

λ

-4 -3 -2 -1 0 1 2 3 4-10

-8

-6

-4

-2

0

2

4

6

8

10 µ = 1,κ = 1

According to this kind of symmetry, we choose the initial guess

θ0(z) = γ cos(κz)

for the 1st-type eigenfunction, and the initial guess

θ0(z) = π − (π − γ)cos(κz)

for the 2nd-type eigenfunction, corresponding toσ = 0 andσ = π in (9.34), re-spectively. By means of the above initial guess and the 3rd-order HAM iterationapproach, we gain the curves of the eigenvalue versusγ (whenµ = 1 andκ = 1) bymeans of theBVPh1.0, as shown in Fig. 9.9.

In addition, the above symmetry of the eigenfunctions and eigenvalues can beproved mathematically. Assume that the eigenfunctionθ (z) and the eigenvalueλsatisfy

θ ′′+λ sinθ = 0, θ ′(0) = θ ′(π) = 0.

Write θ (z) = π −θ (z) andλ =−λ . Then, we have

θ ′′+ λ sinθ =−θ ′′−λ sin(π −θ ) =−(

θ ′′+λ sinθ)

= 0

andθ ′(0) =−θ ′(0) = 0, θ ′(π) =−θ ′(π) = 0.

Besides, it holds

x(z) =∫ z

0cosθ (z) dz=

∫ z

0cos(π −θ )dz=−

∫ z

0cosθ =−x(z)

and

u(z) =∫ z

0sinθ(z) dz=

∫ z

0sin(π −θ )dz=

∫ z

0sinθ = u(z),


say, the displacements corresponding to the two different type eigenfunctions aresymmetric. Therefore, we have the following theorem:

Theorem 9.1.If the eigenfunctionθ (z) and the eigenvalueλ satisfy

θ ′′+λ sin(θ ) = 0, θ ′(0) = θ ′(π) = 0,

then the eigenfunctionπ −θ (z) and the eigenvalue−λ also satisfy the same equa-tion.

Such kind of symmetry holds even for a non-uniform beam with arbitrary µ(z),as described by the following theorem:

Theorem 9.2.If the eigenfunctionθ (z) and the eigenvalueλ satisfy

µ(z)θ ′′(z)+ µ ′(z) θ ′(z)+λ sin(θ ) = 0, θ ′(0) = θ ′(π) = 0,

then the eigenfunctionπ −θ (z) and the eigenvalue−λ also satisfy the same equa-tion.

Mathematically, this symmetry is indeed true. However, the2nd-type eigenfunc-tion with the negative eigenvalue make us confused in physics, because the negativeeigenvalue corresponds to a tensile force! It is a well-known knowledge in mechan-ics that a beam acted by a compressive force (P> 0) may have great deflection if thecompressive force is greater than a critical value. But, it is traditionally believed thatsuch kind of deflection never happens for a beam acted by a tensile force(P < 0).However, from the mathematical points of view, there indeedexists the 2nd-typeeigenfunction with the negative eigenvalue, as proved above. What is the physicalmeaning of the 2nd-type eigenfunction with the negative eigenvalue?

Let us consider again the case ofµ = 1,γ = 1 andκ = 1. The two correspondingeigenfunctions are as shown in Fig. 9.4, and their displacements are as shown inFig. 9.5, respectively. Assume thatθ (z) = 0 before the axial forceP is suddenlyacted att = 0. Then, for a compressive force greater than the critical value, thebeam deflects in such a “natural” way that the end of beam atz= π begins to movefrom the positionx= π to the left untilx= xA. This phenomena has been observedin physical experiments reported in textbooks. However, for a tensile force greaterthan the critical value, the end point of the beam atz= π must be firstsuddenlymoved to the left-hand side of the other end of the beam atz= 0 and then moves tothe positionx= xB. In physics, this kind of sudden deformation is “unnatural”andneeds much more energy than the “natural” ones, and therefore hardly happens inpractice. Even so, the 2nd-type eigenfunction with the negative eigenvalue has stillphysical meanings. Thus, when a beam is acted by a large enough tensile force, thebeam may also have a great deflection in theory, although thisphenomena is hardlyobserved in practice without a sudden, hugh external disturbance.

Note that sinθ ≈ θ for smallθ . However, the linearized equation

9.3 Examples 335

Table 9.3 Multiple eigenvalues of the uniform beam whenγ = 1 andκ > 1.

κ Positive eigenvalue Negative eigenvalue

2 4.548275 -7.8054653 10.233617 -17.562313

Fig. 9.10 Displacement givenby different eigenfunctionswhenµ = 1,γ = 1 andκ =2. Solid line: displacementcorresponding to the positiveeigenvalueλ = 4.548275given by θ0(z) = cos(2z);dashed line: displacementcorresponding to the negativeeigenvalueλ = −7.805465given byθ0(z) = π − (π −1)cos(2z).

x

u

-0.5 0 0.5 1 1.5 2 2.5

-0.75

-0.5

-0.25

0

0.25

0.5

0.75 µ = 1, γ = 1, κ = 2

Fig. 9.11 Displacement givenby different eigenfunctionswhenµ = 1,γ = 1 andκ =2. Solid line: displacementcorresponding to the positiveeigenvalueλ = 10.233617given by θ0(z) = cos(3z);dashed line: displacementcorresponding to the negativeeigenvalueλ = −17.562313given byθ0(z) = π − (π −1)cos(3z).

x

u

-0.5 0 0.5 1 1.5 2 2.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5 µ = 1, γ = 1, κ = 3

θ ′′+λ θ = 0, θ ′(0) = θ ′(π) = 0

has no such kind of symmetry, because its eigenvalues are always positive. Thisillustrates that we might lose a lots of solutions by linearizing a nonlinear equation.In other words, a nonlinear equation may have more interesting properties than itslinearized one.

For other values ofκ , we obtain the convergent eigenfunctions and eigenvaluesin a similar way by means of theBVPh 1.0. For example, in case ofγ = 1, weobtain the convergent positive and negative eigenvalues when κ = 2 andκ = 3,respectively, as listed in Table 9.3. The corresponding eigenfunctions are as shownin Fig. 9.10 and Fig. 9.11, respectively. In theory, givenγ = θ (0), there exist one


positive eigenvalue and one negative eigenvalue for eachκ ≥ 1. Therefore, thereexist an infinite umber of eigenfunctions and eigenvalues for a givenγ = θ (0). Allof these eigenfunctions and eigenvalue can be found by meansof theBVPh1.0 in asimilar way.

Therefore, using theBVPh 1.0, the multiple solutions of nonlinear eigenvalueequations in a finite intervalz∈ [0,a] can be found out by means of different initialguesses and/or different auxiliary linear operators, as illustrated above. Note that,the so-called multiple-solution-control parameterσ provides us a convenient wayto search for multiple eigenfunctions and eigenvalues. Similarly, we can also regardthe positive integerκ in the auxiliary linear operator (9.18) as a kind of multiple-solution-control parameter, too.

It should be emphasized that, to the best of author’s knowledge, the “unnatural”eigenfunctions of (9.32) corresponding to the negative eigenvalues have been neverreported, although it happens hardly in practice. This shows the validity and poten-tial of theBVPh1.0 for nonlinear eigenvalue equations with multiple solutions in afinite intervalz∈ [0,a].

9.3.1.2 Non-uniform beam

To show the general validity of theBVPh1.0 for nonlinear eigenvalue problems, wefurther consider a beam with non-uniform distribution of the inertia moment

µ(z) = 1+cos(4z)

2√

1+exp(z2)+ sin(z2). (9.36)

Note that its averaged moment of inertia reads

1π

∫ π

0

[

1+cos(4z)

2√

1+exp(z2)+ sin(z2)

]

dz≈ 1.002,

which is rather close to that of the uniform beamµ(z) = 1.In this case, the governing equation (9.26) contains varying coefficientsµ(z),

µ ′(z) and thus becomes much more complicated than (9.32) for a uniform beam.Even so, by means of theBVPh 1.0, we still choose the same auxiliary lin-ear operator (9.18) as that for the uniform beam. Besides, the same initial guessθ0(z) = γ cos(κz) andθ0(z) = π − (π − γ)cos(κz) are used to gain the two dif-ferent types of eigenfunctions corresponding to positive and negative eigenvalues,respectively.

Without loss of generality, let us consider the case ofγ = 1 andκ = 1. Twotypes of convergent eigenfunctions are obtained byc0 = −1. The 1st-type eigen-function corresponds to a positive eigenvalueλ = 1.1061, and the 2nd-type to anegative eigenvalueλ =−1.8158, respectively. As proved above, for arbitraryµ(z)and givenγ = θ (0), there exist two types of eigenfunctions with positive and neg-ative eigenvalue. All of them can be gained by means of theBVPh1.0 in a similar

9.3 Examples 337

way. This illustrates the validity and generality of theBVPh 1.0 for complicatedeigenvalue equations with highly nonlinearity.

9.3.2 Gelfand equation

Let us further consider the Gelfand equation [6,9,29]

u′′+(K−1)u′

z+λ eu = 0, u(0) = 0, u(1) = 0, (9.37)

where the prime denotes the differentiation with respectz, K ≥ 1 is a constant,u(z)andλ denote eigenfunction and eigenvalue, respectively.

It is a special case of (9.1) when

F [z,u,λ ] = u′′+(K−1)u′

z+λ eu.

Note that (9.37) is highly nonlinear, since it contains the exponential term exp(u).Besides, it contains a singularity atz= 0 due to the termu′(z)/z. This kind of singu-larity results in difficulty to numerical techniques such asthe shooting method usedin BVP4c, although the limit ofu′(z)/z asz→ 0 is a constant. However, this kindof singularity can be easily resolved by theBVPh1.0, because the computer algebrasystem Mathematica provides us the ability to “compute withfunctions instead ofnumbers” [32].

Since different non-zero eigenfunctions have different values ofu′(0), we add anadditional boundary condition

u′(0) = A, (9.38)

whereA is a given constant, so as to distinguish different eigenfunctions.TheMth-order approximations ofu(z) andλ are given by (9.12), whereum(z)

andλm−1 are determined by themth-order deformation equation

L [um(z)− χm um−1(z)] = c0 δm−1(z), (9.39)


um(0) = 0, um(1) = 0, u′m(0) = 0, (9.40)

where

δk(z) = u′′k(z)+ (K−1)u′k(z)

z+

k

∑i=0

λk−i Di exp[φ(x;q)] , (9.41)

is given by Theorem 4.1, and the termDi exp[φ(x;q)] is gained by a recursionformula described by Theorem 4.7 and Theorem 4.10.

The eigenfunctionu(z) is expressed by the hybrid-base of trigonometric func-tions and polynomial, described in§ 7.2.3. Although (9.37) is quite different from


Fig. 9.12 Squared residualof Gelfand equation (9.37)versusc0 whenK = 1 andA = 1. Solid line: 1st-orderapproximation; Dashed line:2nd-order approximation;Dash-dotted line: 3rd-orderapproximation.

c0

Em

-1 -0.8 -0.6 -0.4 -0.2 010-2

10-1

100

101K = 1, A = 1

Fig. 9.13 Eigenvalue versusA of Gelfand equation (9.37)whenK = 1 andK = 2. Solidline: K = 1; Dashed line:K = 2. Filled circles: exactsolution (9.43).

A

λ

0 2 4 6 8 100

0.5

1

1.5

2

2.5

(9.26), we still choose the same auxiliary linear operator (9.18) withκ = 1. Besides,to satisfy the two original boundary conditions in (9.37) and the additional boundarycondition (9.38), we choose the initial guess

u0(z) =A2[1+ cos(π z)] . (9.42)

This kind of nonlinear eigenvalue equation is solved by means of theBVPh1.0.For givenA, the optimal value of the convergence-control parameterc0 is found bythe minimum of the squared residual of the governing equation (9.37), defined by(9.25).

Without loss generality, let us first consider the case ofK = 1 and A = 1.The squared residual of governing equation at the up-to 3rd-order of homotopy-approximation versusc0 are as shown in Fig. 9.12, which indicates that the optimalconvergence-control parameter is near -0.55. Indeed, we gain the fast convergenteigenfunction and eigenvalueλ = 0.866215 by means ofc0 = −3/5 and the 3rd-order iteration approach withNt = 20, as shown in Table 9.4, which agrees well

9.3 Examples 339

Fig. 9.14 Eigenvalue versusK of Gelfand equation (9.37)whenA= 1 andA= 2. Solidline: A = 1; Dashed line:A= 2.

K

λ

0 2 4 6 8 100

2

4

6

8

10

12

14

16

Table 9.4 Eigenvalue and squared residual of Gelfand equation (9.37)whenK = 1 andA= 1 bymeans ofc0 =−3/5 andNt = 20.

Number of iterationm Eigenvalueλ Squared residualEm

1 0.779831 4.1×10−2

2 0.866491 3.5×10−5

3 0.866221 1.1×10−7

4 0.866215 4.3×10−9

5 0.866215 4.1×10−9

6 0.866215 4.1×10−9

Table 9.5 Eigenvalue of Gelfand equation (9.37) whenK = 2.

A Eigenvalueλ A Eigenvalueλ

0.05 0.192644 2 1.8603530.10 0.371137 2.5 1.6353580.25 0.829569 3 1.3867450.50 1.378161 4 0.9361570.75 1.719382 5 0.6027761.00 1.909210 6 0.3784671.25 1.990053 7 0.2342851.50 1.993891 8 0.14391.75 1.944705 10 0.0535

with the exact eigenvalueλ = 0.866215 given by the closed-form solution [9]

u(z) =12

e−A

ln

[

2eA+2√

eA(eA−1)−1

]2

(9.43)

for K = 1. Given other values ofA, we gain convergent eigenvalue and eigenfunctionin a similar way by means of theBVPh1.0, which agree well with the above exactformula, as shown in Fig. 9.13.


Table 9.6 Eigenvalue of Gelfand equation (9.37) whenA= 1 andA= 2.

K Eigenvalueλ whenA= 1 Eigenvalueλ whenA= 2

1 0.8662 0.74362 1.9092 1.86043 3.0460 3.25224 4.2328 4.80595 5.4471 6.44286 6.6775 8.12097 7.9177 9.81988 9.1642 11.52999 10.4149 13.246210 11.6684 14.9663

In case ofK = 2, using theBVPh1.0, we also gain convergent eigenfunctions andeigenvalues by the 3rd-order HAM iteration approach in a similar way, as shownin Table 9.5 and Fig. 9.13. Similarly, in case ofA = 1 andA = 2, also we obtainconvergent eigenfunctions and eigenvalues for different values ofK, as shown inFig. 9.14 and Table 9.6.

The singularity termu′(z)/z asz→ 0 is easy to resolve by means of computeralgebra system like Mathematica, since it regardszas a function instead of a number,and besides the limit ofu′(z)/zasz→ 0 is a constant.

This example confirms the validity and generality of theBVPh 1.0 for highlynonlinear eigenvalue equations with singularity, since (9.37) contains the exponen-tial term exp(u) and the singularity termu′(z)/z.

9.3.3 Equation with singularity and varying coefficient

Note that theBVPh1.0 is valid for thenth-order nonlinear eigenvalue equation (9.1)subject to then linear boundary conditions (9.2), which are rather generalin form.To show its general validity, we solve here a nonlinear eigenvalue equation withvarying coefficients defined in a finite interval 0< z< π :

√

1+ z2 u′′+cos(πz) u′

z+λ

[

eu

1+ z2 +(1+ z) sinu

]

= sin(z2+e−z), (9.44)

subject to the two boundary conditions

u′(0) = 0, u(π)−u′(π) =35, (9.45)

where the prime denotes the differentiation with respect toz, u(z) andλ are theunknown eigenfunction and eigenvalue, respectively.

The above problem is a special case of (9.1) when

9.3 Examples 341

F [z,u,λ ] =√

1+ z2 u′′+cos(πz) u′

z+λ

[

eu

1+ z2 +(1+ z) sinu

]

− sin(z2+e−z).

Especially, it contains the varying coefficients

√

1+ z2,cos(πz)

z,

11+ z2 , (1+ z), −sin(z2+e−z)

and the highly nonlinear terms exp(u) and sin(u). In addition, it has a singularity atz= 0 due to the termu′(z)/z. Such kind of singularity leads to difficulty to numericaltechniques such as the shooting method used byBVP4c. Thus, this equation israther complicated. Fortunately, the limit ofu′(z)/z asz→ 0 is a constant and thuscan be easily resolved by computer algebra system like Mathematica in the frameof the HAM, which regardszas a function instead of a number.

For a non-zero eigenfunctionu(z), it holdsu(0) 6= 0. So, we use

u(0) = A (9.46)

as the additional boundary condition to distinguish different non-zero eigenfunc-tions.

TheMth-order approximations ofu(z) andλ are given by (9.12), whereum(z)andλm−1 are determined by themth-order deformation equation

L [um(z)− χm um−1(z)] = c0 δm−1(z) (9.47)


u′m(0) = 0, um(π)−u′m(π) = 0, (9.48)

where

δk(z) =√

1+ z2 u′′k +cos(πz) u′k

z− (1− χk+1)sin(z2+e−z)

+k

∑i=0

λk−i

[

Di exp[φ(z;q)]1+ z2 +(1+ z)Di sin[φ(z;q)]

]

(9.49)

is given by Theorem 4.1, and the termsDi exp[φ(z;q)] andDi sin[φ(z;q)] aregiven by recursion formulas described by Theorem 4.7 and Theorem 4.8, or Theo-rem 4.10, respectively.

We successfully solved this highly nonlinear eigenvalue problem with singularityby means of theBVPh1.0. Since it is defined in a finite intervalz∈ [0,a], we expressthe eigenfunction by the hybrid-base mentioned in§ 7.2.3. Thus, although (9.44)is more complicated than (9.26) and (9.37), we still use the same auxiliary linearoperator (9.18) withκ = 1, and besides choose the initial guess

u0(z) =(5A+3)

10+

(5A−3)10

cos(z), (9.50)


Fig. 9.15 Squared residualof (9.44) whenA = 1/2versusc0. Solid line: 1st-orderapproximation; Dashed line:2nd-order approximation;Dash-dotted line: 3rd-orderapproximation.

c0

Em

-1 -0.8 -0.6 -0.4 -0.2 010-5

10-4

10-3

10-2

10-1

100

101

A = 1/2, κ = 1

Table 9.7 The eigenvalue and squared residual given by the 3rd-order HAM iteration approachwith c0 =−2/5, Nt = 30 and the initial guess (9.50) whenA= 1/2 andκ = 1

Number of iterationm Eigenvalueλ Squared residualEm

1 0.373901 3.3×10−4

2 0.380270 5.4×10−7

3 0.379978 5.6×10−8

4 0.379957 5.3×10−8

5 0.379956 5.3×10−8

6 0.379956 5.3×10−8

7 0.379956 5.3×10−8

8 0.379956 5.3×10−8

Fig. 9.16 Eigenfunction of(9.44) and (9.45) by meansof κ = 1. Solid line: 10th-iteration approximation whenA = 1/2 by means ofc0 =−2/5; Dashed line: 10th-iteration approximation whenA = 0 by means ofc0 =−2/5; Dash-dotted line: 15th-iteration approximation whenA= −1/2 by means ofc0 =−1/5. Symbols: the 3rd-iteration approximations (A=1/2 andA = 0) or the 8th-iteration approximation (A=−1/2).

z

u(z

)

0 0.5 1 1.5 2 2.5 3 3.5-1

-0.5

0

0.5

1

1.5

2

2.5

3κ = 1

9.3 Examples 343

Fig. 9.17 Eigenvalue of(9.44) and (9.45) versusAby means ofκ = 1.

A

λ

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1

1.5

2

κ = 1

which satisfies the two original boundary conditions (9.45)and the additionalboundary condition (9.46).

Without loss of generality, let us first consider the case ofA= 1/2. The squaredresiduals of the governing equation (9.44) at up-to the 3rd-order approximation areas shown in Fig. 9.15, which indicates that the optimal convergence-control param-eter is about -0.4. Indeed, choosingc0 = −2/5, we gain fast convergent eigenvalueand eigenfunction by the 3rd-order HAM iteration approach with Nt = 30, as shownin Table 9.7 and Fig. 9.16. Note that the squared residual of the governing equation(9.44) stops decreasing atEm = 5.3×10−8 for m≥ 4. However, if more truncatedterms (i.e. larger value ofNt ) are used, smaller squared residuals are gained.

Given other values ofA, we gain convergent eigenvalues and eigenfunctions ina similar way by means of theBVPh 1.0. For example, the eigenfunctions whenA = 1/2,A= 0 andA = −1/2 are as shown in Fig. 9.16. It is found that there isno symmetry between the eigenfunctions forA= 1/2 andA= −1/2. Besides, thecurves of the eigenvalue versusA by means ofκ = 1 are as shown in Fig. 9.17. It isfound that there are two branches of eigenvalues by means ofκ = 1. The 1st branchof eigenvalues are positive, and tends to zero asA → +∞, as shown in Table 9.8.The 2nd branch of eigenvalues exists in a intervalcL

0 < c0 < cR0 , wherecL

0 is closeto -3.2, and−0.49< cR

0 <−0.33. Different from the 1st branch of eigenvalues, the2nd branch of eigenvalues are negative for some values ofu(0) = A, as shown inTable 9.9. According to our computations, it seems that there exist no convergenteigenfunctions and eigenvalues whenc0 < cR

0 by means ofκ = 1. Figure 9.17 sug-gests that there might exist some kinds of singularity atc0 = cL

0 andc0 = cR0.

All of the above results are gained by means of the auxiliary linear operator(9.18) with κ = 1 and the initial guess (9.50). As mentioned before, multiple so-lution can be found by means of different auxiliary linear operators and differentinitial guesses. Note that the initial guess (9.50) can be generalized by

u0(z) =(5A+3)

10+

(5A−3)10

cos(κ z), (9.51)


Table 9.8 The 1st branch of eigenvalues of (9.44) by means ofκ = 1.


-0.33 1.607740 0.05 0.565095-0.32 1.559068 0.1 0.528119-0.31 1.508751 0.5 0.379956-0.30 1.457289 1 0.311333-0.25 1.205111 2 0.238667-0.20 1.001034 3 0.162146-0.10 0.748010 4 0.089831-0.05 0.670271 6 0.0203060 0.611237 10 0.000797

Table 9.9 The 2nd branch of eigenvalues of (9.44) by means ofκ = 1.


-3.2 2.356321 -2.0 0.355912-3.1 1.517056 -1.5 0.221140-3.0 1.187195 -1.0 0.081523-2.9 0.984646 -0.7 -0.089925-2.75 0.784556 -0.6 -0.212886-2.5 0.578778 -0.5 -0.532902-2.25 0.448467 -0.49 -0.634153

Table 9.10 Eigenvalues and the squared residual of (9.44) whenA= 1 by means ofκ = 3,Nt = 40,c0 =−1/5 andκ = 5, Nt = 50,c0 =−1/5.

m, Number of iteration λ (κ = 3) Em (κ = 3) λ (κ = 5) Em (κ = 5)

1 6.2210 18.96 13.4273 126.43 6.5234 1.57 18.7510 5.735 7.1834 0.25 19.6540 0.377 7.3451 2.1×10−2 19.8553 2.1×10−2

10 7.3555 1.6×10−4 19.9021 1.7×10−4

15 7.3500 7.1×10−7 19.9044 9.1×10−7

20 7.3500 6.3×10−7 19.9043 8.4×10−7

25 7.3500 6.3×10−7 19.9043 8.4×10−7

whereκ ≥ 1 is an odd integer. Using the above initial guess and the auxiliary lin-ear operator (9.18) withκ = 3 andκ = 5, we gain the multiple eigenfunctions andeigenvaluesλ = 7.3500 (whenκ = 3) andλ = 19.9043 (whenκ = 5) in case ofA = 1 by means ofc0 = −1/5, as shown in Fig. 9.18 and Table 9.10. This sug-gests that the eigenvalue equation (9.44) with the boundaryconditions (9.45) mighthave an infinite number of eigenfunctions and eigenvalues. Note that the multipleeigenfunctions and eigenvalues are gained simply by using different values ofκin the initial guess (9.51) and the auxiliary linear operator (9.18). Therefore, us-ing trigonometric functions as base functions, we can find out multiple solutions of

9.3 Examples 345

Fig. 9.18 Multiple eigenfunc-tion of (9.44) and (9.45) whenA = 1 gained by means ofdifferent values ofκ . Solidline: 8th-iteration approxi-mation by means ofκ = 1and c0 = −2/5; Dashedline: 30th-iteration approx-imation by means ofκ = 3andc0 = −1/5; Dash-dottedline: 30th-iteration approx-imation by means ofκ = 5and c0 = −1/5. Symbols:10th-iteration approximations(κ = 3 andκ = 5) or the3rd-iteration approximation(κ = 1).

z

u(z

)

0 0.5 1 1.5 2 2.5 3 3.5-2

-1

0

1

2

A = 1

some nonlinear eigenvalue equations by means of different initial guesses and thedifferent auxiliary linear operators in the frame of the HAM.

Note that the singularity termu′(z)/zasz→ 0 is easily resolved by theBVPh1.0,since the computer algebra system Mathematica regardsz as a function instead ofa number, and besides the limit ofu′(z)/z asz→ 0 is a constant. In addition, themultiple solutions are found out simply by means of different auxiliary linear op-erators (9.18) and different initial guesses (9.51) with different values ofκ . Thus,we can also regardκ as a kind of multiple-solution-control parameter. Finally, theconvergence-control parameterc0 provides us a convenient way to guarantee theconvergence of homotopy-solutions series for the highly nonlinear governing equa-tion (9.44) .

Therefore, this example confirms the general validity of theBVPh1.0 for com-plicated eigenvalue problems with high nonlinearity and singularity, although (9.44)has no physical meanings.

9.3.4 Multipoint boundary-value problem with multiple solutions

Some nonlinear boundary-value problems with multiple solutions can be solved bytransferring them into eigenvalue equations. To show the general validity of theBVPh1.0, let us consider here a 4th-order nonlinear eigenvalue problem with mul-tipoint boundary condition

u′′′′ = βz(1+u2), u(0) = u′(1) = u′′(1) = 0, u′′(0)−u′′ (α) = 0, (9.52)

whereα ∈ (0,1) andβ are given constants. Graef, Qian and Yang [3,4] proved thatthe above equation has at least two positive solutions whenα = 1/5 andβ = 10.Note that, different from the previous three examples, thisboundary-value equation


is 4th-order, and besides its boundary conditions are defined at three separate points,i.e. it is a so-called multipoint boundary-value problem.

Writing u(z) = λ θ (z) with the definitionλ = u(1), the original equation (9.52)becomes

λ θ ′′′′ = βz(1+λ 2 θ 2), θ (0) = θ ′(1) = θ ′′(1) = 0, θ ′′(0)−θ ′′ (α) = 0, (9.53)

with one additional boundary condition

θ (1) = 1. (9.54)

Regardingλ as an unknown eigenvalue, (9.53) is a special case of (9.1) when

F [z,θ ,λ ] = λ θ ′′′′−βz(1+λ 2 θ 2).

TheMth-order approximations ofθ (z) andλ are given by (9.12), whereθm(z)andλm−1 are determined by themth-order deformation equation

L [θm(z)− χm θm−1(z)] = c0 δm−1(z), (9.55)

subject to the multipoint boundary conditions

θm(0) = θ ′m(1) = θ ′′

m(1) = 0, θ ′′m(0)−θ ′′

m(α) = 0, (9.56)


θm(1) = 0, (9.57)

where

δk(z) =k

∑i=0

λk−i θ ′′′′i − (1− χk+1)β z

− β zk

∑i=0

(

i

∑j=0

λ j λi− j

)(

k−i

∑r=0

θr θk−i−r

)

(9.58)

is gained by Theorem 4.1 in Chapter 4.This nonlinear eigenvalue problem with multipoint boundary conditions is solved

by means of theBVPh1.0. Since (9.53) contains the termβz, the polynomial ofz isused to express the eigenfunctionθ (z). Thus, we choose

L θ = θ ′′′′ (9.59)

as the auxiliary linear operator and the polynomial

θ0(z) =1

2α −3

[

2(3α −4) z+6(1−α) z2+2α z3− z4] , (9.60)

9.3 Examples 347

Fig. 9.19 Squared residualEm of the governing equation(9.53) versusc0 whenα =1/5 andβ = 10 by meansof the first initial guessλ0 =0.63313 of the eigenvalue.Dashed line: 1st-order; Dash-dotted line: 3rd-order; Solidline: 5th-order.

c0

Em

-3 -2.5 -2 -1.5 -1 -0.5 010-12

10-10

10-8

10-6

10-4

10-2

100

α = 0.2, β = 10

Fig. 9.20 Squared residualEm of the governing equation(9.53) versusc0 whenα =1/5 andβ = 10 by meansof the 2nd initial guessλ0 =2.14591 of the eigenvalue.Dashed line: 1st-order; Dash-dotted line: 3rd-order; Solidline: 5th-order.

c0

Em

-1 -0.8 -0.6 -0.4 -0.2 010-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

α = 0.2, β = 10

as the initial guess, which satisfies the four original boundary conditions in (9.53)and the additional boundary condition (9.54).

Without loss of generality, let us first consider the case ofα = 1/5 andβ = 10. Itis found that, at the 1st-order of approximation, the additional boundary-condition(9.57), i.e.θ1(1) = 0, gives a nonlinear algebraic equation

λ 20 −2.77904λ0+1.35864= 0, (9.61)

which has two different solutionsλ0 = 0.63313 andλ0 = 2.14591, respectively.Note that the above algebraic equation is independent of theconvergence-controlpa-rameterc0. Choosing one solution of the above algebraic equation as the initial guessof the eigenvalue, we gain themth-order homotopy-approximationof the eigenfunc-tion θ (z) and the(m− 1)th-order homotopy-approximation of the eigenvalueλ :both of them contain the convergence-control parameterc0. It is found that, whenλ0 = 0.63313, the squared residualEm of the governing equation (9.53) decreases inan intervalc0 ∈ (−3,0) asm increases, and besides the optimal convergence-controlparameterc0 is about -3/2, as shown in Fig. 9.19. Indeed, by means ofλ0 = 0.63313


Table 9.11 The eigenvalue and squared residualEm of (9.53) whenα = 1/5 andβ = 10 by meansof λ0 = 0.63313 andc0 =−3/2

Order of approximationm Eigenvalueλ Squared residualEm

2 0.627446 2.3×10−3

4 0.627318 1.0×10−6

6 0.627315 6.9×10−10

8 0.627315 5.7×10−13

10 0.627315 5.3×10−16

Table 9.12 The eigenvalue and squared residualEm of (9.53) whenα = 1/5 andβ = 10 by meansof λ0 = 2.14591 andc0 =−1/2

Order of approximationm Eigenvalueλ Squared residualEm

4 2.24105 6.9×10−3

8 2.24118 1.6×10−6

12 2.24118 6.3×10−10

16 2.24118 3.2×10−13

20 2.24118 1.7×10−16

andc0 = −3/2, the squared residual of (9.53) decreases monotonously toa rathersmall value 5.3×10−16 at the 10th-order homotopy-approximation, so that we gainthe first eigenfunctionθ (z) with the first eigenvalueλ = 0.627315, as shown in Ta-ble 9.11. Similarly, by means ofλ0 = 2.14591, it is found that the squared resid-ual Em decreases in an intervalc0 ∈ (−0.7,0) as m increases, and the optimalconvergence-control parameterc0 is about -1/2, as shown in Fig. 9.20. Indeed, bymeans ofc0 =−1/2 andλ0 =2.14591, the squared residual decreases to 1.7×10−16

at the 20th-order of approximation, and we obtain the 2nd eigenfunction with the2nd eigenvalueλ = 2.24118, as shown in Table 9.12. It is found that, although thetwo eigenvalues are obviously different, the two corresponding eigenfunctions arerather close, as shown in Fig. 9.21. In general, it is rather difficult to distinguish suchkind of very close eigenfunctions by means of numerical methods.

Note that, sinceu(z) = λ θ (z), we have two obviously different solutionsu(z)corresponding to the two different values ofλ , as shown in Fig. 9.22.

Similarly, we can gain the two solutions of the nonlinear multipoint boundary-value equation (9.52) for different values ofα andβ . This illustrates that a nonlinearboundary-value problem with multiple solutions can be transferred into a nonlineareigenvalue problem. Note that, as shown in§ 8.3.2, it can be directly solved byregarding (9.52) as a nonlinear boundary-value equation, too. Note also that (9.52)is a 4th-order nonlinear boundary-value equation, whose boundary conditions aresatisfied at two endpoints and one seperated point in the interval z∈ (0,1). Thus,(9.52) can be solved by means of theBVPh1.0, no matter we regard it either as anormal nonlinear boundary-value problem or a nonlinear eigenvalue problem.

9.3 Examples 349

Fig. 9.21 Comparison oftwo eigenfunctions of (9.53)and (9.54) whenα = 1/5and β = 10 by means ofdifferent initial guessesλ0of the eigenvalue. Solid line:the 1st eigenfunctionθ (z)given byλ0 = 0.63313 andc0 = −3/2; Dashed linewith open circles: the 2ndeigenfunctionθ (z) given byλ0 = 2.14591 andc0 =−1/2.

z

θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

α = 0.2, β = 10

Fig. 9.22 Two solutions ofthe original equation (9.52)whenα = 1/5 andβ = 10Solid line: 1st solutionu(z)given byλ0 = 0.63313 andc0 = −3/2; Dashed line:2nd solutionu(z) given byλ0 = 2.14591 andc0 =−1/2.

z

u

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

α = 0.2, β = 10

Similarly as mentioned in§ 8.3.2, the multipoint boundary conditions can beeasily resolved by theBVPh1.0, since computer algebra system like Mathematicaregards all of these boundary conditions as the same. This ismainly because com-puter algebra system provides us the ability to “compute with functions instead ofnumbers” [32].

So, this example verifies the validity and generality of theBVPh 1.0 for high-order nonlinear boundary-value problems with multipoint boundary conditions.

9.3.5 Orr-Sommerfeld stability equation with complex coefficient

As the final example, let us consider the famous Orr-Sommerfeld stability equation[27,30] for the stability of plane Poiseuille flow

(D2−α2)2u− i(α R)[

(U0−λ ) (D2−α2)−D2U0]

u= 0, (9.62)



u′(0) = u′′′(0) = 0, u(1) = 0, u′(1) = 0, (9.63)

where the prime denotes the differentiation with respect toz, i =√−1 is an imagi-

nary number, the operatorD is defined byDu= u′′, Rdenotes the Reynolds number,λ is the complex eigenvalue,U0 = 1−z2 is the exact solution of the plane Poiseuilleflow, respectively. The two dimensional disturbance of velocity is proportional to

u(z) exp[iα(x−λ t)]

with α real andλ complex number. So, the flow becomes unstable when the imagi-nary part ofλ is positive, i.e. Im(λ )> 0. For details, please refer to [27,30].

This eigenvalue problem with complex coefficient is also a special case of (9.1)when

F [z,u,λ ] = (D2−α2)2u− i(α R)[

(U0−λ ) (D2−α2)−D2U0]

u.

Note that, ifu(z) is an eigenfunction, thenu(z) = u(z)/u(0) is also an eigenfunctionfor the same eigenvalueλ , since (9.62) is linear. So, we have the additional boundarycondition

u(0) = 1.

TheMth-order approximations ofu(z) andλ are given by (9.12), whereum(z)andλm−1 are determined by themth-order deformation equation

L [um(z)− χm um−1(z)] = c0 δm−1(z) (9.64)


u′m(0) = u′′′m(0) = 0, um(1) = 0, u′m(1) = 0, (9.65)


um(0) = 0, (9.66)

wherec0 is the convergence-control parameter,L is the auxiliary linear operator,u0(z) is the initial guess ofu(z), and

δn(z) = (D2−α2)2un(z)− i(α R)[

U0 (D2−α2)−D2U0

]

un(z)

+ i(αR)n

∑k=0

λk(

D2−α2)un−k(z) (9.67)

is given by Theorem 4.1 in Chapter 4.Let us consider here the eigenvalueu(z) with the symmetryu(z) = u(−z) only.

Sinceu(z) is defined in a finite intervalz∈ [−1,1], it can be expressed by polyno-mials ofz. Therefore, we choose the auxiliary linear operator

9.3 Examples 351

Table 9.13 Eigenvalue and the squared residual of (9.62) in case ofR= 100 andα = 1 by meansof the 3rd-order iteration formula withc0 = (−1+ i)/2.

m, times of iteration Eigenvalueλ Squared residualEm

1 0.473522−0.158606i 23303 0.478861−0.161875i 88.85 0.478652−0.163091i 3.7710 0.478490−0.162945i 7.0×10−4

15 0.478494−0.162944i 9.3×10−8

20 0.478494−0.162944i 1.8×10−11

25 0.478494−0.162944i 6.2×10−15

30 0.478494−0.162944i 2.3×10−18

Table 9.14 Convergent eigenvalues for different Reynolds numberR whenα = 1 by means ofthe 3rd-order HAM iteration formula with the complex convergence-control parameterc0. Theeigenfunction is expressed by a polynomial ofzup-too(zNt ).

R λ c0 with i =√−1 Nt

100 0.478494 - 0.162944i (−1+ i)/2 90200 0.430714 - 0.116810i (−1+ i)/4 90500 0.380566 - 0.0704922i (−1+ i)/10 901000 0.346285 - 0.0421283i (−1+ i)/20 1502000 0.312100 - 0.0197987i (−1+ i)/40 2003000 0.292289 - 0.0101846i (−1+ i)/60 3005000 0.268131 - 0.0017503i (−1+ i)/100 3005500 0.263762 - 0.00060763i (−1+ i)/100 3005800 0.261348 - 0.00002691i (−1+ i)/100 3005814 0.261239 - 1.5004×10−6 i (−1+ i)/100 3005814.83 0.261233 + 2.2495×10−9 i (−1+ i)/100 3005815 0.261231 + 3.1000×10−7 i (−1+ i)/100 3005825 0.261154 + 0.00001837i (−1+ i)/100 3006000 0.259816 + 0.00032309i (−1+ i)/120 300

L u= u′′′′. (9.68)

Notice that the critical eigenvalue is often related to the simplest form of the eigen-function, as shown in§ 9.3.1 for the first example. Thus, we choose the initial guess

u0(z) = (1− z2)2, (9.69)

which is the simplest polynomial ofz satisfying all boundary conditions (9.63) andthe additional boundary conditionu(0) = 1.

This problem is successfully solved by means of theBVPh1.0. Without loss ofgenerality, let us first consider the case ofα = 1 with different values of ReynoldsnumberR. Note that (9.62) contains the imaginary numberi =

√−1. Fortunately, in

the frame of the HAM, we have so great freedom to choose the convergence-controlparameterc0 thatc0 can be a complex number, as described in the Theorem§ 5.3


and the Theorem§ 5.4. Thus, we use here the 3rd-order HAM iteration formula witha complex convergence-control parameterc0. For example, in case ofR= 100 andα = 1, we gain the convergent eigenvalue

λ = 0.478494−0.162944i

by means of a complex convergence-control parameterc0 = (−1+ i)/2, as shownin Table 9.13. This verifies that, in the frame of the HAM, the convergence-controlparameterc0 can be indeed a complex number. Furthermore, it is found that, forgivenR, one can always find a properc0 in such the form

c0 =(−1+ i)

ρ, ρ ≥ 1,

that the corresponding 3rd-order HAM iteration approach converges, as shown inTable 9.14. The real and imaginary parts of the convergent eigenfunctions for differ-ent Reynolds numbers fromR= 100 to the critical valueR= 5814.83 are as shownin Fig. 9.23 and Fig. 9.24, respectively. It is found that, for small Reynolds numberR, the imaginary part of the corresponding eigenvalue is negative, i.e. Im(λ ) < 0,corresponding to a stable viscous flow. Whenα = 1 andR= 5814.83, the imaginarypart of the corresponding eigenvalue

0.261233+2.2495×10−9 i

is rather close to zero. Whenα = 1 andR> 5814.83, it holds Im(λ )> 0 so that theflow becomes unstable. So, the above eigenvalue and the corresponding eigenfunc-tion correspond to the most unstable viscous flow. Note that,unlike Orszag [30], weneed not calculate other higher modes of eigenfunctions andeigenvalues, which arenot important to the stability of the flow.

Fig. 9.23 The real part ofthe eigenfunctions of the Orr-Sommerfeld stability equation(9.62) whenα = 1. Dashedline: R= 100; Dash-dottedline: R= 1000; Solid line:R= 5814.83.

z

Re(

u)

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

α = 1


Fig. 9.24 The imaginarypart of the eigenfunctions ofthe Orr-Sommerfeld stabilityequation (9.62) whenα = 1.Dashed line:R= 100; Dash-dotted line:R= 1000; Dash-dot-dotted line:R= 3000;Solid line:R= 5814.83.

z

Im(u

)

-1 -0.5 0 0.5 1-0.03

0

0.03

0.06

0.09

0.12 α = 1

Note that, as the Reynolds numberR increases from 0 to the critical valueRc ≈5814.83, the real part of the eigenfunctions changes very little,as shown in Fig. 9.23,but the imaginary part varies greatly, as shown in Fig. 9.24.Especially, whenα = 1andR= 5814.83, the imaginary part of eigenfunction is very close to zeroin a largeinterval−0.7≤ z≤ 0.7. It would be valuable to reveal the physical meanings of thisinteresting result.

The critical Reynolds numberRc is defined by Orszag [30] as the smallest valueof R for which an unstable eigenmode exists. For the plane Poiseuille flow, Orszag[30] reported the critical Reynolds numberRc = 5772.22 with αc = 1.02056. Bymeans of theBVPh 1.0 with the 3rd-order HAM iteration approach and the com-plex convergence-controlparameterc0 =(−1+ i)/100, we obtain the correspondingeigenvalue

λ = 0.26943−3.085×10−9 i

in case ofRc = 5772.22 andαc = 1.02056, which agrees well with the numericalones given by Orszag [30].

Thus, this example verifies the validity and generality of theBVPh1.0 for stabil-ity equations with complex coefficients.


In this chapter, we illustrate the validity of theBVPh 1.0 for nonlinear eigenvalueequationsF [z,u,λ ] = 0 in a finite interval 0≤ z≤ a, subject to then linear boundaryconditionsBk[z,u] = γk (1≤ k ≤ n), whereF denotes anth-order nonlinear ordi-nary differential operator,Bk is a linear differential operator,γk is a constant,u(z)andλ denote eigenfunction and eigenvalue, respectively. Five different types of ex-amples are used, such as a non-uniform beam acted by axial force, the Gelfand equa-tion, an eigenvalue equation with varying coefficients, a multipoint boundary-valueproblem with multiple solutions, and the famous Orr-Sommerfeld stability equation


with complex coefficients. These examples illustrate that,using theBVPh1.0, mul-tiple solutions of some highly nonlinear eigenvalue equations with singularity andmultipoint boundary conditions can be found by means of different initial guessesand different types of base functions.

Using the first example, we illustrate that multiple solutions of nonlinear eigen-value equations can be found out by means of different initial guesses. Especially,we successfully gained, maybe for the first time, the eigenfunctions with theneg-ative eigenvalues of the beam equation (9.26), and even proved that such kind of“unnatural” eigenfunctions with the negative eigenvaluesindeed exist for the non-uniform beam equation (9.26) in general. This kind of eigenfunctions with the neg-ative eigenvalues indicate that a beam acted by a large-enough tensile force mayalso have a large deflection, just like a beam acted by a large enough compressiveforce. However, such kind of “unnatural” deflection needs a sudden and huge dis-turbance att = 0, which requires much larger energy than the “natural” deflection,and thus hardly happens in practice. To the best of the author’s knowledge, suchkind of “unnatural” eigenfunctions with negative eigenvalues of the non-uniformbeam equation hasneverbeen reported.Indeed, any a truly new method alwaysgives something new and/or different. This shows the great potential and validity oftheBVPh1.0 for highly nonlinear eigenvalue equations with multiple solutions andsingularity.

In addition, we verifies the generality and validity of theBVPh 1.0 for eigen-value problems with highly nonlinearity and singularity (Example 9.3.2), and/orseparate multiple boundary conditions (Example 9.3.3), and/or varying coefficientsand high-order of derivatives (Example 9.3.4), and/or the complex coefficients (Ex-ample 9.3.5), respectively.

In the frame of the HAM, we have extremely large freedom to choose differentauxiliary linear operators and initial guesses so as to gainconvergent eigenfunctionsin different base functions, as illustrated by these examples. In the first three exam-ples, the combination of trigonometric functions and polynomial are used as hybrid-base functions to express eigenfunctions, together with the auxiliary linear operator(9.18) and the initial guesses like (9.34) and (9.51), whichcontain a positive integerκ . The parameterκ can be also regarded as a multiple-solution-control parameter,too, since multiple eigenfunctions and eigenvalues can be obtained simply by usingdifferent values ofκ .

The power polynomial ofz is used as base function to express the eigenfunctionsof the last two 4th-order eigenvalue equations, together with the simple auxiliarylinear operator

L u= u′′′′.

Example 9.3.4 has only finite number of eigenfunctions, since it is originally a non-linear boundary-value problem with multiple boundary conditions. Note note themultiple solutions of Example 9.3.4 are gained by means of the multiple initialguessλ0, governed by a nonlinear algebraic equation (9.61). This shows anotherway to find out multiple solutions of nonlinear BVPs. The power polynomial is alsoused in Example 9.3.5, mainly because we are only interestedin the most important


eigenfunction corresponding to the eigenvalue whose Im(λ ) is the maximum. Suchkind of unique eigenfunction can be gained easily by means ofthe polynomial asthe base function, as shown in Example 9.3.5.

All of these examples verify the validity and generality of theBVPh1.0 for com-plicated, highly nonlinear BVPs with singularity, and/or multipoint boundary con-ditions, and /or complex coefficients.

Finally, although theBVPh1.0 is developed for the general eigenvalue equationF [z,u,λ ] = 0 in a finite intervalz∈ [0,a], it doesnotmean thatall eigenvalue equa-tions in such a form can be solved by it. As mentioned in Chapter 7, our aim isto develop a Mathematica package valid for as many nonlinearBVPs as possible.Certainly, further modifications (see the Problems of this chapter) and more appli-cations are needed in future. Even so, the Mathematica packageBVPh(version 1.0)provides us an useful and alternative tool to investigate many nonlinear eigenvalueproblems in science and engineering.






( * Define the governing equation * )mu[z_] := 1;f[z_,u_,lambda_] := mu[z] * D[u,z,2]

+ D[mu[z],z] * D[u,z]+lambda * Sin[u];

( * Define Boundary conditions * )zR = Pi;OrderEQ = 2;BC[0,z_,u_,lambda_] := Limit[u - gamma, z -> 0];BC[1,z_,u_,lambda_] := Limit[D[u,z], z -> 0];BC[2,z_,u_,lambda_] := D[u,z] /. z -> Pi ;

( * Define initial guess * )u[0] = sigma - (sigma - gamma) * Cos[kappa * z];kappa = 1;sigma = Pi;gamma = 1;


D[u[k],z] /. z->0//N];





];temp[1]//Expand];



( * Use 3rd-order iteration approach * )iter[1,6,3];







( * Define the governing equation * )f[z_,u_,lambda_] := D[u,z,2]

+ (K-1) * D[u,z]/z + lambda * Exp[u] ;K = 1;

( * Define Boundary conditions * )zR = 1;OrderEQ = 2;BC[0,z_,u_,lambda_] := Limit[D[u,z] - A, z -> 0];BC[1,z_,u_,lambda_] := Limit[u, z -> 0];BC[2,z_,u_,lambda_] := u /. z -> 1 ;A = 1;

( * Define initial guess * )u[0] = A/2 * (1 + Cos[Pi * z]);


D[u[k],z] /. z->0//N];





];temp[1]//Expand];



( * Use 3rd-order iteration approach * )iter[1,6,3];







( * Define the governing equation * )L0[z_, u_]:= Sqrt[1+zˆ2] * D[u,z,2]

+ Cos[Pi * z] * D[u,z]/z;F[z_,u_] := Exp[u]/(1+zˆ2) + (1+z) * Sin[u];g[z_] := Sin[zˆ2 + Exp[-z]];f[z_,u_,lambda_]:= L0[z,u] + lambda * F[z,u] - g[z];

( * Define Boundary conditions * )zR = Pi;OrderEQ = 2;BC[0,z_,u_,lambda_] := Limit[u - A, z->0] ;BC[1,z_,u_,lambda_] := Limit[D[u,z], z->0 ];BC[2,z_,u_,lambda_] := u-D[u,z] - 3/5 /. z->zR;A = 1/2;

( * Define initial guess * )u[0] = (5 * A+3)/10 + (5 * A-3)/10 * Cos[kappa * z];kappa = 1;


u[k] /. z -> zR //N ];





];temp[1]//Expand];



( * Gain approximations by 3rd-order iteration * )iter[1,6,3];







( * Define the governing equation * )f[z_,u_,lambda_] := lambda * D[u,z,4]-beta * z

-beta * z* uˆ2 * lambdaˆ2;beta = 10 ;

( * Define Boundary conditions * )zR = 1;OrderEQ = 4;BC[0,z_,u_,lambda_] := u - 1 /. z -> 1 ;BC[1,z_,u_,lambda_] := Limit[u, z -> 0 ];BC[2,z_,u_,lambda_] := D[u,z] /. z -> 1 ;BC[3,z_,u_,lambda_] := D[u,z,2] /. z -> 1;BC[4,z_,u_,lambda_] := Module[temp,

temp[1] = D[u,z,2] /. z -> 0;temp[2] = D[u,z,2] /. z -> alpha;temp[1]-temp[2]//Expand];

alpha = 1/5;

( * Define initial guess * )u[0] = sigma/(2 * alpha-3) * ((6 * alpha-8) * z

+ 6* (1-alpha) * zˆ2+2 * alpha * zˆ3-zˆ4);sigma = 1;


D[u[k],z] /. z->0//N];

( * Define the auxiliary linear operator * )omega[1] = Pi/zR;omega[2] = Pi/zR;L[f_] := Module[temp,numA,numB,i,If[TypeL == 1,





];temp[1]//Expand];


( * Gain HAM approx. by 3rd-order iteration * )iter[1,6,3];






( * Define the physical and control parameters * )TypeEQ = 2;TypeL = 1;TypeBase = 2;ApproxQ = 0;ErrReq = 10ˆ(-20);Nupdate = 10000;NtermMax = 90;ComplexQ = 1;

( * Define the governing equation * )L0[u_,z_] := D[u,z,2]-alphaˆ2 * u;L02[u_,z_] := L0[L0[u,z],z];U0 = 1 - zˆ2;U02 = D[U0,z,2];f[z_,u_,lambda_] := L02[u,z]

- I * alpha * R* (U0 * L0[u,z]-U02 * u)+ I * lambda * R* L0[u,z];

alpha = 1 ;R = 100 ;

( * Define Boundary conditions * )zR = 1;OrderEQ = 4;BC[0,z_,u_,lambda_] := Limit[u - 1 , z->0 ];BC[1,z_,u_,lambda_] := Limit[D[u,z], z->0 ];BC[2,z_,u_,lambda_] := Limit[D[u,z,3], z->0 ];BC[3,z_,u_,lambda_] := u /. z->zR;BC[4,z_,u_,lambda_] := D[u,z] /. z->zR;

( * Define initial guess * )U[0] = u[0];u[0] = ( 1 - zˆ2 )ˆ2;

( * Define the auxiliary linear operator * )L[f_] := D[f,z,4];


D[u[k],z,2] /. z->0//N];


( * Set convergence-control parameter * )c0 = (-1+I)/2;


( * Gain HAM approx. by 3rd-order iteration * )iter[1,31,3];




Problems

9.1. Eigenvalue problems with nonlinear boundary conditionsDevelop a HAM-based analytic approach for thenth-order nonlinear eigenvalueequation

F [z,u,λ ] = 0, 0≤ z≤ a,

subject to then nonlinear multipoint boundary conditions

Bk[z,u,λ ] = γk,

whereBk is a nonlinear operator andγk is a constant. Assume that the above equa-tion has at least one smooth solution. Modify the Mathematica packageBVPh(ver-sion 1.0) given in Chapter 7 for this kind of problems in general.

9.2. Coupled nonlinear eigenvalue problemsDevelop a HAM-based analytic approach forn coupled nonlinear eigenvalue equa-tions in a finite intervalz∈ [0,a]:

Fk[z,u,λ1,λ2, · · · ,λn] = 0, 1≤ k≤ n,

subject to some linear/nonlinear multipoint boundary conditions, wheren≥ 2. As-sume that the above equation has at least one smooth solution. Give a Mathematicapackage for this kind of problems in general.

9.3. Eigenvalue problems in an infinite intervalDevelop a HAM-based analytic approach for thenth-order nonlinear eigenvalueequation in an infinite interval

F [z,u,λ ] = 0, 0≤ z<+∞,

subject to then multipoint nonlinear boundary conditions

Bk[z,u,λ ] = γk,

whereBk is a nonlinear operator andγk is a constant. Assume that the above equa-tion has at least one smooth solution. Give a Mathematica package for this kind ofproblems in general.

9.4. Coupled nonlinear eigenvalue problems in an infinite intervalDevelop a HAM-based analytic approach forn coupled nonlinear eigenvalue equa-tions in an infinite intervalz∈ [0,+∞):

Fk[z,u,λ1,λ2, · · · ,λn] = 0, 1≤ k≤ n,

subject to some linear/nonlinear multipoint boundary conditions, wheren≥ 2. As-sume that the above equations have at least one smooth solution. Give a Mathemat-ica package for this kind of problems in general.

References 367

References

1. Abbasbandy, S., Shirzadi, A.: A new application of the homotopy analysis method: Solvingthe Sturm – Liouville problems. Commun. Nonlinear Sci. Numer. Simulat.16, 112 – 126(2011)


3. Graef, J.R., Qian. C. Yang, B.: A three point boundary value problem for nonlinear forth orderdifferential equations. J. Mathematical Analysis and Applications.287, 217 – 233 (2003)

4. Graef, J.R., Qian. C. Yang, B.: Multiple positive solutions of a boundary value prolem forordinary differential equations. Electronic J. of Qualitative Theory of Differential Equations.11, 1 – 13 (2004)

5. Boley, A.B.: On the accuracy of the Bernoulli-Euler theory for beams of variable section. J.Appl. Mech. ASME30, 373 – 378 (1963)

6. Boyd, J.P.: An analytical and numerical Study of the two-dimensional Bratu equation. Journalof Scientific Computing.1, 183-206 (1986)

7. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. DOVER Publications, Inc. New York(2000)

8. Chang, D., Popplewell, N.: A non-uniform, axially loadedEuler-Bernoulli beam having com-plex ends. Q.J. Mech. Appl. Math.49, 353 – 371 (1996)

9. Jacobsen, J., Schmitt, K.: The Liouville-Bratu-Gelfandproblem for radial operators. Journalof Differential Equations.184, 283-298 (2002)

10. Katsikadelis, J.T., Tsiatas, G.C.: Non-linear dynamicanalysis of beams with variable stiff-ness. J. Sound and Vibration270, 847-863 (2004)

11. Lee, B.K., Wilson, J.F., Oh, S.J.: Elastica of cantilevered beams with variable cross sections.Int. J. Non-linear Mech.28, 579 – 589 (1993)

















27. Lin, C.C.: The Theory of Hydrodynamics Stability. Cambridge University Press. Cambridge(1955).

28. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall/CRC Press, BocaRaton (2003)

29. McGough, J.S.: Numerical continuation and the Gelfand problem. Applied Mathematics andComputation.89, 225-239 (1998)

30. Orszag, S.A.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech.50,689 – 703 (1971)

31. Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge Uni-versity Press. Cambridge (2003)



Chapter 10A boundary-layer flow with an infinite numberof solutions

Abstract In this chapter, the Mathematica packageBVPh(version 1.0) based on thehomotopy analysis method (HAM) is used to gain exponentially and algebraicallydecaying solutions of a nonlinear boundary-value equationin an infinite interval.Especially, an infinite number of algebraically decaying solutions were found forthe first time by means of the HAM, which illustrate the originality and validity ofthe HAM for nonlinear boundary-value problems.

10.1 Introduction

In this chapter, we illustrate the validity of the Mathematica packageBVPh(version1.0) for nonlinear boundary-value problems in an infinite interval, governed by anth-order nonlinear ordinary differential equations (ODEs)

F [z,u] = 0, 0≤ z<+∞, (10.1)

subject to some linear boundary conditions, whereF denotes a nonlinear differen-tial operator,u(z) is a smooth solution, respectively. Assume thatu(z) decays eitherexponentially or algebraically asz→+∞.

In Chapter 8 and Chapter 9, we illustrate that theBVPh 1.0 provides us anuseful tool to gain multiple solutions ofnth-order highly nonlinear boundary-value/eigenvalue problems with singularity and multipoint boundary conditions ina finite interval z∈ [0,a]. In this chapter, we further illustrate the validity of theBVPh1.0 for such kind of nonlinear ODEs in aninfinite interval.

In 2005, Liao [11] successfully applied the the homotopy analysis method(HAM) [2–15, 18, 21] to solve the nonlinear boundary-value equation in an infiniteinterval

F ′′′+12

F F ′′−βF ′2 = 0, F(0) = 0, F ′(0) = 1, F ′(+∞) = 0,

369

370 10 A boundary-layer flow with an infinite number of solutions

where−1< β < +∞ is a constant. Introducing an unknown quantityδ = F(+∞),Liao [11] found one new branch of exponentially decaying solutions whenβ > 1,which had beenneverreported by other analytic methods and even neglected bynumerical methods, mainly because the difference between the values ofF ′′(0) ofthe two branches of the exponentially decaying solutions isso small that it is hard todistinguish them. Besides, by means of the HAM, Liao and Magyari [17] found in2006 a new branch of algebraically decaying solutions of a kind of boundary-layerflow. Indeed,a truly new method always gives something new and/or different. Allof these illustrate the originality and validity of the HAM.In addition, they alsoillustrate that the HAM can be used to solve nonlinear boundary-value problemswith either exponentially or algebraically decaying solutions in an infinite interval.

Without loss of generality, let us consider here a two-dimensional boundary-layerviscous flow in the regionx > 0 andy> 0, where(x,y) denotes a Cartesian coor-dinate system and the flow results solely from the movement ofan impermeableflat plate aty = 0 in its plane. LetUw(x) = a(x+b)κ denote the speed of the flatplate, wherea> 0,b> 0 are given constants. Assume that the boundary-layer equa-tions are appropriate so that such kind of flow is described bythe partial differentialequations (PDEs)

u∂u∂x

+ v∂u∂y

= ν∂ 2u∂y2 ,

∂u∂x

+∂v∂y

= 0,


u= a(x+b)κ , v= 0, at y= 0,

andu= 0, aty→+∞,

whereν is the kinematic viscosity andu,v are the velocity components in the direc-tions of increasingx,y, respectively.

Let ψ denote the stream function. Using the similarity transformation

ψ =√

a ν(x+b)(κ+1)/2 f (η), η =

√

aν

y (x+b)(κ−1)/2, (10.2)

the original PDEs become the following nonlinear ODE

f ′′′+(1+κ)

2f f ′′−κ f ′2 = 0, f (0) = 0, f ′(0) = 1, f ′(+∞) = 0. (10.3)

For details, please refer to Banks [1]. The above equation has the close-form solution

f (η) = 1−exp(−η), whenκ = 1, (10.4)

10.2 Exponentially decaying solutions 371

and

f (η) =√

6 tanh

(

η√6

)

, whenκ =−1/3, (10.5)

respectively. Besides, there exist solutions when−1/2< κ <+∞.The nonlinear boundary-value equation (10.3) in the infinite intervalz∈ [0,+∞)

has two types of solutions: one decays exponentially at infinity [16], the other decaysalgebraically [17]. These two types of solutions of (10.3) can be obtained by meansof theBVPh1.0, as shown below.

10.2 Exponentially decaying solutions

Physically, most of boundary-layer flows exponentially tend to a uniform flow atinfinity. Mathematically, this is confirmed by the close-form solutions (10.4) and(10.5). Such kinds of exponentially decaying solutions canbe gained by meansof the HAM, as shown by Liao and Pop [16]. Here, it is solved by means of theBVPh1.0, which is given in the Appendix 7.1 and free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.The corresponding input data file is given in the Appendix 10.1 and free availableat the above website.

For the sake of simplicity, define a nonlinear operator

N [φ(η ;q)] = φ ′′′+(1+κ)

2φ φ ′′−κ (φ ′)2. (10.6)

In the frame of the HAM, we first construct the so-called zeroth-order deformationequation

(1−q)L [φ(η ;q)− f0(η)] = q c0 N [φ(η ;q)] , (10.7)


φ(η ;q) = 0, φ ′(η ;q) = 0, φ ′(+∞;q) = 0, (10.8)

where the prime denotes the differentiation with respect toη , f0(η) is an initialguess,L is an auxiliary linear operator, andc0 is the so-called convergence-controlparameter, respectively. The above equation defines a kind of continuous variationφ(η ;q) from the initial guessf0(η) (at q= 0) to the solutionf (η) of the originalequation (10.3) (atq= 1). The homotopy-Maclaurin series ofφ(η ;q) reads

φ(η ;q) = f0(η)++∞

∑n=1

fn(η) qn,

where

fn(η) = Dn [φ(η ;q)] =1n!

∂ nφ(η ;q)∂qn

∣

∣

∣

∣

q=0(10.9)

andDn is called thenth-order homotopy-derivative operator.


If the initial guessf0(η), the auxiliary linear operatorL , and especially theconvergence-control parameterc0 are properly chosen so that the above homotopy-Maclaurin series absolutely converges atq= 1, we have the homotopy-series solu-tion

f (η) = f0(η)++∞

∑n=1

fn(η), (10.10)

where fn(η) is governed by thenth-order deformation equation

L [ fn(η)− χn un−1(η)] = c0 δn−1(η), (10.11)


fn(0) = 0, f ′n(0) = 0, f ′(+∞) = 0, (10.12)

where

χm =

0, m≤ 1,1, m> 1

and

δm(η) = DmN [φ(η ;q)]= f ′′′m +(1+κ)

2

m

∑i=0

fm−i f ′′i −κm

∑i=0

f ′m−i f ′i (10.13)

is gained by Theorem 4.1. Equation (10.11) is given by Theorem 4.15. For details,please refer to Chapter 4.

The exponentially decaying solution of (10.3) can be expressed in the form

f (η) =+∞

∑n=1

An(η)exp(−nη), (10.14)

whereAn(η) is a polynomial to be determined. It provides us the so-called solution-expression off (η). To gain such kind of exponentially decaying solution by meansof theBVPh1.0, we choose the initial guess

f0(η) = σ +(1−2σ) e−η − (1−σ)e−2η (10.15)

and the auxiliary linear operator

L f = f ′′′− f ′, (10.16)

respectively, whereσ = f (+∞) is a unknown parameter. Note that the initial guess(10.15) satisfies all boundary-conditions(10.3) and besides f0(+∞) =σ . Obviously,a better value ofσ corresponds to a better initial guess. So, the unknownσ providesus one additional degree of freedom in the initial guessf0(η). Note also that theauxiliary linear operator (10.16) has the property

L [B0+B1 exp(−η)+B2 exp(+η)] = 0. (10.17)

10.2 Exponentially decaying solutions 373

In addition, for any a given functionf ∗(η) that decays exponentially at infinity, theunknown integral coefficientsB0,B1,B2 of the following function

f ∗(η)+B0+B1 exp(−η)+B2 exp(+η)

can be uniformly determined by the three boundary conditions (10.3). In otherwords, the auxiliary linear operatorL is chosen in such a way that the solutionu(z) is expressed in the form of (10.14) for the exponentially decaying solution, andbesides all integral coefficients are uniformly determined. For the choice of the ini-tial guess and the auxiliary linear operatorL in general, please refer to§ 7.1.3 and§ 7.1.4.

Letf ∗n (η) = χn un−1(η)+ c0L

−1 [δn−1(η)]

denote a special solution of (10.11). Its general solution reads

fn(η) = χn un−1(η)+ c0L−1 [δn−1(η)]

+ B0+B1 exp(−η)+B2 exp(+η), (10.18)

where the integral coefficientsB0,B1 and B2 are uniformly determined by threeboundary conditions (10.12). For details, please refer to Liao and Pop [16].

Fig. 10.1 The exponentiallydecaying solutionsf ′(η)of (10.3). Filled circle: the10th-order homotopy approx-imation whenκ = −1/3 bymeans ofc0 = −5/4 andσ = 3; Solid line: the exactsolution whenκ = −1/3;Open circle: the 10th-orderhomotopy approximationwhenκ = −1/4 by meansof c0 = −5/4 andσ = 11/4;Dashed line: the 30th-orderhomotopy approximationwhenκ =−1/4. η

f’

0 2 4 6 8

0.2

0.4

0.6

0.8

1

First, let us consider the case ofκ = −1/3. The corresponding homotopy-approximations are obtained by means of theBVPh 1.0, which contain two un-known parameters:σ in the initial guess (10.15) and the convergence-control pa-rameterc0. Using the Mathematica commandMinimize , it is found that, at the5th-order homotopy approximation, we have the minimum 6.8×10−6 of the aver-aged squared residual of (10.3) over the intervalη ∈ [0,10], corresponding to theoptimal convergence-control parameterc∗0 = −1.2411 and the optimal parameterσ∗ = 3.0435. Indeed, by means ofc0 = −5/4 andσ = 3, the averaged squaredresidual of (10.3) over the intervalη ∈ [0,10] decreases quickly, as shown in Ta-


Table 10.1 The averaged squared residual of (10.3) at themth-order (exponentially decaying)homotopy approximation over the intervalη ∈ [0,10] whenκ =−1/3 by means ofc0 =−5/4 andσ = 3 .

Order of approximation,m f′′(0) Squared residualEm

2 7.0×10−2 5.2×10−3

4 -2.1×10−2 3.7×10−5

6 -1.4×10−2 3.8×10−6

8 -8.9×10−3 1.8×10−6

10 -4.8×10−3 6.8×10−7

15 -6.6×10−4 3.9×10−8

20 8.2×10−5 1.4×10−9

ble 10.1. Note that,f ′′(0) quickly converges to the exact valuef ′′(0) = 0, and the10th-order homotopy-approximation off (η) agrees well with the exact solution, asshown in Fig. 10.1.

Note that the unknown parameterσ in the initial guess (10.15) is used here tosearch for the optimal initial guess, which can be regarded as a kind of convergence-control parameter. So, we have here two convergence-control parameters:c0 in thezeroth-order deformation equation (10.7) andσ in the initial guess (10.15). Notethat, in Chapter 8 and Chapter 9, the unknown parameterσ in initial guesses areused as the so-called multiple-solution-control parameter to find out multiple solu-tions of nonlinear ODEs in a finite intervalz∈ [0,a]. So, an unknown parameter ininitial guesses can be used either asthe multiple-solution-control parameterto findout multiple solutions, or asthe convergence-control parameterto control the con-vergence of homotopy-series solutions. This is mainly because, based on the HAM,theBVPh1.0 provides us large freedom to choose initial guesses.

Similarly, we can gain accurate homotopy-approximationf (η) of (10.3) for−1/2 < κ < +∞ by means of theBVPh 1.0. For example, whenκ = −1/4, atthe 5th-order homotopy approximation, we have the minimum 6.5× 10−6 of theaveraged squared residual over the intervalη ∈ [0,10], corresponding to the twooptimal convergence-control parametersc∗0 =−1.2133 andσ∗ = 2.7369. By meansof c0 =−5/4 andσ = 11/4, we indeed gain the accurate homotopy-series solutionwhosef ′′(0) converges to -0.1620, as shown in Table 10.2. Besides, the 10th-orderapproximation of f ′(η) is rather accurate, as shown in Fig. 10.1. Therefore, bymeans of theBVPh 1.0, we successfully gain the exponential decaying solutionsof (10.3). For simplicity, we denote this kind of exponentially decaying solution byfexp(η) in the following section.

Note that, using the transformation

f (η) =√

2κ +1

F(z), z=

√

κ +12

η ,

Equation (10.3) becomes

10.3 Algebraically decaying solutions 375

Table 10.2 The averaged squared residual of (10.3) at themth-order (exponentially decaying)homotopy approximation over the intervalη ∈ [0,10] whenκ =−1/4 by means ofc0 =−5/4 andσ = 11/4.

Order of approximation,m f′′(0) Squared residualEm

5 -0.1692 2.9×10−6

10 -0.1639 1.4×10−7

15 -0.1623 4.5×10−9

20 -0.1620 8.7×10−11

25 -0.1620 7.8×10−12

30 -0.1620 1.1×10−12

F ′′′+F F ′′− εF ′2 = 0, F(0) = 0, F ′(0) = 1, F ′(+∞) = 0,

whereε = 2 κ/(1+κ), and the prime denotes the differentiation with respect toz.By means of the HAM, Liao and Pop [16] obtained a 3rd-order approximation

F ′′(0) =−(

145293+231153ε+94999ε2+12395ε3)

15120(3+ ε)5/2, (10.19)

which agrees well with numerical results in the whole interval 0≤ ε < +∞. Fordetails, please refer to Liao and Pop [16]. Thus, for the exponentially decaying so-lutions fexp(η) of (10.3), we have the 3rd-order homotopy-approximation

f ′′exp(0) =

√

κ +12

F ′′(0)

= −(

145293+898185κ+1740487κ2+1086755κ3)

15120√

2(3+5κ)5/2, (10.20)

which is valid in an infinite intervalκ ∈ (0,+∞).This example verifies the validity of theBVPh1.0 for nonlinear boundary-value

ODEs in an infinite interval, whose solutions decay exponentially at infinity.

10.3 Algebraically decaying solutions

As pointed out by Magyari, Pop and Keller [19], whenκ = −1/3, (10.3) has aninfinitenumber of solutions in the closed-form

f (η) = (36µ)1/3[

Bi′(t0)Ai′(t)−Ai′(t0)Bi′(t)Bi′(t0)Ai(t)−Ai′(t0)Bi(t)

]

, (10.21)

whereAi(t) andBi(t) are two kinds of Airy functions,


t0 =(√

6 µ)−2/3

, t = t0 (1+ µ η),

andµ = f ′′(0)≥ f ′′exp(0), (10.22)

respectively. Here,fexp(η) denotes the solution of (10.3) exponentially decaying atinfinity, mentioned in the above section. Sincef ′′exp(0) = 0 whenκ = −1/3, thenthe solution (10.21) satisfies (10.3) (whenκ = −1/3) for arbitrary values ofµ =f ′′(0) ≥ 0. It was proved [19] that the solution (10.21) is equal to theclose-formsolution (10.5) asµ = f ′′(0) tends to zero, but for any other values ofµ > 0, thesolution (10.21) decaysalgebraicallyat infinity. In other words, whenκ = −1/3,the exponentially decaying solution

fexp(η) =√

6tanh(η/√

6)

is the limit of the algebraically decaying solutions (10.21) as f ′′(0)→ f ′′exp(0) = 0.The BVPh 1.0 is also valid for such kind of algebraically decaying boundary-

layer flows, too. By means ofBVPh1.0 as a tool, it is found that (10.3) has indeedan infinite number of algebraically decaying solutions not only atκ =−1/3 but alsoin thewholeinterval−1/2< κ < 0, as shown below.

Let us first consider the asymptotic property of the algebraically decaying solu-tion f (η) at infinity. Write the asymptotic expression

f ′ ∼ ηb, i.e. f ∼ 1(b+1)

ηb+1, asη →+∞,

whereb is a constant to be determined. Substituting these asymptotic expressionsinto (10.3) and balancing the dominant terms, we have

b=2κ

1−κ, i.e. 1+b=

1+κ1−κ

= β .

Obviously, to satisfy the boundary conditionf ′(+∞) = 0, b must be negative, cor-responding to−1/2< κ < 0, which gives

13< β < 1.

Then, the algebraically decaying solution has the asymptotic property

f ∼ ηβ ,13< β < 1, asη →+∞.

To avoid the singularity of the above asymptotic expressionat η = 0, we use thetransformation

ξ = 1+α η , f (η) = α−1 g(ξ ),

whereα > 0 is a constant. Then, the original equation (10.3) becomes


α2 g′′′(ξ )+(

1+κ2

)

g g′′−κ(g′)2 = 0, g(1)= 0,g′(1) = 1,g′(+∞) = 0, (10.23)

where the prime denotes the differentiation with respect toξ . Since (10.3) has aninfinitenumber of solutions dependent uponµ = f ′′(0)> f ′′exp(0), we should add anadditional boundary conditionf ′′(0) = µ , i.e.

g′′(0) =µα

(10.24)

so as to distinguish these different algebraically decaying solutions. Considering theasymptotic property mentioned above, our aim is to gain the algebraically decayingsolutions in the form

g(ξ ) = a0,0 ξ β ++∞

∑m=0

+∞

∑n=0

am,n ξ−(1−β )m−n, (10.25)

which provides us the so-calledsolution-expressionof the algebraically decayingsolutiong(ξ ).

Such kind of algebraically decaying solutions of the nonlinear boundary-valueODE (10.23), with the additional boundary conditiong′′(0) = µ/α, can be gainedby means of theBVPh 1.0, as shown below. The corresponding input data file isgiven in the Appendix 10.2 and free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

Define a nonlinear operator

ˇN [u] = α2 u′′′+

(

1+κ2

)

u u′′−κ(u′)2,

where the prime denotes the differentiation with respect toξ . Let q∈ [0,1] denotethe embedding-parameter. In the frame of the HAM, we should first construct sucha continuous variationφ (ξ ;q) (or deformation) thatφ(ξ ;q) = g0(ξ ) at q= 0 andφ (ξ ;q) = g(ξ ) atq= 1, respectively. Such a kind of continuous variation is definedby the zeroth-order deformation equation

(1−q)L[

φ(ξ ;q)−g0(ξ )]

= c0 q H(ξ ) ˇN[

φ (ξ ;q)]

, (10.26)


φ (ξ ;q) = 0, φ ′(ξ ;q) = 1, φ ′′(ξ ;q) = µ/α, atξ = 1, (10.27)

andφ ′(ξ ;q) = 0, asξ →+∞, (10.28)

whereL is the auxiliary linear operator,g0(ξ ) is the initial guess ofg(ξ ), c0 is theconvergence-control parameter,H(ξ ) is an auxiliary function, the prime denotes thedifferentiation with respect toξ , respectively. Assuming that the initial guessg0(ξ ),the auxiliary functionH(ξ ), and especially the convergence-control parameterc0

are so properly chosen that the homotopy-Maclaurin series


φ (ξ ;q) = g0(ξ )++∞

∑m=1

gm(ξ ) qm,

absolutely converges atq= 1, we have the homotopy-series solution

g(ξ ) = g0(ξ )++∞

∑m=1

gm(ξ ), (10.29)

wheregm(ξ ) is governed by themth-order deformation equation

L [gm(ξ )− χm gm−1(ξ )] = c0 H(ξ ) δm−1(ξ ) (10.30)


gm(1) = 0, g′m(1) = 0, g′′m(1) = 0, g′m(+∞) = 0, (10.31)

where

δn(ξ ) = α2 g′′′m +(1+κ)

2

m

∑i=0

gm−i g′′i −κm

∑i=0

g′m−i g′i (10.32)

is gained by Theorem 4.1. Equation (10.30) is given by Theorem 4.15. For details,please refer to Chapter 4.

The solution-expression (10.25 ) plays an important role inchoosing the ini-tial guessg0(ξ ) and the auxiliary linear operator (10.34). To satisfy the solution-expression (10.25), we choose the initial guess in the form

g0(ξ ) = a0ξ β +a1ξ β−1+a2 ξ 2(β−1),

where the unknown coefficientsa0,a1 anda2 are determined by the two boundaryconditionsg(1) = 0,g′(1) = 1 and the additional boundary conditiong′′(1) = µ/α.Thus, we have

g0(ξ ) =(4−3β +α−1µ)

2−βξ β − (3−3β +α−1µ)

1−βξ β−1

+(2−2β +α−1µ)(1−β )(2−β )

ξ 2β−2, (10.33)

which automatically satisfies the boundary conditiong′(+∞) = 0, and besides de-caysalgebraicallyat infinity.

To satisfy the solution-expression (10.25 ), we choose suchan auxiliary linearoperatorL in the form

L u= u′′′+B1(ξ )u′′+B2(ξ )u′+B3(ξ )u

that the 3rd-order linear differential equation

L u= 0


Fig. 10.2 The algebraicallydecaying solutionsf ′(η) of(10.3) whenκ = −1/3 andµ = 1. Solid line: the exact so-lution (10.21); Filled circles:the 20th-order homotopy-approximation by meansof c0 = −8,α = 2/5 andH(ξ ) = ξ .

η

f’

0 5 10 15 200

0.5

1

1.5

2

µ = 1, κ =-1/3

has the general solutions algebraically decaying at infinity, i.e.

L

[

C1 ξ β +C2 ξ β−1+C3 ξ 2(β−1)]

= 0.

Substitutingu = ξ β , u = ξ β−1 and u = ξ 2(β−1) into L u = 0 gives three linearalgebraic equations ofB1(ξ ), B2(ξ ) andB3(ξ ), which uniformly determine them.In this way, we gain the auxiliary linear operator

L u= ξ 3 u′′′−2(2β −3)ξ 2u′′+(β −1)(5β −6)ξ u′−2β (β −1)2u. (10.34)

Letg∗m(ξ ) = χm gm−1(ξ )+ c0L

−1[

H(ξ )δm−1(ξ )]

denote a special solution of (10.30), whereL −1 is the inverse operator ofL . Then,its general solution reads

gm(ξ ) = g∗m(ξ )+Cm,1 ξ β +Cm,2 ξ β−1+Cm,3 ξ 2(β−1),

where the integral coefficientsCm,1,Cm,2 and Cm,3 are determined by the threeboundary conditions (10.31) atξ = 0. Note that the boundary condition at infinity,i.e. g′m(+∞) = 0, is automaticallysatisfied. Thus, by means of proper base func-tions, it is very easy to satisfy the boundary conditions at infinity. This is mainlybecause the computer algebra system like Mathematica provides us the ability to“compute with functions instead of numbers” [20].

Besides, to satisfy the so-called solution-expression (10.25), the termH(ξ )δm−1(ξ )on the right-hand side of (10.30) should not contain the following terms

ξ β , ξ β−1, ξ 2(β−1).

For this reason, we must choose the auxiliary functionH(ξ ) = ξ . For details, pleaserefer to Liao and Magyari [17].


Table 10.3 The averaged squared residual of (10.23) at themth-order (algebraically decaying)homotopy-approximation over the intervalξ ∈ [1,20] when κ = −1/3 andµ = 1 by means ofc0 =−8,α = 2/5 andH(ξ ) = ξ .

Order of approximation,m α2 g′′′(1) Squared residualEm

2 -0.4546 9.8×10−3

4 -0.3428 1.6×10−3

6 -0.3341 1.7×10−4

8 -0.3334 2.8×10−5

10 -0.3333 1.1×10−5

20 -0.3333 6.1×10−7

30 -0.3333 6.2×10−8

50 -0.3333 1.4×10−9

Table 10.4 The averaged squared residual of (10.23) at themth-order (algebraically decaying)homotopy approximation over the intervalξ ∈ [1,20] whenκ = −1/4 andµ = 1 by means ofc0 =−7,α = 2/5 andH(ξ ) = ξ .

Order of approximation,m α2 g′′′(1) Squared residualEm

2 -0.2699 2.6×10−2

4 -0.2503 5.1×10−3

6 -0.2500 9.0×10−4

8 -0.2500 2.4×10−4

10 -0.2500 1.0×10−4

20 -0.2500 9.9×10−6

30 -0.2500 1.1×10−6

40 -0.2500 1.6×10−7

50 -0.2500 3.7×10−8

To show the validity of theBVPh1.0 for algebraically decaying solutions of non-linear ODEs in an infinite interval, let us first consider the caseκ = −1/3, whoseclose-form solution (10.21) is known. According to (10.23), we have the exact re-lationshipα2g′′′(1) = κ , which is used to check the accuracy of the homotopy-approximation. Note that both ofc0 and α are unknown, whose optimal valuesare determined by the minimum of the squared residual of the governing equa-tion (10.23). For example, whenµ = f ′′(0) = 1, the averaged squared residual of(10.23) at the 5th-order homotopy-approximation over the intervalξ ∈ [1,20] hasthe minimum 4.2× 10−4 by means of the optimal convergence-control parameterc∗0 =−8.1789 and the optimal parameterα∗ = 0.4185. Indeed, by means ofc0 =−8andα = 2/5, the corresponding homotopy-approximations converge quickly to theexact solution (10.21), as shown in Table 10.3 and Fig. 10.2.This illustrates the va-lidity of the BVPh1.0 for algebraically decaying solutions of nonlinear boundary-value ODEs in an infinite intervalη ∈ [0,+∞).

Note that the unknown parameterα in (10.33) provides us one additional degreeof freedom in initial guess, which supplies a convenient wayto find out the optimalinitial guess by means of the minimum of the squared residualof the governing


Fig. 10.3 The algebraicallydecaying solutionsf ′(η)of (10.3) whenκ = −1/4.Line: 50th-order homotopy-approximation; Open cir-cles: 20th-order homotopy-approximations; Filled cir-cles: 30th-order homotopy-approximations; Solid line:µ = −0.15; Dashed-line:µ = 0; Dash-dotted line:µ = 1; Dash-dot-dotted line:µ = 3.

η

f’

0 5 10 15 20-0.5

0

0.5

1

1.5

2

2.5

3

3.5

κ =-1/4

Fig. 10.4 f ′′(0) versusκ forthe exponentially and alge-braically decaying solutionsof (10.3). Solid line: expo-nentially decaying solutions;Dashed area: algebraicallydecaying solutions.

-0.4 -0.2 0 0.2 0.4

-1

-0.5

0

0.5

1

κ

f ’’ (0)

algebraicallydecayingsolutions

exponentiallydecayingsolution

equation. Therefore, we can also regardα as a convergence-control parameter, too.Obviously, the use of the two convergence-control parameters c0 andα improvesour ability to guarantee the convergence of homotopy-series solution in the frameof the HAM, as shown above.

Similarly, theBVPh 1.0 can be used to gain the algebraically decaying solu-tions for other values ofκ in the whole intervalκ ∈ (−1/2,0). For example, whenκ = −1/4 andµ = 1, it is found that the averaged squared residual of (10.23) atthe 5th-order homotopy-approximation over the intervalξ ∈ [1,20] has the min-imum 1.7× 10−3 by means of the two optimal convergence-control parameterc∗0 = −6.8702 andα∗ = 0.4133. Usingc0 = −7 andα = 2/5, the correspondinghomotopy-approximations converge quickly, as shown in Table 10.4 and Fig. 10.3.Similarly, whenκ = −1/4, we gain the algebraically decaying solutions for othervaluesµ = f ′′(0)> f ′′exp(0) =−0.1620, as shown in Fig. 10.3. It is found that, whenκ = −1/4, there exist an infinite number of algebraically decaying solutions f (η)of the boundary-layer equation (10.3) with the propertyf ′′(0)≥ f ′′exp(0)=−0.1620,where fexp(η) is the corresponding exponentially decaying solution.


The above conclusion has general meanings: as reported by Liao and Mag-yari [17], when−1/2 < κ < 0, there exist an infinite number of algebraicallydecaying solutionsf (η) of the boundary-layer equation (10.3) with the propertyf ′′(0) ≥ f ′′exp(0), where fexp(η) is the corresponding exponentially decaying solu-tion. In other words, not only atκ =−1/3 but in thewholeinterval−1/2< κ < 0,the exponentially decaying solutionfexp(η) of the boundary-layer equation (10.3)is the limiting case of an infinite number of algebraically decaying solutionsf (η),as shown in Fig. 10.4.

Note that these algebraically decaying solutions have close-form expression onlywhenκ = −1/3. It should be emphasized that, the new algebraically decaying so-lutions of the boundary-layer equation (10.3) in the whole interval−1/2< κ < 0(exceptκ = −1/3) are found [17], for the first time, by means of the HAM. Thisshows the great potential and general validity of the HAM.

All of above results are obtained by means of theBVPh1.0, which is free avail-able athttp://numericaltank.sjtu.edu.cn/BVPh.htm. In addition, the correspondinginput data is given in the Appendix 10.2 and free available atthe same website.


In this chapter, theBVPh1.0 is applied to gain the exponentially and algebraicallydecaying solutions of boundary-layer flows, governed by a nonlinear ODE in aninfinite interval. Thus, a lots of boundary-layer flows can besolved by means of theBVPh1.0 in a similar way.

Note that it isimpossiblefor numerical techniques to resolve the infinite intervalwith boundary conditions at infinity exactly. Many numerical packages likeBVP4regard the infinite interval as a kind of singularity and replace it by a finite onein practice. However, based on computer algebra system, theBVPh 1.0 can solvenonlinear ODEs in an infinite interval exactly. This is mainly because the computeralgebra system provides us the ability to “compute with functions instead of num-bers” [20], so that boundary conditions at infinity can be easily resolved by choosingproper base functions, as shown in this chapter.

Note that theBVPh1.0 can be used to gain both exponentially and algebraicallydecaying solutions of nonlinear boundary-value ODEs in an infinite interval. Thisis mainly because, based on the HAM, theBVPh 1.0 provides us extremely largefreedom and great flexibility to choose different types of auxiliary linear operatorsand different initial guesses. Besides, the convergence-controlparameterc0 providesus a convenient way to guarantee the convergence of homotopy-series solution.

In addition, the unknown parameterσ in (10.15) andα in (10.33) also provide usone additional degree of freedom in initial guess, which supplies a convenient wayto find out the optimal initial guess. Likec0, both ofσ andα can be regarded asa convergence-control parameter: combined withc0 together, each of them furtherimproves our ability to guarantee the convergence of homotopy-series solution inthe frame of the HAM, as illustrated in this chapter.


It should be emphasized that the infinite number of algebraically decaying so-lutions of (10.3) have close-form solution only whenκ = −1/3. It is by means ofthe HAM that new algebraically decaying solutions of (10.3)were found, for thefirst time, by Liao and Magyari [17] in the whole interval−1/2< κ < 0 (exceptκ =−1/3). In fact, by means of the HAM, new solutions of some other boundary-layer flows have been found by Liao [11], which had been never reported and evenneglected by numerical methods. Indeed,a truly new method always gives some-thing new and/or different. All of these show the originality of the HAM.

The BVPh 1.0 can be used as an analytic tool to solve many nonlinear ODEsin an infinite interval, especially those related to boundary-layer flows. It can beeven applied to solve some types of nonlinear PDEs in an infinite interval, such asthe non-similarity boundary-layer flow as shown in Chapter 11, and the unsteadyboundary-layer flow as shown in Chapter 12. However, these donot mean that theBVPh1.0 is valid forall boundary-value problems in an infinite interval. As men-tioned in Chapter 7, our aim is to develop a package for as manynonlinear boundary-value problems as possible. Thus, further modifications andmore applications areneeded in future.



Appendix 10.1 Input data ofBVPh for exponentially decayingsolution


( * Define the physical and control parameters * )TypeEQ = 1;ApproxQ = 0;ErrReq = 10ˆ(-10);zRintegral = 10;

( * Define the governing equation * )kappa = -1/4;f[z_,u_,lambda_] := D[u,z,3]

+(1+kappa)/2 * u* D[u,z,2]-kappa * D[u,z]ˆ2 ;

( * Define Boundary conditions * )zR = infinity;OrderEQ = 3;BC[1,z_,u_,lambda_] := Limit[u, z -> 0 ];BC[2,z_,u_,lambda_] := Limit[D[u,z] - 1, z -> 0 ];BC[3,z_,u_,lambda_] := Limit[D[u,z], z -> zR ];

( * Define initial guess * )temp[1] = (1-2 * sigma);temp[2] = (1-sigma);u[0] = sigma+temp[1] * Exp[-z]-temp[2] * Exp[-2 * z];


D[u[k],z,2] /. z->0//N];

( * Defines the auxiliary linear operator * )L[u_] := D[u,z,3] - D[u,z];


( * Set optimal c0 and sigma * )c0 =-5/4 ;sigma = 3;Print[" c0 = ",c0, " sigma = ",sigma];

( * Gain up to 10th-order approxiamtion * )BVPh[1,10];

The above input data of theBVPh (version 1.0) for the exponentially decayingsolution of (10.3) is free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

Appendix 10.2 Input data ofBVPh for algebraically decaying solution 385

Appendix 10.2 Input data ofBVPh for algebraically decayingsolution


( * Define the physical and control parameters * )TypeEQ = 1;ApproxQ = 0;ErrReq = 10ˆ(-10);zRintegral = 10;H[z_] := z;

( * Define the governing equation * )kappa = -1/3;alpha = 2/5;f[z_,u_,lambda_]:=alphaˆ2 * D[u,z,3]

+(1+kappa)/2 * u* D[u,z,2]-kappa * D[u,z]ˆ2;

( * Define Boundary conditions * )zL = 1;zR = infinity;OrderEQ = 3;BC[1,z_,u_,lambda_] := Limit[u, z -> 1 ];BC[2,z_,u_,lambda_] := Limit[D[u,z] - 1, z -> 1 ];BC[3,z_,u_,lambda_] := Limit[D[u,z,2] - mu/alpha,

z -> 1 ];

( * Define initial guess * )mu = 1;beta = (1+kappa)/(1-kappa);u[0] = Module[temp,temp[1] = (4-3 * beta+mu/alpha)/(2-beta);temp[2] = (3-3 * beta+mu/alpha)/(1-beta);temp[3] = (2-2 * beta+mu/alpha)/(1-beta)/(2-beta);temp[1] * zˆbeta - temp[2] * zˆ(beta-1)

+ temp[3] * zˆ(2 * beta-2)];


alphaˆ2 * D[u[k],z,3] /. z->1//N];

( * Define the auxiliary linear operator * )(L[u_] := Module[temp,temp[1] = 2 * (2 * beta-3);temp[2] = (beta-1) * (5 * beta-6);temp[3] = 2 * beta * (beta-1)ˆ2;zˆ3 * D[u,z,3] - temp[1] * zˆ2 * D[u,z,2]

+ temp[2] * z* D[u,z] - temp[3] * u];



( * Exact solution when kappa = -1/3 * )Uexact = Module[temp,t,t0,Ai,Bi,Ait,Bit,Ai = AiryAi[t];Bi = AiryBi[t];Ait = D[Ai,t];Bit = D[Bi,t];Ait0 = Ait /. t -> t0;Bit0 = Bit /. t -> t0;temp[1] = Bit0 * Ait - Ait0 * Bit;temp[2] = Bit0 * Ai - Ait0 * Bi ;t0 = (Sqrt[6] * mu)ˆ(-2/3);t = t0 * (1 + mu* x);(36 * mu)ˆ(1/3) * temp[1]/temp[2]//Expand];Uzexact = D[Uexact,x];

( * Coordinate transform * )Wz[k_] := Uz[k] /. z -> 1 + alpha * x;

( * Set optimal c0 * )c0 = -8;Print[" c0 = ",c0];

( * Gain up to 10th-order approximation * )BVPh[1,30];

The above input data of theBVPh (version 1.0) for the algebraically decayingsolution of (10.3) is free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm.

Note that, for nonlinear boundary-value problems defined inan infinite interval,we must setzR = infinity in the input data ofBVPh(version 1.1).

References 387

References

1. Banks, W.H.H.: Similarity solutions of the boundary-layer equations for a stretching wall.Journal de Mecanique theorique et appliquee.2, 375-392 (1983)















16. Liao, S.J., Pop, I.: Explicit analytic solution for similarity boundary layer equations. Int. J.Heat and Mass Transfer.47, 75-85 (2004)



19. Magyari, E., Pop, I., Keller, B.: New analytical solutions of a well known boundary valueproblem in fluid mechanics. Fluid Dyn. Res.33, 313 – 317 (2003)



Chapter 11Non-similarity boundary-layer flows

Abstract In this chapter, we illustrate the validity of the Mathematica packageBVPh (version 1.0) for nonlinear partial differential equations (PDEs) related tonon-similarity boundary-layerflows. We show that, usingBVPh1.0, a non-similarityboundary-layer flow can be solved in a rather similar way to that for similarity onesgoverned by nonlinear ODEs. In other words, in the frame of the HAM, solvingnon-similarity boundary-layer flows is as easy as similarity ones. This shows thevalidity of the BVPh 1.0 for some nonlinear PDEs, especially for those related toboundary-layer flows.

11.1 Introduction

In the previous chapters of Part II, we illustrate that the Mathematica packageBVPh(version 1.0) provides us an analytic tool to solve boundary-value problems gov-erned by a nonlinear ordinary differential equation (ODE) either in a finite intervalz∈ [0,a] or in an infinite intervalz∈ [b,+∞), wherea> 0 andb≥ 0 are boundedconstants. In this and the next chapter, we illustrate that the BVPh (version 1.0)can be even applied to solve some nonlinear partial differential equations (PDEs),especially those related to boundary-layer flows.

Without loss of generality, let us consider here a nonlinearPDE arising from akind of non-similarity boundary-layer flow. Since Prandtl [39] proposed the revolu-tionary concept of boundary-layer flows of viscous fluid in 1904, the boundary-layertheory [4,11,37,43,44,49] has been developing greatly andapplied in nearly all re-gions of fluid mechanics [48]. When similarity solutions exist, boundary-layer flowsare governed by nonlinear ODEs. However, when there exist nosuch kind of sim-ilarity, one had to solve nonlinear PDEs, which are much moredifficult to solvethan ODEs. Mainly due to this reason, most of researchers in this field focused onsimilarity boundary-layerflows: thousands of articles related to similarity boundary-layer flows have been published, but in contrast to the large number of publications

389

390 11 Non-similarity boundary-layer flows

in similarity boundary-layer flows [4, 11, 37, 43, 44, 49], articles on non-similarityflows [5,6,8,9,38,47] are much less.

For example, let us consider here a non-similarity boundary-layer flow of New-tonian fluid over a stretching flat sheet [3,7]. Two forces in opposite directions withsame magnitude are added along the sheet. Thus, there is a rest point on the sheet,which is defined as the origin of the coordinate system. Thex andy axes are alongand perpendicular to the sheet, respectively. The fluid is atrest far from the sheet(i.e asy→ +∞). Due to the symmetry of flows, we can only consider the flows inthe upper quarter planex ≥ 0 andy ≥ 0. LetUw(x) denote the stretching velocityof the sheet,(u,v) the velocity components,ν the kinematic viscosity of the fluid,respectively. As mentioned by Prandtl [39], the velocity variation across the flowdirection is much larger than that in the flow direction, so that there exists a thinboundary-layer near the sheet. In the frame of the boundary-layer theory, this kindof viscous flow is governed by

∂u∂x

+∂v∂y

= 0, (11.1)

u∂u∂x

+ v∂u∂y

= ν∂ 2u∂y2 , (11.2)


u=Uw(x), v= 0 aty= 0 (11.3)

and

u= 0,∂v∂x

= 0 atx= 0, (11.4)

u→ 0 asy→+∞. (11.5)

The similarity solutions exist only in some special cases ofUw(x). FollowingGortler [10], we define the so-called principal function

∆(x) =U ′

w(x)U2

w(x)

∫ x

0Uw(ξ )dξ . (11.6)

When∆(x) equals to a constantβ , i.e.

U ′w(x)

U2w(x)

∫ x

0Uw(ξ )dξ = β , (11.7)

there exists the similarity solution

ψ =

√

ν∫ x

0Uw(ξ )dξ g(η), η =

Uw(x)√

ν∫ x

0 Uw(ξ )dξy, (11.8)

whereψ is a stream function, andg(η) is governed by a nonlinear ODE


g′′′+12

gg′′−β g′2 = 0, g(0) = 0, g′(0) = 1, g′(+∞) = 0. (11.9)

The corresponding local coefficient of skin friction of the similarity boundary-layerflows reads

Cf =τw

12ρU2

w(x)= 2g′′(0)

√

ν∫ x

0 Uw(ξ )dξ. (11.10)

For example, in case ofUw(x) = a xλ , wherea(1+λ )> 0, it holds

β = λ/(1+λ ),

so that the similarity criteria (11.7) is satisfied and therefore there exists the similar-ity solution

ψ =

√

a ν(1+λ )

xλ+1

2 g(η), η =

√

a(1+λ )ν

xλ−1

2 y. (11.11)

In this case, the system of the two coupled PDEs (11.1) and (11.2) becomes oneODE (11.9) that is much easier to solve than the original ones. So, from the view-point of mathematics, the problem is greatly simplified whensimilarity solutionsexist.

For similarity boundary-layer flows, all velocity profiles at differentx are similar.However, such kind of similarity is lost for non-similarityflows [5,6,8,9,38,45–47,50]. Physically speaking, the non-similarity boundary-layer flows are more generalin practice, and thus are more important than similarity ones.

When similarity solutions do not exist, one had to solve a nonlinear PDE. Tradi-tionally, there are two different approaches: analytic andnumerical ones. Numericalmethods are widely applied to investigate non-similarity boundary-layer flow. Asshown in [40, 42], one can use numerical methods to obtain approximate results ata large number of discretized points. However, one had to replace an infinite inter-val by a finite ones, and this results in some additional inaccuracy and uncertaintyinto numerical results. On the other hand, by means of analytical methods, one cansolve nonlinear PDEs in an infinite interval. However, it is apity that, using thetraditional analytic techniques such as perturbation techniques, it is hard to get an-alytic approximations that are valid and accurate forall physical variables in thewhole interval. This is mainly because perturbation methods are often dependenton small variables or parameters, and thus perturbation results are often invalid forall physical parameters/variables. Currently, Cimpean et al.[6] applied the pertur-bation techniques, combined with numerical techniques, tosolve a free convectionnon-similarity boundary-layer problem over a vertical flatsheet in a porous medium.Like most of perturbation solutions, their results are valid only for small or largex,which are regarded as perturbation quantities.

In addition, the so-called “method of local similarity” [38,46] for non-similarityboundary-layer problems is based on such an assumption thatnon-similarity termsin governing equations are so small that they can be regardedas zero and thus theoriginal PDEs become an ODE. However, the results given by “the method of local


similarity” are of “uncertain accuracy”, as pointed out by Sparrow et al. [46], andvalid only for small variables in general, as pointed out by Massoudi [38], respec-tively. This is easy to understand, because non-similarityterms are certainlynotzero and must be considered. Sparrow et al. [45, 46, 50] introduced the so-called“method of local non-similarity”, which was applied by Massoudi [38] in 2001to solve a non-similarity flow of non-Newtonian fluid over a wedge. Differentiat-ing the original governing equations by a dimensionless variable ξ along the freestream velocity, Massoudi [38] gave two additional auxiliary nonlinear PDEs forboth momentum and energy equations, then regarded the variable ξ in these twoPDEs to be aconstantso as to reduce them as a system of ODEs, and finally usednumerical techniques to solve the more complicated system of four equations. It isa pity that Massoudi [38] only gave numerical results for small ξ , although it wasreported [47,50] that the results given by “the method of local non-similarity” agreewell with numerical or series solutions in some cases.

The above-mentioned attempts reveal the mathematical difficulties for non-similarity boundary-layer flows. This might be the reason why the publicationsabout non-similarity boundary-layer flows are much less than those for similarityones, although the former is more important than the latter,not only in theory butalso in applications.

By means of the stream functionψ , (11.1) is automatically satisfied, and then(11.2) becomes such a nonlinear PDE

ν∂ 3ψ∂y3 +

∂ψ∂x

∂ 2ψ∂y2 − ∂ψ

∂y∂ 2ψ∂x∂y

= 0, (11.12)


ψ = 0,∂ψ∂y

=Uw(x) at y= 0,∂ψ∂y

→ 0 asy→+∞. (11.13)

Using the transformation

η =y

ν1/2σ(x), ψ = ν1/2 σ(x) f (x,η), (11.14)

whereσ(x)> 0 is a real function to be chosen later, we have the velocities

u=∂ f∂η

, v= ν1/2[

σ ′(x)

(

η∂ f∂η

− f

)

−σ(x)∂ f∂x

]

.

Then, the governing equation (11.12) becomes

∂ 3 f∂η3 +

12[σ2(x)]′ f

∂ 2 f∂η2 +σ2(x)

(

∂ f∂x

∂ 2 f∂η2 − ∂ f

∂η∂ 2 f

∂x∂η

)

= 0, (11.15)



f (x,0) = 0, fη (x,0) =Uw(x), fη (x,+∞) = 0, (11.16)

where fη denotes the partial differentiation off (x,η) with respect toη . Note that(11.15) is a nonlinear PDE with variable coefficients[σ2(x)]′/2 andσ2(x).

There exist an infinite number of sheet stretching velocityUw(x) which do notsatisfy the similarity-criteria (11.7). Here, without loss of generality, let us considerthe caseUw(x) = Uw(ξ ), whereξ =Γ (x) defines a kind of transform. Then, (11.15)becomes

∂ 3 f∂η3 +σ1(ξ ) f

∂ 2 f∂η2 +σ2(ξ )

(

∂ f∂ξ

∂ 2 f∂η2 −

∂ f∂η

∂ 2 f∂ξ ∂η

)

= 0, (11.17)


f (0,ξ ) = 0, fη (0,ξ ) = Uw(ξ ), fη (+∞,ξ ) = 0, (11.18)

where

σ1(ξ ) =12[σ2(x)]′, σ2(ξ ) = Γ ′(x) σ2(x), (11.19)

in which x is expressed byξ , i.e. x = Γ −1(ξ ). For example, whenσ(x) =√

1+ xandξ = Γ (x) = x/(1+ x), we haveσ1(ξ ) = 1/2 andσ2(ξ ) = 1− ξ . For details,please refer to Liao [29].

The corresponding local coefficient of skin friction of the non-similarity boundary-layer flows is given by

Cf (x) =τ(x)

12ρU2

w(x)=

2ν1/2

σ(x)U2w(x)

∂ 2 f∂η2

∣

∣

∣

∣

η→0. (11.20)

So, it is important to get accurate results offηη (0,ξ ). The replacement boundary-layer thicknessδ (x) is given by

δ (x) =1

Uw(x)

∫ +∞

0u(x,y) dy. (11.21)


In the frame of the homotopy analysis method (HAM) [17–35], the non-similarityflow governed by the nonlinear PDE (11.17) can be solvedwithout any additionalassumptions, as shown by Liao [29]. This is completely different from all analyticmethods mentioned above. As shown below, this kind of non-similarity boundary-layer flow can be solved even by means of theBVPh1.0 in a rather similar way tothe similarity flows mentioned in Chapter 10.

According to (11.17), we define a nonlinear operator


N f =∂ 3 f∂η3 +σ1(ξ ) f

∂ 2 f∂η2 +σ2(ξ )

(

∂ f∂ξ

∂ 2 f∂η2 − ∂ f

∂η∂ 2 f

∂ξ ∂η

)

. (11.22)

Let q ∈ [0,1] denote the homotopy-parameter,c0 6= 0 the convergence-control pa-rameter,H(η) 6= 0 an auxiliary function,L an auxiliary linear operator with theproperty

L [0] = 0, (11.23)

and f0(η ,ξ ) an initial guess that satisfies the boundary conditions (11.18), respec-tively. Note that the HAM provides us extremely large freedom to choose the aux-iliary linear operatorL and the initial guessf0(η ,ξ ). In the frame of the HAM,we first construct such a continuous variation (or deformation)φ(η ,ξ ;q) that, asqincreases from 0 to 1,φ(η ,ξ ;q) varies from the initial guessf0(η ,ξ ) to the solutionf (η ,ξ ) of (11.17) and (11.18). Such kind of continuous variation (or mapping) isgoverned by the so-called zeroth-order deformation equation

(1−q)L [φ(η ,ξ ;q)− f0(η ,ξ )] = q c0 H(η) N [φ(η ,ξ ;q)], (11.24)

subject to the boundary conditions on the sheet

φ(0,ξ ;q) = 0, φη (0,ξ ;q) = Uw(ξ ), (11.25)

and the boundary condition at infinity

φη (+∞,ξ ;q)→ 0. (11.26)

Note that the initial guessf0(η ,ξ ) satisfies the boundary conditions (11.18), andbesidesL has the property (11.23). Thus, whenq= 0, we have the initial guess

φ(η ,ξ ,0) = f0(η ,ξ ). (11.27)

Whenq= 1, sincec0 6= 0, the zeroth-order deformation equations (11.24) to (11.26)are equivalent to the original equations (11.17) and (11.18), provided

φ(η ,ξ ;1) = f (η ,ξ ). (11.28)

Thus, as the embedding parameterq increases from 0 to 1,φ(η ,ξ ;q) indeed variescontinuouslyfrom the initial guessf0(η ,ξ ) to the exact solutionf (η ,ξ ) of theoriginal equations (11.17) and (11.18).

Then, expandingφ(η ,ξ ;q) in Maclaurin series with respect toq and using(11.27), we have the homotopy-Maclaurin series

φ(η ,ξ ;q) = f0(η ,ξ )++∞

∑m=1

fm(ξ ,η) qm, (11.29)

where

fm(η ,ξ ) = Dm[φ(η ,ξ ;q)] =1m!

∂ mφ(η ,ξ ;q)∂qm

∣

∣

∣

∣

q=0


is themth-order homotopy-derivativeofφ(η ,ξ ;q), andDm is themth-order homotopy-derivative operator, respectively. Note that the convergenceof the homotopy-Maclaurinseries (11.29) depends on the initial guessf0(η ,ξ ), the auxiliary linear operatorL ,the auxiliary functionH(η), and the convergence-control parameterc0. Assumingthat all of them are so properly chosen that the homotopy-Maclaurin series (11.29)absolutely converges atq= 1, we have due to (11.28) the homotopy-series solution

f (η ,ξ ) = f0(η ,ξ )++∞

∑m=1

fm(ξ ,η). (11.30)

According to Theorem 4.15, we have themth-order deformation equation

L [ fm(η ,ξ )− χm fm−1(η ,ξ )] = c0 H(η) δm−1(η ,ξ ), (11.31)

subject to the boundary conditions on the sheet

fm = 0,∂ fm∂η

= 0, aty= 0 (11.32)

and the boundary condition at infinity

∂ fm∂η

→ 0, asy→+∞, (11.33)

where

χm =

0, m≤ 1,1, m> 1,

(11.34)

and

δn(η ,ξ ) =∂ 3 fn∂η3 +σ1(ξ )

n

∑k=0

fn−k∂ 2 fk∂η2

+ σ2(ξ )n

∑k=0

(

∂ fk∂ξ

∂ 2 fn−k

∂η2 − ∂ fk∂η

∂ 2 fn−k

∂ξ ∂η

)

(11.35)

is gained by Theorem 4.1. For details, please refer to Chapter 4.It should be emphasized here that the high-order deformation equations (11.31)

to (11.33) arelinear. Besides, unlike perturbation techniques, we donot need anysmall/large parameters to obtain these linear differential equations. Furthermore,different from “the method of local similarity” [38, 46] and“the method of localnon-similarity” [45, 46, 50], weneitherenforce the non-similarity terms to be zero,nor regard the variableξ as a constant. In a word, different from all other previ-ous analytic methods for the non-similarity flows, our approach doesnot need anyadditional assumptions. More importantly, as mentioned before, we have extremelylarge freedom to chooseL : this freedom is so large that, in the frame of the HAM,a nonlinear PDE can be (although sometimes) transferred into an infinite number oflinear ODEs, as shown below.


Mathematically, the essence to approximate a nonlinear differential equation isto find a set ofproperbase functions to fit its solutions. Physically, it is well-knownthat most of viscous flows decay exponentially at infinity (i.e.η →+∞). So, for thenon-similarity boundary-layer flows over a stretching flat sheet, the velocitiesu andv should decay exponentially asη →+∞. Therefore,f (η ,ξ ) should be in the form

f (η ,ξ ) =+∞

∑m=0

+∞

∑n=0

am,n(ξ ) ηn exp(−mη), (11.36)

wheream,n(ξ ) is a polynomial ofξ . This expression is calledthe solution-expressionof f (η ,ξ ), which plays an important role in the frame of the HAM, as shown below.

To satisfy the solution-expression (11.36) and the boundary conditions (11.18),we choose the initial guess

f0(η ,ξ ) = Uw(ξ )(

1−e−η) , (11.37)

which contains the simplest but leading terms of (11.36) asη → +∞. Note thatf0(η ,ξ ) satisfies the boundary conditions (11.18) and decays exponentially at infin-ity.

As mentioned before, we have extremely large freedom to choose the auxiliarylinear operatorL . This freedom is however restricted by the solution-expression(11.36) and the boundary conditions (11.18). Note that the original governing equa-tion (11.17) is a nonlinear PDE with variable coefficients. So, if we choose a partialdifferential operator asL , the high-order deformation equation (11.31) is a PDE.It is well-known that a PDE with variable coefficients is moredifficult to solvethan an ODE with constant coefficients. So, mathematically,it is much easier tosolve (11.31) ifL is an linear differential operator which contains derivatives withrespect to eitherη or ξ only, and besides without any variable coefficients. Phys-ically, for boundary-layer flows, the velocity variation across the flow direction ismuch larger than that in the flow direction. Therefore, the derivatives

∂ f∂η

,∂ 2 f∂η2 ,

∂ 3 f∂η3

across the flow direction are considerably larger and thus physically more importantthan the derivatives

∂ f∂ξ

,∂ 2 f

∂ξ ∂ηin the flow direction. Considering all of these mentioned above, we choose the aux-iliary linear operator

L f =∂ 3 f∂η3 −

∂ f∂η

, (11.38)

which is independent ofξ , and besides does not contain any variable coefficients.For details, please refer to Liao [29]. Notice that the auxiliary linear operator (11.38)

11.3 Homotopy-series solution 397

is exactly thesameas the auxiliary linear operator (10.16) used in§ 10.2 to solve akind of similarity boundary-layer flow!

Using the initial guess (11.37) and the auxiliary linear operator (11.38), it is easyto solve thelinear ODEs (11.31) to (11.33). The special solution of (11.31) reads

f ∗m(η ,ξ ) = χm fm−1(η ,ξ )+ c0 L−1[H(η) δm−1(η ,ξ )], (11.39)

whereL −1 denotes the inverse operator ofL . Then, the solution of the high-orderdeformation equations (11.31) to (11.33) is

fm(η ,ξ ) = f ∗m(η ,ξ )+C0(ξ )+C1(ξ )exp(−η),

where

C0(ξ ) =− f ∗m(0,ξ )−∂ f ∗m∂η

∣

∣

∣

∣

η=0, C1(ξ ) =

∂ f ∗m∂η

∣

∣

∣

∣

η=0

are determined by the boundary conditions (11.32) and (11.33). In this way, it iseasy to solve the high-order deformation equations (11.31)to (11.33), especially bymeans of computer algebra system such as Mathematica. For details about mathe-matical formulas, please refer to Liao [29].

In this way, the originalnonlinearPDE withvariablecoefficients is transferredinto an infinite number oflinear ODEs withconstantcoefficients. It should be em-phasized that the high-order deformation equation (11.31)for the non-similarityboundary-layerflow is quite similar to the high-order deformation equations for sim-ilarity boundary-layer flows with exponentially decaying solutions in Chapter 10:both of them use thesameauxiliary linear operator. In other words, in the frameof the HAM, the non-similarity boundary-layer flow can be solved in the similarway as similarity ones. This greatly simplifies solving the nonlinear PDEs related tonon-similarity boundary-layer flows.

Finally, it should be emphasized thatfm(η ,ξ ) contains the convergence-controlparameterc0. As pointed out before, it is the convergence-control parameter c0

which provides us with a simple way to guarantee the convergence of the homotopy-series solution forall physical variables/parameters, as shown below.

11.3 Homotopy-series solution

Without loss of generality, let us consider such a sheet stretching velocity

Uw(x) =x

1+ x

that the corresponding flow is a non-similarity ones, because it does not satisfy thesimilarity criteria (11.7).

In this case, the stretching velocityUw(x) increases monotonously from 0 to 1along the sheet. In addition,Uw → x asx→ 0, andUw → 1 asx→+∞, respectively.


Fig. 11.1 Squared residualof (11.17) versusc0. Dashedline: 5th-order approxima-tion; Dash-dotted line: 10th-order approximation; Dash-dot-dotted line: 15th-orderapproximation; Solid line:20th-order approximation.

c0

Squ

ared

resi

dual

-1 -0.8 -0.6 -0.4 -0.2 010-8

10-7

10-6

10-5

10-4

10-3

10-2

Physically, the flow nearx = 0 should be close to the similarity ones withUw =x, and the flow atx → +∞ should be close to the similarity ones withUw = 1,respectively. It is well-known that, for similarity boundary-layer flows related toUw = x andUw = 1, the corresponding similarity variables arey/

√ν andy/

√νx,

respectively. Therefore, according to the definition (11.14) of η , we choose

σ(x) =√

1+ x

so thatη tends to the corresponding similarity variabley/√

ν asx → 0 and to thecorresponding similarity variabley/

√νx as x → +∞, respectively. Besides, it is

natural for us to defineξ = Γ (x) =

x1+ x

,

which gives, according to (11.19), that

Uw(ξ ) = ξ , σ1(ξ ) =12, σ2(ξ ) = 1− ξ .

Like similarity boundary-layerflows, the high-order deformation equations (11.31)to (11.33) of the non-similarity flow are linear ODEs. Therefore, we can solve iteven by means of theBVPh 1.0 directly, which is given in the Appendix 7.1 andfree available athttp://numericaltank.sjtu.edu.cn/BVPh.htm. The corresponding in-put data for theBVPh1.0 is given in the appendix of this chapter and free availableat the above-mentioned website.

Up to now, only the auxiliary functionH(η) and the convergence-control param-eterc0 are not determined. For the sake of simplicity, we choose

H(η) = 1.

Then, the solutionfm(η ,ξ ) of the high-order deformation equations (11.31) to(11.33) is dependent upon the convergence-control parameter c0 only, which pro-vides us a convenient way to guarantee the convergence of thehomotopy-series


Table 11.1 Averaged squared residual of the governing equation (11.17) overη ∈ [0,10] andξ ∈[0,1] by means ofc0 =−1/2

Order of approximation Averaged squared residual of (11.17)

1 2.3×10−3

2 7.4×10−4

4 8.5×10−5

6 1.1×10−5

8 2.3×10−6

10 8.9×10−7

15 1.7×10−7

20 4.2×10−8

25 1.8×10−8

30 1.0×10−8

solution, as mentioned in Chapter 2. The averaged squared residual of (11.17)over η ∈ [0,10] at thekth-order approximation is gained by means of the moduleGetErr[k] of theBVPh1.0, which is then integrated in the intervalξ ∈ [0,1]. Thecorresponding squared residuals of (11.17) versusc0 are as shown in Fig. 11.1. Notethat, at the 10th and 15th-order approximation, the optimalconvergence-control pa-rameterc0 is about -0.8 and -0.6, respectively. At the 20th-order of approximation,the minimum of the squared residual is 3.6× 10−8, corresponding to the optimalconvergence-control parameterc∗0 =−0.5554. It is found that, whenc0 =−1/2, thehomotopy-series solution (11.30) converges in thewholespatial interval 0≤ ξ ≤ 1and 0≤ η < +∞, corresponding to 0≤ x < +∞ and 0≤ y < +∞, as shown inTable 11.1 and Fig. 11.2.

Fig. 11.2 Homotopy-approximation offηη (0,ξ )by means ofc0 =−1/2. Solidline: 30th-order approxima-tion; Symbols: 20th-orderapproximation.

ξ

f ηη(0

,ξ)

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

According to Fig. 11.1, the homotopy-series (11.30) diverges whenc0 =−1. Asproved by Liao [23], some non-perturbation techniques, such as Lyapunov’s artifi-cial small parameter method [36], Adomian’s decompositionmethod [1, 2] and the


δ -expansion method [13], are special cases of the HAM in case of c0 = −1. Be-sides, it is proved [16, 41] that the so-called “homotopy perturbation method” [12]is also a special case of the HAM whenc0 =−1. So, if any of the above-mentionedmethods are used, one can not gain convergent results. This illustrates once againthe importance of the convergence-control parameterc0, and also the validity of theHAM for highly nonlinear problems.

It is important to give accurate local coefficient of skin friction of the non-similarity boundary-layer flow, which is related tofηη (0,ξ ) via (11.20). Whenc0 =−1/2, the 15th-order HAM approximation reads

fηη (0,ξ ) = −ξ +0.357142ξ 2+0.0745784ξ 3+0.0329563ξ 4+0.0187480ξ 5

+ 0.0121641ξ 6+0.00856336ξ 7+0.00638095ξ 8+0.00468866ξ 9

+ 0.0125286ξ 10−0.108975ξ 11+0.729854ξ 12−2.509780ξ 13

+ 4.702786ξ 14−4.475992ξ 15+1.690259ξ 16, (11.40)

which agrees well with the 20th-order approximations and isaccurate in the wholeregion 0≤ ξ ≤ 1, as shown in Fig. 11.2. Then, it is straightforward to calculate thelocal coefficientCf of the skin friction by (11.20). It is found thatCf → −2

√ν/x

asx→ 0, andCf →−0.8875√

ν/x asx→+∞, respectively, as shown in Fig. 11.3.Besides, the boundary-layer thicknessδ (x) of the non-similarity flow tends to

√ν

asx → 0 and 1.61613√

ν x asx → +∞, respectively, as shown in Fig. 11.4. Notethat the boundary-layer thickness of the corresponding similarity flow is just

√ν

in case ofUw = x (nearx = 0) and 1.61613√

ν x in case ofUw = 1 (at x → +∞),respectively. Thus, the homotopy-series solution of the non-similarity flow gives

Cf (x)→−2√

ν/x, δ (x)→√

ν , asx→ 0

andCf (x)→−0.8875

√

ν/x, δ (x)→ 1.61613√

ν x, asx→+∞,

respectively. Therefore, physically speaking, the non-similarity boundary-layer flowin the regionx→ 0 andx→+∞ is very close to the corresponding similarity onesin case ofUw = x andUw = 1, respectively. However, the flows in other regions arenon-similarity ones, as shown in Fig. 11.3 and Fig. 11.4, respectively. The velocityprofile u∼ y/

√ν of the non-similarity boundary-layer flow at differentx are given

in Fig. 11.5. For more discussions on the physical meanings of the homotopy-seriessolution, please refer to Liao [29].

Therefore, by means of theBVPh 1.0, we successfully gain the convergent so-lution of the non-similarity boundary-layer flow governed by the nonlinear PDE(11.17), which is valid in thewholearea 0≤ x<+∞ and 0≤ y<+∞.

All of above results are given by means ofH(η) = 1. Note that we have greatfreedom to choose the auxiliary functionH(η). For example, using

H(η) = exp(−η),


Fig. 11.3 The local co-efficient of skin frictionCf (x)/

√ν of the non-

similarity flow in case ofUw = x/(1+ x). Solid-line:30th-order HAM result; Sym-bols: 20th-order HAM re-sult; Dashed-line:Cf (x) =−0.8875

√

ν/x; Dash-dottedline:Cf (x) =−2

√ν/x.

x

-Cf

/ν1

/2

10-2 10-1 100 101 102 10310-2

10-1

100

101

102

Fig. 11.4 The boundary-layer thicknessδ (x)/

√ν

of the non-similarity flowin case ofUw = x/(1+ x).Solid-line: 30th-order HAMresult; Symbols: 20th-orderHAM result; Dashed-line:δ (x) = 1.61613

√ν x; Dash-

dotted line:δ (x) =√

ν .

x

δ/ν1/

2

10-3 10-2 10-1 100 101 102 103

100

101

Fig. 11.5 The velocity pro-file u ∼ y/

√ν of the non-

similarity flow at differentx.Solid line:x= 1/4; Dashedline: x = 1/2; Dash-dottedline: x= 1; Dash-dot-dottedline: x= 5; Long-dashed line:x= 10.

u

y/ν

1/2

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

we gain the series solution in the form


f (η ,ξ ) =+∞

∑m=0

bm(ξ ) exp(−mη),

wherebm(ξ ) is a polynomial ofξ . In this case, we can gain convergent series solu-tion by means ofc0 = −3/2 in a similar way. This illustrates once again the flexi-bility of the HAM.


We illustrate in this chapter that, in the frame of the HAM, the nonlinear PDEdescribing a kind of non-similarity boundary-layer flow canbe solved directly bymeans of theBVPh1.0 in a rather similar way to that for similarity ones. This showsthe validity of theBVPh1.0 for some nonlinear PDEs, especially for those relatedto boundary-layer flows.

Note that the original nonlinear PDE (11.17) contains the derivatives of f (η ,ξ )with respect to bothη andξ . However, we choose here the auxiliary linear operator(11.38), which only contains the derivatives with respect to η . Mathematically, thisis mainly because the HAM provides us extremely large freedom to choose theauxiliary linear operator, as mentioned in Chapter 2. More importantly, the HAMalso provides us a convenient way to guarantee the convergence of the homotopy-series solution by means of choosing a proper convergence-control parameterc0:the freedom on the choice of the auxiliary linear operatorL has no meanings if onecan not guarantee the convergence of homotopy-series solutions. It is interestingthat thesameauxiliary linear operator (11.38) is used by Liao and Pop [32] to solvethe similarity boundary-layer flows, too. In addition, the auxiliary linear operator(11.38) is exactly thesameas the auxiliary linear operator (10.16) used in§ 10.2to solve a kind ofsimilarity boundary-layer flow. Thus, in the frame of the HAM,non-similarity boundary-layer flows can be solved in a similar way like similarityones. Physically, it is mainly due to the existence of the boundary-layer: the velocityvariation across the flow direction is much larger than that in the flow direction.Thus, theBVPh 1.0 can be applied to solve other non-similarity boundary-layerflows [14,15,51] in a rather similar way.

It is well-known that a nonlinear PDE is much more difficult tosolve than alinear PDE. So, it is a good idea to replace a nonlinear PDE by asequence of linearODEs, if possible. Thus, this approach has general meanings. For example, in theframe of the HAM, a nonlinear PDE describing unsteady boundary-layer flows canbe transferred into an infinite number of linear PDEs similarto (11.31), as shownby Liao [26, 27]. Besides, a nonlinear PDE describing an unsteady heat transfercan be replaced by an infinite number of linear ODEs, as shown by Liao, Su andChwang [33]. All of these kinds of nonlinear PDEs can be solved by means of theBVPh1.0 in a similar way. This show the general validity of theBVPh1.0 .

Finally, it should be pointed out that, although this approach has some generalmeanings, it is howevernot valid for all types of nonlinear PDEs, especially for


those related to waves. As mentioned in Chapter 7, our aim is to develop a pack-age valid for as many nonlinear boundary-value problems as possible. Even so, theMathematica packageBVPh(version 1.0) can be used as a tool to solve many non-linear PDEs, especially those related to non-similarity and/or unsteady boundary-layer flows and heat transfer.


Appendix 11.1 Input data ofBVPh


( * Define the physical and control parameters * )TypeEQ = 1;ApproxQ = 0;ErrReq = 10ˆ(-10);NgetErr = 100;zRintegral = 10;

( * Define the governing equation * )f[z_,u_,lambda_] := Module[temp,temp[1] = D[u,z,3] + u * D[u,z,2]/2;temp[2] = D[u,t] * D[u,z,2] - D[u,z] * D[u,z,t];temp[1] + (1-t) * temp[2] // Expand];

( * Define boundary conditions * )zR = infinity;OrderEQ = 3;BC[1,z_,u_,lambda_] := u /. z-> 0 ;BC[2,z_,u_,lambda_] := D[u,z] - t /. z -> 0 ;BC[3,z_,u_,lambda_] := Limit[D[u,z], z -> zR ];

( * Define initial guess * )u[0] = t * (1 - Exp[-z]);

( * Define the auxiliary linear operator * )L[u_] := D[u,z,3] - D[u,z];


D[u[k],z,2]/.z->0,t->1//N];

( * Define Getdelta[k] * )Getdelta[k_]:=Module[temp,i,uz[k] = D[u[k],z]//Expand;uzz[k] = D[uz[k],z]//Expand;uzzz[k] = D[uzz[k],z]//Expand;ut[k] = D[u[k],t]//Expand;uzt[k] = D[uz[k],t]//Expand;uzzu[k] = Sum[uzz[i] * u[k-i],i,0,k]//Expand;uzuzt[k] = Sum[uz[i] * uzt[k-i],i,0,k]//Expand;uzzut[k] = Sum[uzz[i] * ut[k-i],i,0,k]//Expand;temp[1] = uzzz[k] + uzzu[k]/2 //Expand;temp[2] = (1-t) * (uzzut[k] - uzuzt[k]);delta[k] = temp[1] + temp[2]//Expand;];

Appendix 11.1 Input data ofBVPh 405

( * Print input and control parameters * )PrintInput[u[z,t]];

( * Set convergence-control parameter c0 * )c0 = -1/2;Print["c0 = ",c0];


( * Calculate the squared residual * )For[k=2, k<=10, k=k+2,GetErr[k];err[k] = Integrate[Err[k],t,0,1];Print[" k = ", k, " Squared residual = ",

err[k]//N];];


Note that, for nonlinear boundary-value problems defined inan infinite interval,we must setzR = infinity in the input data ofBVPh (version 1.0). Besides,we must define the moduleGetdelta in the input date, when a nonlinear PDE issolved by means of theBVPh(version 1.0).


References



3. Banks, W.H.H.: Similarity solutions of the boundary-layer equations for a stretching wall.Journal de Mecanique theorique et appliquee.2, 375-392 (1983)

4. Blasius, H.: Grenzschichten in Fussigkeiten mit kleiner Reibung. Z. Math. Phys.56, 1-37(1908)

5. Cheng, W.T., Lin, H.T.: Non-similarity solution and correlation of transient heat transfer inlaminar boundary layer flow over a wedge. Int. J. EngineeringScience.40, 531-548 (2002)

6. Cimpean, D., Merkin, J.H., Ingham, D.B.: On a free convection problem over a vertical flatsurface in a porous medium. Transport in Porous.64, 393-411 (2006)

7. Crane, L.: Flow past a stretching plate. Z. Angew. Math. Phys.21, 645-647 (1970)8. Duck, P.W., Stow, S.R., Dhanak, M.R.: Non-similarity solutions to the corner boundary-layer

equations (and the effects of wall transpiration). J. FluidMechanics.400, 125-162 (1999)9. Gorla, R.S.R., Kumari, M.: Non-similar solutions for mixed convection in non-Newtonian

fluids along a vertical plate in a porous medium. Transport inPorous.33, 295-307 (1998)10. Gortler, H.: Eine neue Reihenentwicklung fur laminare Grenzschichten. ZAMM .32, 270-

271 (1952)11. Howarth, L.: On the solution of the laminar boundary layer equations. Proc. Roy. Soc. London

A. 164, 547-579 (1938)12. He, J.H.: Homotopy perturbation technique. Comput. Method. Appl. M. 178, 257 – 262

(1999)13. Karmishin, A.V., Zhukov, A.T., Kolosov, V.G.: Methods of Dynamics Calculation and Testing

for Thin-walled Structures (in Russian). Mashinostroyenie, Moscow (1990)14. Kousar, N., Liao, S.J.: Series solution of non-similarity boundary flows over a porous wedge.

Transport Porous Media.83, 397 – 412 ( 2010)15. Kousar, N., Liao, S.J.: Unsteady non-similarity boundary-layer flows caused by an impul-

sively stretching flat sheet. Nonlinear Analysis B.12, 333 – 342 (2011)16. Liang, S.X., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy pertur-











26. Liao, S.J.: An analytic solution of unsteady boundary-layer flows caused by an impulsivelystretching plate. Communications in Nonlinear Science andNumerical Simulation.11, 326 –339 (2006)

References 407



29. Liao, S.J.: A general approach to get series solution of non-similarity boundary layer flows.Commun. Nonlinear Sci. Numer. Simulat.14, 2144 - 2159 (2009)



32. Liao, S.J., Pop, I.: Explicit analytic solution for similarity boundary layer equations. Int. J.Heat and Mass Transfer.47, 75-85 (2004)

33. Liao, S.J., Su, J., Chwang, A.T.: Series solutions for a nonlinear model of combined convec-tive and radiative cooling of a spherical body. Int. J. Heat and Mass Transfer.492437 – 2445(2006)




37. Magyari, E., Keller, B.: Exact solutions for self-similar boundary-layer flows induced by per-meable stretching walls. Eur. J. Mech. B-Fluids.19, 109-122 (2000)

38. Massoudi, M.: Local non-similarity solutions for the flow of a non-Newtonian fluid over awedge. Int. J. Non-Linear Mech.36, 961-976 (2001)

39. Prandtl, L.:Uber Flussigkeitsbewegungen bei sehr kleiner Reibung. Verhandlg. Int. Math.Kongr. Heidelberg. 484-491 (1904)

40. Roy, S., Datta, P., Mahanti, N.C.: Non-similar solutionof an unsteady mixed convection flowover a vertical cone with suction or injection. Int. J. of Heat and Mass Transfer.50, 181-187(2007)

41. Sajid, M., Hayat, T.: Comparison of HAM and HPM methods for nonlinear heat conductionand convection equations. Nonlin. Anal. B.9, 2296 – 2301 (2008)

42. Sahu, A.K., Mathur, M.N., Chaturani, P., Bharatiya, S.S.: Momentum and heat transfer froma continuous moving surface to a power-law fluid. Acta Mechanica.142, 119-131 (2000)

43. Schlichting, H., Gersten, K.: Boundary Layer Theory. Springer, Berlin (2000)44. Sobey, I.J.: Introduction to Interactive Boundary Layer Theory. Oxford University Press, Ox-

ford (2000)45. Sparrow, E.M., Quack, H.: Local non-similarity boundary-layer solutions. AIAA J.8, No.8,

1936-1942 (1970)46. Sparrow, E.M., Yu, H.S.: Local non-similarity thermal boundary-layer solutions. J. Heat

Transfer Trans. ASME. 328-334 (1971)47. Stewartson, C.D., Williams: Viscous flow past a flat platewith uniform injection. Proceedings

of Royal Society (A).284, 370-396 (1965)48. Tani, I.: Histry of boundary-layer theory. Ann. Rev. Fluid Mech.9, 87-111 (1977)49. Van Dyke, M.: Higher approximations in boundary-layer theory. Part 1: General analysis.

Journal of Fluid Mechanics.14, 161-177 (1962)50. Wanous, K.J., Sparrow, E.M.: Heat transfer for flow longitudinal to a cylinder with surface

mass transfer. J. Heat Transfer Trans. Ser. C.87, 317-319 (1965)51. You, X.C., Xu, H., Liao, S.J.: On the non-similarity boundary-Layer flows of second-order

fluid over a stretching sheet. ASME J App Mech.77: 021002-1 (2010)

Chapter 12Unsteady boundary-layer flows

Abstract In this chapter, we illustrate the validity of the Mathematica packageBVPh(version 1.0) for nonlinear partial differential equations (PDEs) related to un-steady boundary-layer flows. We show that, usingBVPh1.0, an unsteady boundary-layer flow can be solved in a rather similar way to that for steady-state similarityones governed by nonlinear ODEs. In other words, in the frameof the HAM, solv-ing unsteady boundary-layer flows is as easy as steady-stateones. This shows thevalidity of the BVPh 1.0 for some nonlinear PDEs, especially for those related toboundary-layer flows.

12.1 Introduction

In Chapter 11, we illustrate that the Mathematica packageBVPh (version 1.0) canbe used to solve a kind ofnon-similarityboundary-layer flow governed by a nonlin-ear partial differential equation (PDE) in an infinite area.In this chapter, we furtherillustrate that theBVPh1.0 can be used to gain accurate approximations of anun-steadysimilarity boundary-layer flow governed by a nonlinear PDE in the infinitespatial and temporal interval.

Without loss of generality, let us consider here an unsteadyboundary-layer flowof viscous fluid developed by an impulsively stretching plate [1,2,19–25], governedby the PDE

∂u∂ t

+u∂u∂x

+ v∂u∂y

= ν∂ 2u∂y2 , (12.1)

∂u∂x

+∂v∂y

= 0, (12.2)


t > 0 : u= a x, v= 0 aty= 0, (12.3)

409

410 12 Unsteady boundary-layer flows

t > 0 : u→ 0 asy→+∞ (12.4)

and the initial conditions

t = 0 : u= v= 0 at all points(x,y), (12.5)

whereν is the kinematic viscosity,t denotes the time,(u,v) are the velocity com-ponents in the directions of increasingx,y, respectively. Here, we consider the casea> 0 only, corresponding to a stretching plate.

There exist similarity variables for this kind of flow. Following Seshadri et al.[21] and Nazar et al. [19], we use Williams and Rhyne’s similarity transformation[26]:

ψ =√

aνξ x f(η ,ξ ), η =

√

aνξ

y, ξ = 1−exp(−τ), τ = a t, (12.6)

whereψ denotes the stream-function. Note that the new dimensionless timeξ isbounded in a finite interval

0≤ ξ ≤ 1,

corresponding to 0≤ τ <+∞. Besides, the similarity variableη is dependent uponnot only the spatial variabley but also the timet so that the initial condition (12.5)is automatically satisfied. Thus, at any timeτ ∈ [0,+∞), i.e. ξ ∈ [0,1], it is still asimilarity boundary-layer flow.

Using the above similarity transformation, the original PDEs become

∂ 3 f∂η3 +

12(1− ξ ) η

∂ 2 f∂η2 + ξ

[

f∂ 2 f∂η2 −

(

∂ f∂η

)2]

= ξ (1− ξ )∂ 2 f

∂η∂ξ, (12.7)


f (0,ξ ) = 0,∂ f∂η

∣

∣

∣

∣

η=0= 1,

∂ f∂η

∣

∣

∣

∣

η=+∞= 0. (12.8)

This is still a nonlinear PDE. However, since the initial conditions (12.5) are au-tomatically satisfied so that there exist only boundary conditions (12.8), the abovenonlinear PDE is easier to solve than the original two coupled PDEs.

Whenξ = 0, corresponding toτ = 0, (12.7) becomes the Rayleigh type of equa-tion

∂ 3 f∂η3 +

12

η∂ 2 f∂η2 = 0, (12.9)

subject to

f (0,0) = 0,∂ f∂η

∣

∣

∣

∣

η=0,ξ=0= 1,

∂ f∂η

∣

∣

∣

∣

η=+∞,ξ=0= 0. (12.10)

The above equation has the closed-form solution


f (η ,0) = η erfc(η/2)+2√π[

1−exp(−η2/4)]

, (12.11)

where erfc(η) is the complementary error function defined by

erfc(η) =2√π

∫ +∞

ηexp(−z2) dz.

Whenξ = 1, corresponding toτ →+∞, we have from (12.7) that

∂ 3 f∂η3 + f

∂ 2 f∂η2 −

(

∂ f∂η

)2

= 0, (12.12)

subject to

f (0,1) = 0,∂ f∂η

∣

∣

∣

∣

η=0,ξ=1= 1,

∂ f∂η

∣

∣

∣

∣

η=+∞,ξ=1= 0. (12.13)

The above equation has the closed-form solution

f (η ,1) = 1−exp(−η). (12.14)

So, as the time variableξ increases from 0 to 1,f (η ,ξ ) varies from the initial solu-tion (12.11) to the steady-state solution (12.14). Note that, althoughf ′(+∞,ξ )→ 0exponentially for allξ , where the prime denotes the differentiation with respectto η , the initial solution (12.11) decays much more quickly asη → +∞ than thesteady-state solution (12.14). So, mathematically, the initial solution (12.11) is dif-ferent from the steady-state solution (12.14) in essence. This might be the reasonwhy it is so hard to give an accurate analytic solution uniformly valid for all time0≤ τ <+∞.

Whenξ = 0 andξ = 1, we have

∂ 2 f∂η2

∣

∣

∣

∣

η=0,ξ=0=− 1√

π(12.15)

and∂ 2 f∂η2

∣

∣

∣

∣

η=0,ξ=1=−1, (12.16)

respectively. The skin friction coefficient is given by

cxf (x,ξ ) = (ξ Rex)

−1/2 f ′′(0,ξ ), 0≤ ξ ≤ 1, (12.17)

whereRex = ax2/ν is the local Reynolds number.


12.2 Perturbation approximation

Like Seshadri et al. [21] and Nazar et al. [19], one can regardξ as a small parameterto search for the perturbation approximation in the form

f (η ,ξ ) = g0(η)+g1(η) ξ +g2(η) ξ 2+ · · ·=+∞

∑m=0

gm(η) ξ m. (12.18)

Substituting it into (12.7) and (12.8), and balancing the coefficients of the like-powerof ξ , we have the zeroth-order perturbation equation

g′′′0 (η)+η2

g′′0(η) = 0, g0(0) = 0, g′0(0) = 1, g′0(+∞) = 0, (12.19)

and themth-order (k≥ 1) perturbation equation

g′′′m(η)+η2

g′′m(η) − m g′m(η) =η2

g′′m−1(η)− (m−1)g′m−1(η)

−m−1

∑i=0

[

gi(η) g′′m−1−i(η)−g′i(η)g′m−1−i(η)

]

, (12.20)


gm(0) = 0, g′m(0) = 0, g′m(+∞) = 0. (12.21)

All of the above perturbation equations are linear ODEs withrespect toη only.The solution of the zeroth-order perturbation equation (12.19) reads

g0(η) = f (η ,0) = η erfc(η/2)+2√π[

1−exp(−η2/4)]

, (12.22)

where erfc(η/2) is a complementary error function and exp(−η2/4) is a Gaussiandistribution function, respectively. Substituting the above expression into (12.20)and (12.21) gives the 1st-order perturbation equation

g′′′1 (η)+η2

g′′1(η)−g′1(η) =[

2π− η

2√

π+

η√π

erfc(η

2

)

]

exp

(

−η2

4

)

− 2π

exp

(

−η2

2

)

+[

erfc(η

2

)]2, (12.23)

subject to the boundary condition

g1(0) = 0, g′1(0) = 0, g′1(+∞) = 0. (12.24)

Although the above equation is a line ODE, it is however difficult to solve. Firstly,even the homogeneous equation

12.2 Perturbation approximation 413

g′′′1 (η)+η2

g′′1(η)−g′1(η) = 0

has a rather complicated special solution

g∗1(η) =(

1+η2

4

)

exp

(

−η2

4

)

− 3√

π4

(

η +η3

6

)

erfc(η

2

)

,

although two other special solutions

g∗1(η) = 1, g∗1(η) = η +η3

6

are simple. Therefore, it seems impossible for the perturbation approximations toavoid the complementary error function erfc(η/2) and the Gaussian distributionfunction exp(−η2/4). Secondly, the right-hand side of the first-order perturbationequation (12.23) contains the complementary error function erfc(η/2), the Gaussiandistribution function exp(−η2/4) and their combinations such as

η erfc(η

2

)

exp

(

−η2

4

)

.

These might be the reasons why Seshadri et al. [21] and Nazar et al. [19] reportedonly the 1st-order perturbation approximation

g1(η) =(

12− 2

3π

)[(

1+η2

2

)

erfc(η/2)− η√π

e−η2/4]

−12

(

1− η2

2

)

erfc2(η/2)− 3η2√

πe−η2/4 erfc(η/2)

− 1√π

(

43√

π− η

4

)

e−η2/4+2π

e−η2/2. (12.25)

Notice that the right-hand side of the high-order perturbation equations (12.20)becomes more and more complicated as the order of approximation increases. So,although we gain, very luckily, the solution (12.25) of the first-order perturbationequation (12.23) and (12.24), it becomes more and more difficult to solve the high-order ones. More importantly, even if we could solve all high-order perturbationequations efficiently, we still can not guarantee the convergence of the perturbationseries (12.18).

Therefore, although the original nonlinear PDE (12.7) can be transferred into aninfinite number of linear ODEs (12.19) and (12.20) by regarding ξ as a small pa-rameter and expandingf (η ,ξ ) into the perturbation series (12.18 ), we still can notgain high-order perturbation approximations efficiently,because the correspondinglinear perturbation equations (12.20) become more and moredifficult to solve. Notethat the perturbation equations (12.19) and (12.20) are completely determined bythe perturbation quantityξ and the original nonlinear PDE (12.7) so that we haveno freedom to choose either an initial solution better than (12.22), or a linear oper-


ator better thanLp[ f ] = f ′′′+

η2

f ′′−m f′, (12.26)

where the prime denotes the differentiation with respect toη .The skin friction coefficient given by the 1st-order of perturbation approximation

reads

Cxf (x,ξ )≈− 1

√

π ξ Rex

[

1+

(

54− 4

3π

)

ξ]

, (12.27)

which is not a good approximation in the whole time interval,as shown in Fig. 12.5.So, by means of perturbation approach, one can not gain accurate enough approxi-mation valid in the whole temporal and spatial interval for this unsteady boundary-layer flow.

12.3 Homotopy-series solution


As shown by Liao [13, 14], some unsteady similarity boundary-layer flows can besolved by means of the homotopy analysis method (HAM) [3–18,27]. Here, weillustrate that the unsteady boundary-layer flow, governedby the nonlinear PDE(12.7) with the boundary conditions (12.8), can be solved bymeans of theBVPh1.0in a similar way for steady similarity boundary-layer flows.

According to (12.7), we define a nonlinear operator

N [φ ] =∂ 3φ∂η3 +

12(1− ξ ) η

∂ 2φ∂η2 + ξ

[

φ∂ 2φ∂η2 −

(

∂φ∂η

)2]

− ξ (1− ξ )∂ 2φ

∂η∂ξ. (12.28)

Let f0(η ,ξ ) denote the initial guess off (η ,ξ ), q∈ [0,1] the embedding parameter,respectively. In the frame of the HAM, we should first construct such a kind ofcontinuous variation (deformation)φ(η ,ξ ;q) that, as the embedding parameterqincreases from 0 to 1,φ(η ,ξ ;q) varies (or deforms) continuously from the initialguessf0(η ,ξ ) to the solutionf (η ,ξ ). Such kind of continuous variationφ(η ,ξ ;q)is governed by the so-called zeroth-order deformation equation

(1−q)L [φ(η ,ξ ;q)− f0(η ,ξ )] = q c0 H(η) N [φ(η ,ξ ;q)] , (12.29)


φ(0,ξ ;q) = 0,∂φ(η ,ξ ;q)

∂η

∣

∣

∣

∣

η=0= 1,

∂φ(η ,ξ ;q)∂η

∣

∣

∣

∣

η=+∞= 0, (12.30)


whereL is an auxiliary linear operator with the propertyL [0] = 0, c0 6= 0 is theconvergence-control parameter,H(η) 6= 0 is an auxiliary function, respectively.Note that, unlike perturbation techniques, we have extremely large freedom tochooseL , c0 andH(η). Obviously, whenq= 0 andq= 1, we have

φ(η ,ξ ;0) = f0(η ,ξ ) (12.31)

andφ(η ,ξ ;1) = f (η ,ξ ), (12.32)

respectively. Thus, asq increases from 0 to 1,φ(η ,ξ ;q) indeed varies from theinitial approximationf0(η ,ξ ) to the solutionf (η ,ξ ) of the original equations (12.7)and (12.8).

Expandingφ(η ,ξ ;q) into the Maclaurin series with respect toq and then using(12.31), we have the homotopy-Maclaurin series

φ(η ,ξ ;q) = f0(η ,ξ )++∞

∑n=1

fn(η ,ξ ) qn, (12.33)

where

fn(η ,ξ ) = Dn [φ(η ,ξ ;q)] =1n!

∂ nφ(η ,ξ ;q)∂qn

∣

∣

∣

∣

q=0(12.34)

andDn is thenth-order homotopy-derivative operator.As mentioned above, we have extremely large freedom to choose the initial guess

f0(η ,ξ ), the auxiliary linear operatorL , and especially the convergence-controlparameterc0. Assuming that all of them are so properly chosen that the homotopy-Maclaurin series (12.33) absolutely converges atq = 1, we have from (12.32) thehomotopy-series solution

f (η ,ξ ) = f0(η ,ξ )++∞

∑n=1

fn(η ,ξ ). (12.35)

According to Theorem 4.15, we have themth-order deformation equation

L [ fm(η ,ξ )− χm fm−1(η ,ξ )] = c0 H(η) δm−1(η ,ξ ), (12.36)


fm(0,ξ ) = 0,∂ fm(η ,ξ )

∂η

∣

∣

∣

∣

η=0= 0,

∂ fm(η ,ξ )∂η

∣

∣

∣

∣

η=+∞= 0, (12.37)

where

χn =

1, n> 1,0, n= 1,

(12.38)

and


δk(η ,ξ ) = DkN [φ(η ,ξ ;q)]

=∂ 3 fk∂η3 +

12(1− ξ )η

∂ 2 fk∂η2 − ξ (1− ξ )

∂ 2 fk∂η∂ξ

+ ξk

∑n=0

[

fk−n∂ 2 fn∂η2 − ∂ fk−n

∂η∂ fn∂η

]

(12.39)

is given by Theorem 4.1. For details, please refer to Chapter4 and Liao [14].Since the high-order deformation equations (12.36) and (12.37) are linear, the

original nonlinear problem is transferred into an infinite number of linear sub-problems. However, unlike perturbation approach mentioned above, such kind oftransformation doesnot need any small perturbation quantities. More importantly,we have extremely large freedom to choose the initial guessf0(η ,ξ ), the auxil-iary linear operatorL , the auxiliary functionH(η), and the convergence-controlparameterc0: it is due to such kind of freedom that we can gain accurate analyticapproximations valid in the whole spatial and temporal interval, as shown below.

In general, a continuous function can be approximated by different base func-tions. Mathematically, according to the above discussionsabout the failure of theperturbation approach,f (η ,ξ ) should not contain the complementary error functionerfc(η/2) and the Gaussian distribution function exp(−η2/4), otherwise it is verydifficult to solve the high-order deformation equations (12.36) and (12.37). Phys-ically, at arbitrary timeτ ∈ [0,+∞), corresponding toξ ∈ [0,1], (12.7) and (12.8)describe a similarity boundary-layer flow, which decays exponentially asη →+∞.Therefore, it is reasonable to assume thatf (η ,ξ ) could be expressed by the basefunctions

ξ k ηm exp(−nη) | k≥ 0,m≥ 0,n≥ 1

(12.40)

in the form

f (η ,ξ ) = a0,0(ξ )++∞

∑m=0

+∞

∑n=1

am,n(ξ ) ηm exp(−nη), (12.41)

wheream,n(ξ ) is a polynomial ofξ to be determined. It provides us the so-calledsolution-expressionof f (η ,ξ ), which plays an important role in the frame of theHAM, as shown below.

Unlike perturbation methods, we have now extremely large freedom to choosethe initial guessf0(η ,ξ ), the auxiliary linear operatorL and the auxiliary functionH(η) in the zeroth-order deformation equation: all of them should be chosen insuch a way that the high-order deformation equations (12.36) and (12.37) are easyto solve, and besides that the homotopy-series (12.35) converges in the whole spatialand temporal interval.

The initial guessf0(η ,ξ ) should satisfy the boundary conditions (12.8) and obeythe solution-expression (12.41). Therefore, we choose theinitial guess

f0(η ,ξ ) = 1−exp(−η). (12.42)


Note that, this initial guess exactly satisfies the governing equation (12.7) atξ = 1,corresponding toτ → +∞, although it is not a good approximation off (η ,ξ ) atξ = 0.

Note that the governing equation (12.7) contains a linear operator

L0[ f ] =∂ 3 f∂η3 +

12(1− ξ ) η

∂ 2 f∂η2 − ξ (1− ξ )

∂ 2 f∂η∂ξ

.

However, if we choose the above linear operator as the auxiliary linear operatorL , the high-order deformation equation (12.36) becomes a PDEwith the variablecoefficients

12(1− ξ ) η , −ξ (1− ξ ),

and thus is rather difficult to solve. If we chooseLp defined by (12.26) as the aux-iliary linear operator, the high-order deformation equation is indeed an ODE, butits solution contains the complementary error function erfc(η/2) and the Gaussiandistribution function exp(−η2/4), which disobeys the solution expression (12.41)of f (η ,ξ ). So, neitherL0 nor Lp is a good choice for us. Fortunately, the HAMprovides us extremely large freedom to choose the auxiliarylinear operatorL . Notethat, whenξ = 1, the corresponding steady-state similarity boundary-layer flow issolved in§ 10.2 by means of the auxiliary linear operator

L u=∂ 3u∂η3 − ∂u

∂η, (12.43)

which has the property

L [C1+C2exp(−η)+C3exp(η)] = 0. (12.44)

Besides, for an arbitrary polynomialb(ξ ) of ξ , the linear ODE

∂ 3u∂η3 − ∂u

∂η= b(ξ ) ηm exp(−nη)

can be solved quickly by computer algebra system like Mathematica. So, if wechoose (12.43) as the auxiliary linear operator, the high-order deformation equation(12.36) can be easily solved: it is quite interesting that the accurate approximationsvalid in the whole temporal and spatial interval can be obtained even by such asimple auxiliary linear operator, as shown below.

Therefore, thanks to such kind of freedom provided by the HAM, we simplychoose (12.43) as the auxiliary linear operatorL . Let

f ∗m(η ,ξ ) = χm fm−1(η ,ξ )+ c0 L−1 [H(η) δm−1(η ,ξ )]

denote a special solution of (12.36), whereL −1 is the inverse operator of the aux-iliary linear operatorL defined by (12.43). According to the property (12.44), itscommon solution reads


fm(η ,ξ ) = f ∗m(η ,ξ )+C1+C2exp(−η)+C3exp(η),

where the coefficientsC1,C2, andC2 are uniformly determined by the boundaryconditions (12.37).

Note that the auxiliary linear operator (12.43) has nothingto do with the temporalvariableξ . Besides, the right-hand side termδm−1(η ,ξ ) of themth-order deforma-tion equation (12.36) is always known. Mathematically, themth-order deformationequation (12.36) is a linear ODE, and thus can be solved in a similar way like thesteady-state similarity boundary-layer flows described in§ 10.2, since the tempo-ral variableξ can be regarded as a constant. The only difference is thatξ must beregarded as a variable when calculating the termδm−1(η ,ξ ) by means of (12.39).In this way, it is rather easy to solve thelinear high-order deformation equations(12.36) and (12.37), successively, especially by means of computer algebra systemMathematica.

In essence, the above approach transfers a nonlinear PDE into an infinite num-ber of linear ODEs. More importantly, unlike perturbation techniques, the HAMprovides us great freedom to choose the initial guessf0(η ,ξ ) and the auxiliary lin-ear operatorL so that the high-order approximations are easy to obtain by meansof computer algebra system like Mathematica. In this way, wegreatly simplifiessolving the original nonlinear PDE, and thus can obtain accurate analytic approxi-mations valid in the whole spatial and temporal interval, asshown below.

12.3.2 Homotopy-approximation

As mentioned above, in the frame of the HAM, the original nonlinear PDE (12.7)describing the unsteady boundary-layer flow is transferredinto an infinite number oflinear ODEs in a similar way like the steady-state boundary-layer flow consideredin § 10.2. Note that, the high-order deformation equation (12.36) for the unsteadyflow is quite similar to the high-order deformation equation(10.11) for the steady-state flow in§ 10.2: the auxiliary linear operator (12.43) for theunsteadyflow isexact thesameas the auxiliary linear operator (10.16) for thesteady-stateflow!Therefore, like the non-similarity boundary-layer flow mentioned in Chapter 11, theunsteady boundary-layer flow governed by the nonlinear PDE (12.7) can be alsosolved by means of theBVPh1.0, which is given in Chapter 7 and free available athttp://numericaltank.sjtu.edu.cn/BVPh.htm. The corresponding input data is givenin the Appendix of this chapter and free available at the above website.

Note that we have great freedom to choose the auxiliary function H(η). Forsimplicity, let us choose

H(η) = 1.

Then, fm(η ,ξ ) contains the so-called convergence-control parameterc0 only, whichhas no physical meanings but provides us a convenient way to guarantee the con-vergence of the homotopy-series solution (12.35). The averaged squared residualEm of the governing equation (12.7) at themth-order homotopy-approximation is


Fig. 12.1 Averaged squaredresidual of governing equationat themth-order homotopyapproximation versusc0by means of the auxiliaryfunction H(η) = 1 and theinitial guess (12.42). Dashedline: m= 5; Dash-dotted line:m= 10; Dash-dot-dotted line:m= 15; Solid line:m= 20.

c0

Ave

rage

dsq

uare

dre

sidu

al

-1 -0.8 -0.6 -0.4 -0.2 010-6

10-5

10-4

10-3

10-2

10-1

100

Table 12.1 Averaged squared residual of the governing equation (12.7)over η ∈ [0,10] andξ ∈[0,1] by means ofc0 =−1/4 andH(η) = 1

Order of approximation Averaged squared residual of (12.7)

1 6.5×10−3

3 2.5×10−3

5 9.5×10−4

10 9.6×10−5

15 1.1×10−5

20 1.4×10−6

25 3.5×10−7

gained by first integrating the squared residual of (12.7) inthe intervalη ∈ [0,10]by means of the moduleGetErr[m] of the Mathematica packageBVPh (version1.0), which is then further integrated in the intervalξ ∈ [0,1]. The curves of the av-eraged squared residualEm versusc0 are as shown in Fig. 12.1. It indicates that, atthe 20th-order homotopy-approximation, the optimal convergence-control parame-ter c∗0 is about - 0.3. It is found that, by means ofc0 = −1/4, the corresponding25th-order homotopy-approximation is accurate enough with the averaged squaredresidual 3.5×10−7, as shown in Table 12.1.

It is found that, whenc0 =−1/4, the homotopy-approximationsoff ′′(0,0) agreewell with the exact resultf ′′(0,0) =−1/

√π ≈ −0.56419, as shown in Table 12.2,

where the prime denotes the differentiation with respect toη . Besides, the accu-racy of the homotopy-approximationf ′′(0,0) is greatly modified by means of theso-called Homotopy-Pade method [10], as shown in Table 12.3. Furthermore, the20th-order homotopy-approximation and its[3,3] homotopy-Pade approximation ofthe velocity profilef ′(η ,0) agree well with the exact solution (12.11) in the wholeregion 0≤ η < +∞, as shown in Fig. 12.2. This indicates that the initial solution(12.11), which contains the complementary error function erfc(η/2) and the Gaus-sian distribution function exp(−η2/4), can be well approximated by the base func-tions (12.40).


Table 12.2 The homotopy-approximations off ′′(0,0) by means ofc0 =−1/4 andH(η) = 1.

Order of approximation f ′′(0,0)

5 -0.6930310 -0.6011415 -0.5744020 -0.5669325 -0.5649130 -0.5643835 -0.5642440 -0.5642045 -0.5641950 -0.56419

Table 12.3 The[m,m] homotopy-Pade approximations off ′′(0,0) by means ofH(η) = 1.

m f ′′(0,0)

5 -0.5641510 -0.5641815 -0.5641920 -0.5641925 -0.5641930 -0.56419

Fig. 12.2 The comparison ofthe exact solution (12.11) withthe 20th-order homotopy-approximation atξ = 0 bymeans ofc0 =−1/4, H(η) =1 and its[3,3] homotopy-Padeapproximation. Solid line:exact solution (12.11); Opencircles: 20th-order homotopy-approximation atξ = 0; Filledcircles: the [3,3] homotopy-Pade approximation.

η

f’(η

,0)

0 1 2 3 4-0.2

0

0.2

0.4

0.6

0.8

1

H(η) = 1

Note thatf ′′(0,ξ ) is related to the skin friction. It is found that, whenc0 =−1/4,the 20th-order homotopy-approximation off ′′(0,ξ ) agrees well with the 30th-orderone in the whole regionξ ∈ [0,1], as shown in Fig. 12.3. Besides, the correspondingvelocity profiles are accurate in the whole intervalξ ∈ [0,1] and 0≤ η < +∞, asshown in Fig. 12.4. Note that, the velocity profile varies smoothly asτ increase from0 to+∞.

The local skin friction is related tof ′′(0,ξ ). The 30th-order homotopy approxi-mation of f ′′(0,ξ ) reads


f ′′(0,ξ )= −0.5643747892−0.4653303619ξ +2.998049008×10−2 ξ 2

−2.518392990×10−3 ξ 3−2.561860658×10−5 ξ 4

−2.531901893×10−5 ξ 5+3.073353805×10−5 ξ 6

+5.063224875×10−5 ξ 7+5.780083670×10−5 ξ 8

−3.019750875×10−4 ξ 9+0.2746188078ξ 10−48.017634463ξ 11

+3.4358227736×103 ξ 12−1.2769915485×105 ξ 13

+2.8257114566×106 ξ 14−4.0636817506×107 ξ 15

+4.0325552732×108 ξ 16−2.8809318182×109 ξ 17

+1.5277371296×1010ξ 18−6.1471279622×1010 ξ 19

+1.9058081506×1011ξ 20−4.5980815420×1011 ξ 21

+8.6766946108×1011ξ 22−1.2809186178×1012 ξ 23

+1.4720293398×1012ξ 24−1.3019682527×1012 ξ 25

+8.6859786813×1011ξ 26−4.2255442112×1011 ξ 27

+1.4139635601×1011ξ 28−2.9087215325×1010 ξ 29

+2.7723411925×109 ξ 30. (12.45)

The corresponding local skin friction at the dimensionlesstimeτ ∈ [0,+∞) is shownin Fig. 12.5. Using the first four-terms of (12.45), we have the simplified local skinfriction formula

Cxf (x,ξ ) = (ξ Rex)

−1/2 (−0.5643747892−0.4653303619ξ

+2.998049008×10−2 ξ 2−2.518392990×10−3 ξ 3) , (12.46)

which agrees well with the exact 30th-order homotopy-approximation in thewholetime interval 0≤ τ < +∞, as shown in Fig. 12.5. Thus, by means of a properconvergence-control parameterc0, we can obtain accurate analytic approximationof the unsteady boundary-layer flow governed by the nonlinear PDE (12.7), whichis uniformly valid not only in the whole temporal interval 0≤ τ < +∞ but also inthe whole spatial interval 0≤ η < +∞. To the best of our knowledge, such a kindof simple and accurate analytic approximation of the skin friction for the unsteadyboundary-layer flow has never been reported. This verifies once again the validityof the HAM for complicated nonlinear problems.

Note that all of the above homotopy-approximationsare obtained by means of thesimplest auxiliary functionH(η) = 1. However, in the frame of the HAM, we havegreat freedom to choose the auxiliary functionH(η). For example, when we chooseH(η) = exp(−η), the corresponding averaged squared residual of the governingequation (12.7) at the 20th-order of approximation has the minimum 1.0× 10−4,corresponding to the optimal convergence-control parameterc∗0 = −0.44, as shownin Fig. 12.6. Indeed, using theBVPh 1.0 as a tool, we can gain accurate analytic


Fig. 12.3 The homotopy-approximations off ′′(0,ξ )by means ofc0 = −1/4and H(η) = 1. Solid line:30th-order approximation;Symbols: 20th-order approxi-mation.

ξ

f’’(0

,ξ)

0 0.2 0.4 0.6 0.8 1-1

-0.9

-0.8

-0.7

-0.6

-0.5

Fig. 12.4 The velocity pro-files by means ofc0 = −1/4and H(η) = 1 at differentdimensionless timeτ = a t.Solid line:τ = 0.01; Dashedline: τ = 0.1; Dash-dottedline: τ = 0.25; Long-dashedline: τ = 1; Dash-dot-dottedline: τ = 10.

f ’(η,ξ)

y(

a/ν

)1

/2

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

a t = 0.01, 0.1, 0.25, 1, 10

approximation by means ofH(η) = exp(−η) and c0 = −2/5, too. This verifiesonce again the validity and flexibility of theBVPh1.0.


We illustrate in this chapter that, in the frame of the HAM, the nonlinear PDE (12.7)describing an unsteady boundary-layer flow can be transferred into an infinite num-ber of linear ODEs, and thus can be solved by means of theBVPh 1.0 in a rathersimilar way to that for steady-state similarity ones. In other words, by means of theHAM, solving unsteady boundary-layer flows is as easy as steady-state ones. Thisverifies the validity of theBVPh1.0 for some nonlinear PDEs, especially for thoserelated to boundary-layer flows.

Note that, one can not gain such kind of accurate approximations by means ofperturbation techniques, although the original nonlinearPDE (12.7) can be also


Fig. 12.5 The comparisonof the four-term approxi-mation (12.46) ofCx

f

√Rex

with the exact 30th-orderhomotopy-approximation(12.45) and the perturbationapproximation (12.27). Solidline: the exact 30th-orderhomotopy-approximation(12.45); Dashed line: theperturbation approximation(12.27); Symbols: the simpli-fied homotopy-approximation(12.46).

τ

Cx f

Re x1

/2

0 2 4 6 8 10-1.2

-1.1

-1

-0.9

Fig. 12.6 The averagedsquared residual of the gov-erning equation (12.7) overη ∈ [0,10] andξ ∈ [0,1] bymeans ofH(η) = exp(−η).

c0

Ave

rage

dsq

uare

dre

sidu

al

-1 -0.8 -0.6 -0.4 -0.2 0

10-4

10-3

10-2

10-1

H(η) = exp(-η)

transferred into an infinite number of linear ODEs by regarding ξ as a small per-turbation quantity. This is mainly because, using perturbation methods, we havenofreedom to choose the initial approximation (12.22) and thecorresponding linear op-erator (12.26) in the high-order perturbation equations sothat these high-order per-turbation equations are rather difficult to solve. However,unlike perturbation tech-niques, the HAM provides us extremely large freedom to choose the initial guessf0(η ,ξ ) and the auxiliary linear operatorL , therefore we can choose such a sim-ple initial guess (12.42) and such a simple auxiliary linearoperator (12.43) that thecorresponding high-order deformation equation (12.36) can be easily solved. Moreimportantly, in the frame of the HAM, the accuracy of the high-order homotopy-approximations is guaranteed by means of the convergence-control parameterc0.So, unlike perturbation methods, the HAM is valid for more complicated nonlinearproblems.

As illustrated in Chapter 11, in the frame of the HAM, a nonlinear PDE de-scribing a steady-statenon-similarityboundary-layer flow can be transferred into aninfinite number of linear ODEs governed by the auxiliary linear operator


L f =∂ 3 f∂η3 −

∂ f∂η

. (12.47)

Here, we further show that, in the frame of the HAM, the nonlinear PDE describingan unsteadysimilarity boundary-layer flow can be also transferred intoan infinitenumber of linear ODEs, which are governed by thesameauxiliary linear operatoras above! It should be emphasized that the above auxiliary linear operator is alsoexactly thesameas the auxiliary linear operator (10.16) used in§ 10.2 to solve a kindof state-state similarityboundary-layer flow with exponentially decaying solutions!Notice that the boundary-layer flows considered in these three chapters are quitedifferent, not only physically but also mathematically. However, using theBVPh1.0,all of them can be solved by thesameauxiliary linear operator (12.47) in the frameof the HAM! These show the generality of the HAM andBVPh1.0.

Therefore, theBVPh1.0 can be used as a tool to solve many nonlinear PDEs ina similar way, especially those related to unsteady and/or non-similarity boundary-layer flows and heat transfer, although we should keep in mindthat it does not workfor all nonlinear PDEs.

Indeed, it might be impossible to develop an analytic approach valid forall non-linear boundary-value problems. However, as pointed out byRabindranth Tagore(1861 – 1941), “if you shut your door to all errors, truth willbe shut out”. There-fore, our strategy is to develop an analytic approach valid for as many nonlinearboundary-value problems as possible. TheBVPh1.0 and its successful applicationsin some nonlinear ODEs and PDEs mentioned in Part II suggest that such a strategyshould have a good prospect, although further modificationsand more applicationsare needed for the development of the Mathematica packageBVPh.

Anyway, we start out on a promising march, no matter how far wecan go!

Appendix 12.1 Input data ofBVPh 425

Appendix 12.1 Input data ofBVPh


( * Define the physical and control parameters * )TypeEQ = 1;ApproxQ = 0;ErrReq = 10ˆ(-10);NgetErr = 100;zRintegral = 10;

( * Define the governing equation * )f[z_,u_,lambda_] := Module[temp,temp[1] = D[u,z,3] + (1-t) * z/2 * D[u,z,2]

- t * (1-t) * D[u,z,t];temp[2] = u * D[u,z,2] - D[u,z]ˆ2;temp[1] + t * temp[2] // Expand];

( * Define boundary conditions * )zR = infinity;OrderEQ = 3;BC[1,z_,u_,lambda_] := Limit[u, z -> 0];BC[2,z_,u_,lambda_] := Limit[D[u,z] - 1, z -> 0];BC[3,z_,u_,lambda_] := Limit[D[u,z] , z -> zR ];

( * Define initial guess * )u[0] = 1 - Exp[-z];

( * Define the auxiliary linear operator * )L[u_] := D[u,z,3] - D[u,z];


D[u[k],z,2]/.z->0,t->0//N];

( * Define Getdelta[k] * )Getdelta[k_]:=Module[temp,i,uz[k] = D[u[k],z]//Expand;uzz[k] = D[uz[k],z]//Expand;uzzz[k] = D[uzz[k],z]//Expand;uzuz[k] = Sum[uz[i] * uz[k-i],i,0,k]//Expand;uzzu[k] = Sum[uzz[i] * u[k-i],i,0,k]//Expand;uzt[k] = D[uz[k],t]//Expand;temp[1] = uzzz[k] + (1-t) * z/2 * uzz[k]

- t * (1-t) * uzt[k]//Expand;temp[2] = t * (uzzu[k] - uzuz[k]);delta[k] = temp[1] + temp[2]//Expand;];


( * Print input and control parameters * )PrintInput[u[z,t]];

( * Set convergence-control parameter c0 * )c0 = -1/4;Print["c0 = ",c0];

( * Gain 10th-order HAM approximation * )Print[" c0 = ",c0];BVPh[1,10];

( * Calculate the squared residual * )For[k=2, k<=10, k=k+2,GetErr[k];err[k] = Integrate[Err[k],t,0,1];Print[" k = ", k, " Squared residual = ",

err[k]//N];];


Note that, for nonlinear boundary-value problems defined inan infinite interval,we must setzR = infinity in the input data ofBVPh (version 1.0). Besides,we must define the moduleGetdelta in the input date, when a nonlinear PDE issolved by means of theBVPh(version 1.0).

References 427

References

1. Dennis, S.C.R.: The motion of a viscous fluid past an implusively started semi-infinite flatplate. J. Inst. Math. Its Appl.10, 105 – 117 (1972)

2. Hall, M.G.: The boundary layer over an impulsively started flat plate. Proc. R. Soc. A.310,401-414 (1969)












14. Liao, S.J.: An analytic solution of unsteady boundary-layer flows caused by an impulsivelystretching plate. Communications in Nonlinear Science andNumerical Simulation.11, 326 –339 (2006)





19. Nazar, N., Amin, N., Pop, I.: Unsteady boundary layer flowdue to stretching surface in arotating fluid. Mechanics Research Communications.31, 121-128 (2004)

20. Pop, I., Na, T.Y.: Unsteady flow past a stretching sheet. Mech. Research Comm.23, 413(1996)

21. Seshadri, R., Sreeshylan, N., Nath, G.: Unsteady mixed convection flow in the stagnationregion of a heated vertical plate due to impulsive motion. Int. J. Heat and Mass Transfer.45,1345-1352 (2002)

22. Stewartson, K.: On the impulsive motion of a flat plate in aviscous fluid (Part I). Quart. J.Mech. Appl. Math.4, 182-198 (1951)

23. Stewartson, K.: On the impulsive motion of a flat plate in aviscous fluid (Part II). Quart. J.Mech. Appl. Math.22, 143-152 (1973)

24. Wang, C.Y., Du, G., Miklavcic, M., Chang, C.C.: Impulsive stretching of a surface in a viscousfluid. SIAM J. Appl. Math.57, 1 (1997)


25. Watkins, C.B.: Heat transfer in the boundary layer over an implusively started flat plate. J.Heat Transfer.97, 482-284 (1975)

26. Williams, J.C., Rhyne, T.H.: Boundary layer development on a wedge impulsively set intomotion. SIAM J. Appl. Math.38, 215-224 (1980)


Part IIIApplications in Nonlinear Partial

Differential Equations

“The small truth has words which are clear; the great truth hasgreat silence.”

by Rabindranth Tagore (1861 – 1941)

Chapter 13Applications in finance: American put options

Abstract The homotopy analysis method (HAM) is successfully combined withthe Laplace transform to solve the famous American put option in finance. Unlikeasymptotic and/or perturbation formulas that are often valid only a couple of days orweeks prior to expiry, our homotopy-approximation of the optimal exercise bound-ary B(τ) in polynomials of

√τ to o(τM) may be valid a couple of dozen years,

or even a half century, as long asM is large enough. It is found that the homotopy-approximation ofB(τ) in polynomial of

√τ to o(τ48) is often valid in so many years

that the well-known theoretical perpetual optimal exercise price is accurate enoughthereafter, so that the combination of them can be regarded as an analytic formulavalid in the whole time interval 0≤ τ < +∞. A practical Mathematica codeAPOhis provided in the Appendix 13.2 for businessmen to gain accurate enough optimalexercise price of American put option at large expiration-time by a laptop only in afew seconds, which is free available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

13.1 Mathematical modeling

There are many nonlinear partial differential equations (PDEs) in the field of fi-nance. Traditionally, analytic approximations of these nonlinear PDEs are given byasymptotic methods. However, most of these asymptotic approximations are validonly for small time and/or some small quantities. In this chapter, we illustrate that,by means of the homotopy analysis method (HAM) [28–42,44] , we can gain muchbetter analytic approximations of some nonlinear PDEs in finance.

Let us consider a famous problem in finance: an American put option with strikeprice X that expires at timeT. Let V(S, t) denote the value of an American putoption,S the price of the underlying asset andt the time, respectively. Besides, letσ denote the volatility of the underlying,r the risk-free interest rate, respectively,and both of them are constants. At any moment, there exists anoptimal exerciseboundaryB(t) such that it is optimal to exercise the put option whenS is at or belowB(t). Hence, whenS≤ B(t), the put option is of value

431

432 13 Applications in finance: American put options

V(S, t) = X−S, (13.1)

whereX is the strike price. WhenS> B(t), V(S, t) satisfies the famous Black-Scholes equation

∂V∂ t

+12

σ2 S2 ∂ 2V∂S2 + r S

∂V∂S

− r V = 0, (13.2)

subject to the smooth pasting conditions at the exercise boundaryB(t):

limS→B(t)

V(S, t) = X−B(t), limS→B(t)

∂V∂S

(S, t) =−1, (13.3)

the upper boundary condition:

limS→+∞

V(S, t)→ 0. (13.4)

and the terminal condition:

limt→T

V(S, t) = maxX−S,0, (13.5)

respectively.Define the variableτ ≡ T − t. Whenr > 0, the terminal condition (13.5) can be

further simplified as

limτ→0

V(S,τ) = 0. (13.6)

in the rangeΣ1 = (S,τ)| B(τ)≤ S<+∞, 0≤ τ ≤ T.

Kim [23] and Carr et al. [9] derived the formula

V(S,τ) =VE(S,τ)+X∫ τ

0r exp(−rξ )N(−dξ ,2)dξ (13.7)

for the option price, where

VE(S,τ) = X exp(−rτ)N(−d2)+S N(−d1)

is the price of the European put option with the following definitions

d1 =ln(S/X)+ (r +σ2/2)τ

σ√

τ,

d2 = d1−σ√

τ,

dξ ,1 =ln[S/B(τ − ξ )]+ (r +σ2/2)ξ

σ√

ξ,

dξ ,2 = dξ ,1−σ√

ξ ,

13.1 Mathematical modeling 433

andN(x) was a cumulative distribution function for a standardized normal randomvariable defined by

N(x) =1√2π

∫ x

−∞exp

(

−w2

2

)

dw. (13.8)

Hence, by means of (13.7), it is very easy to gain the option priceV(S,τ), as longas the optimal exercise boundaryB(τ) is known. So, the optimal exercise boundaryB(τ) is the key point of this problem.

Due to the existence of the unknown moving boundaryB(τ), this problem is non-linear in essence, although the governing equation (13.2) is linear. The above PDEwith a unknown moving boundary can be solved by means of numerical methods,such as the binomial/trinomial methods [6, 15], the Monte Carlo simulation [19],the least squares method [27], the variational inequalities [16, 22], and the tech-niques based on solving PDEs [1, 5, 7, 11, 20, 43]. However, since most of marketpractitioners are not familiar with numerical methods, analytical approximations areextremely valuable in practice and theory.

Most of traditional analytic approaches are based on the perturbation or asymp-totic methods, such as Barles [3], Kuske and Keller [25], Alobaidi and Mallier [2],Evans, Kuske and Keller [17], Zhang and Li [45] and Knessl [24]. As reported byChen et al. [11] and Chen and Chadam [12], all of these approximations were validfor very short time prior to expiry, usually on the order of days and weeks.

Zhu [46] was the first who applied the HAM to the American put option in 2006.Using the Landau transform [26] and the HAM, Zhu [46] gave a solution in theform of infinite recursive series involving double integrals. With a 30th-order ap-proximation through numerical integration, Zhu [46] numerically demonstrated theconvergence of his results. This is a remarkable contribution to find an analytic for-mula without extra parameters involved. Besides, Zhu [47] also applied the HAMto gain an analytical solution for the valuation of convertible bonds with constantdividend yield. In 2010, Cheng, Zhu and Liao [14] further applied the HAM to givean explicit analytic approximation of the optimal exerciseboundaryB(τ) in poly-nomial of

√τ to o(τ6), which is valid in a much longer time prior to expiry, usually

on the order of years, and is as accurate as many numerical results. Their approachis based on the Laplace transform and has nothing to do with the Landau trans-form [26]. By means of the HAM, Cheng [13] gave an analytic approximation ofB(τ) in the polynomial of

√τ to o(τ7). These successful applications in American

put option show the potential and validity of the HAM in finance.In this chapter, we further modify the HAM-based approach ofCheng, Zhu and

Liao [14] and give more accurate explicit expressions of theoptimal exercise bound-ary B(τ). Especially, we investigate the influence of the ordero(τM) of the optimalexercise boundaryB(τ) in polynomials of

√τ up to o(τM), and illustrate that the

maximum valid time ofB(τ) given by the HAM is directly proportional to the ordero(τM). Therefore, whenM is large enough,B(τ) given by the HAM can be valideven up to a half century prior to expiry, which is about 1000 times longer thanthose given by perturbation or asymptotic methods! Based onthis explicit approx-


imations, a short Mathematica codeAPOhis given in the appendix of this chapterand available athttp://numericaltank.sjtu.edu.cn/HAM.htm, which can be used bybusinessmen to gain, only in a few seconds by a laptop, accurate optimal exerciseprice of American put option for a rather large expiry time.


Whenσ 6= 0, we introduce the following dimensionless variables

V∗ =VX, S∗ =

SX, τ∗ =

σ2

2τ, γ =

2rσ2 . (13.9)

Dropping all stars from now on, the dimensionless governingequation becomes

−∂V∂τ

+S2 ∂ 2V∂S2 + γ S

∂V∂S

− γ V = 0, (13.10)


V(B(τ),τ) = 1−B(τ), (13.11)

∂V∂S

(B(τ),τ) = −1, (13.12)

V(S,τ) → 0 as S→+∞, (13.13)

V(S,0) = 0. (13.14)

Let V0(S,τ) andB0(τ) denote the initial approximations ofV(S,τ) and the opti-mal exercise boundaryB(τ), respectively. Letq∈ [0,1] denote the embedding pa-rameter. In the frame of the HAM, we first construct such two continuous variations(or deformations)φ(S,τ;q) andΛ(τ;q) that, asq increases from 0 to 1,φ(S,τ;q)varies continuously from the initial guessV0(S,τ) to the solutionV(S,τ), so doesΛ(τ;q) from the initial guessB0(τ) to the optimal exercise boundaryB(τ), respec-tively. Such kind of continuous variations are governed by the zeroth-order defor-mation equation

−∂φ(S,τ;q)∂τ

+S2 ∂ 2φ(S,τ;q)∂S2 + γ S

∂φ(S,τ;q)∂S

− γ φ(S,τ;q) = 0 (13.15)

defined in the domain

Λ(τ;q) ≤ S<+∞, 0≤ τ ≤ τexp,

subject to the initial/boundary conditions

φ(S,0;q) = 0, (13.16)


∂φ(S,τ;q)∂S

= −1 atS= Λ(τ;q), (13.17)

φ(+∞,τ;q) → 0, (13.18)

and

(1−q) [ Λ(τ;q)−B0(τ) ] = c0 q Λ(τ;q)+φ [Λ(τ;q),τ;q]−1 , (13.19)

wherec0 6= 0 is a convergence-control parameter and

τexp=12

σ2 T

is the dimensionless expiring time.Whenq= 1, the zero-order deformation equations (13.15) to (13.19)are equiv-

alent to the original equations (13.10 ) to (13.14), thus

φ(S,τ;1) =V(S,τ), Λ(τ;1) = B(τ). (13.20)

Whenq= 0, we have from (13.19) that

Λ(τ;0) = B0(τ), (13.21)

and from (13.15) to (13.18 ), we further have the governing equation

−∂V0(S,τ)∂τ

+S2 ∂ 2V0(S,τ)∂S2 + γ S

∂V0(S,τ)∂S

− γ V0(S,τ) = 0, (13.22)


V0(S,0) = 0, (13.23)

∂V0(S,τ)∂S

= −1 atS= B0(τ), (13.24)

V0(+∞,τ) → 0, (13.25)

where

V0(S,τ) = φ(S,τ;0). (13.26)

For the sake of simplicity, we choose

B0(τ) = 1

as the initial guess of the optimal exercise boundaryB(τ). Then, the initial guessV0(S,τ) is governed by the linear PDE (13.22), subject to the linear initial condi-tion (13.23), the linear boundary condition (13.24) at afixedboundaryS= 1, andthe linear boundary condition (13.25) at infinity. Therefore, the zero-order defor-mation equations (13.15) to (13.19) indeed construct two continuous variations (or


mappings)φ(S,τ;q) andΛ(τ;q), which are two homotopies

φ(S,τ;q) : V0(S,τ)∼V(S,τ), Λ(τ;q) : B0(τ) ∼ B(τ)

in topology.Expandingφ(S,τ;q) andΛ(τ;q) in Maclaurin series with respect toq ∈ [0,1],

we obtain the homotopy-Maclaurin series

φ(S,τ;q) = V0(S,τ)+∞

∑n=1

Vn(S,τ) qn, (13.27)

Λ(τ;q) = B0(τ)+∞

∑n=1

Bn(τ) qn, (13.28)

where

Vn(S,τ) =1n!

∂ nφ(S,τ;q)∂qn

∣

∣

∣

∣

q=0= Dn [φ(S,τ;q)] ,

Bn(τ) =1n!

∂ nΛ(τ;q)∂qn

∣

∣

∣

∣

q=0= Dn [Λ(τ;q)] , (13.29)

are thenth-order homotopy-derivatives ofφ(S,τ;q) andΛ(q), respectively, andDn

is thenth-order homotopy-derivative operator. Note that the zeroth-order deforma-tion equation contains the convergence-control parameterc0. Assuming thatc0 ischosen properly so that the above series are absolutely convergent atq= 1, we havethe homotopy-series solution

V(S,τ) = V0(S,τ)+∞

∑n=1

Vn(S,τ), (13.30)

B(τ) = B0(τ)+∞

∑n=1

Bn(τ). (13.31)

The equations forVn(S,τ) andBn(τ) can be derived directly from the zeroth-order deformation equations (13.15) to (13.19). Substituting the series (13.27) and(13.28) into (13.15), (13.16) and (13.18), then equating the like-power ofq, we havethe so-callednth-order deformation equation (n≥ 1)

−∂Vn(S,τ)∂τ

+S2 ∂ 2Vn(S,τ)∂S2 + γ S

∂Vn(S,τ)∂S

− γ Vn(S,τ) = 0, (13.32)


Vn(S,0) = 0, (13.33)

Vn(+∞,τ) → 0. (13.34)

For details about above formulas, please refer to Cheng, Zhuand Liao [14].


Note that (13.17) and (13.19) are defined at the moving boundary S= Λ(τ;q),which itself is dependent uponq, so that (13.27) is invalid for them. Cheng, Zhu andLiao [14] developed a Mathematica code to gain the Maclaurinseries ofφ(S,τ;q)on the moving boundaryS= Λ(τ;q) with respect toq. Unlike Cheng, Zhu andLiao [14], we give in this chapter theexplicit expression of it.

Let the prime denote the differentiation with respect toS. On the moving bound-aryS=Λ(τ;q), using (13.28) and expandingφ(S,τ;q), φ ′(S,τ;q) into the Maclau-rin series with respect toq, we have

φ(S,τ;q) = V0(B0,τ)++∞

∑n=1

[Vn(B0,τ)+ fn(τ)]qn (13.35)

and

∂φ(S,τ;q)∂S

= V ′0(B0,τ)+

+∞

∑n=1

[

V ′n(B0,τ)+gn(τ)

]

qn (13.36)

where

fn(τ) =n−1

∑j=0

α j ,n− j(τ), gn(τ) =n−1

∑j=0

β j ,n− j(τ), (13.37)

with the following explicit definitions

αn,i(τ) =i

∑m=1

ψn,m(τ) µm,i(τ), i ≥ 1, (13.38)

βn,i(τ) =i

∑m=1

(m+1)ψn,m+1(τ) µm,i(τ), i ≥ 1, (13.39)

ψn,0(τ) = Vn(B0,τ), ψn,m(τ) =1m!

∂ mVn(S,τ)∂Sm

∣

∣

∣

∣

S=1, (13.40)

and the recursion formula

µ1,n(τ) = Bn(τ), µm+1,n(τ) =n−1

∑i=m

µm,i(τ) Bn−i(τ). (13.41)

The detailed derivation of the above explicit expressions is given in Appendix 13.1.These explicit expressions greatly modify the computational efficiency. More im-portantly, by means of these explicit formulas, it becomes easier to investigate theinfluence of the ordero(τM) of the optimal exercise boundaryB(τ) in polynomialsof

√τ, as shown later.

Then, substituting (13.36) into (13.17) and equating the like-power ofq, we havethe boundary condition

∂Vn(S,τ)∂S

=−gn(τ) at S= B0(τ) = 1 for n≥ 1. (13.42)


Similarly, substituting (13.35) into (13.19) and equatingthe like-power ofq, wehave

Bn(τ) =

c0 V0(B0,τ), n= 1,Bn−1(τ)+ c0 [Bn−1(τ)+Vn−1(B0,τ)+ fn−1(τ)] , n> 1,

(13.43)

whereB0 = 1.For the sake of simplicity, defineg0(τ) = 1. Then, (13.22) to (13.25) for the

initial guessV0(S,τ) have the same forms as thenth-order deformation equations(13.32) to (13.34) and (13.42) forVn(S,τ), respectively. Using the explicit recursionformula mentioned above, we have

g0(τ) = 1,

g1(τ) = 2ψ0,2(τ) B1(τ),g2(τ) = 2ψ0,2(τ) B2(τ)+3ψ0,3(τ) B2

1(τ)+2ψ1,2(τ) B1(τ),g3(τ) = 2ψ0,2(τ) B3(τ)+6ψ0,3(τ) B1(τ) B2(τ)+4ψ0,4(τ) B3

1(τ)+ 2ψ1,2(τ) B2(τ)+3ψ1,3(τ) B2

1(τ)+2ψ2,2(τ) B1(τ),...

and

f0(τ) = 0,

f1(τ) = ψ0,1(τ) B1(τ),f2(τ) = ψ0,1(τ) B2(τ)+ψ0,2(τ) B2

1(τ)+ψ1,1(τ) B1(τ),f3(τ) = ψ0,1(τ) B3(τ)+2ψ0,2(τ) B1(τ) B2(τ)+ψ0,3(τ) B3

1(τ)+ ψ1,1(τ) B2(τ)+ψ1,2(τ) B2

1(τ)+ψ2,1(τ) B1(τ),...

Note thatgn(τ) depends only uponVm(S,τ) andBm(τ), wherem= 0,1,2, · · · ,n−1. So, for thenth-order deformation equation,gn(τ) is always known. Therefore,Vn(S,τ) for n≥ 0 is governed by the linear PDE (13.32) subject to the linear initialcondition (13.33), the linear boundary condition (13.34) at infinity, and the linearcondition (13.42) at the fixed boundaryS= B0(τ) = 1. As long asVn(S,τ) at S=B0(τ) = 1 is known, it is convenient to gainfn(τ) by means of the explicit formula(13.37) with the recursion formulas mentioned above, and thenBn+1(τ) by meansof (13.43).

However, the linear PDE (13.32) contains variable coefficients and thus is noteasy to solve. Cheng, Zhu and Liao [14] solved such kind of system of linear PDEswith variable coefficients by means of the Laplace transform. Let

Vn(S,ζ ) = LT [Vn(S,τ)]


andgn(ζ ) = LT [gn(τ)]

denote the Laplace transforms ofVn(S,τ) andgn(τ), respectively, whereLT is theoperator of Laplace transform,ζ is a complex number corresponding toτ. By meansof the Laplace transform and using the initial condition (13.23), one has

LT

[

∂Vn(S,τ)∂τ

]

= ζ LT [Vn(S,τ)]−Vn(S,0) = ζ Vn(S,ζ )

and

LT

[

∂ mVn(S,τ)∂Sm

]

=∂ m

∂Sm LT [Vn(S,τ)]=∂ mVn(S,ζ )

∂Sm

for m≥0. Therefore, by means of the Laplace transform, the high-order deformationequation (13.32) with the initial/boundary conditions (13.33), (13.34) and (13.42)become a linear ordinary differential equation (ODE)

S2 ∂ 2Vn

∂S2 + γ S∂Vn

∂S− (γ + ζ )Vn = 0, (13.44)


∂Vn

∂S=−gn(ζ ), at S= 1, (13.45)

Vn(S,ζ )→ 0, asS→+∞. (13.46)

The above linear ordinary ODE has the closed-form solution

Vn(S,ζ ) = K(S,ζ ) gn(ζ ), (13.47)

where

K(S,ζ ) =−Sλ

λ, λ =

1− γ −√

4ζ +(1+ γ)2

2. (13.48)

Note that ˆgn(ζ ) = LT [gn(τ)] is the Laplace transform ofgn(τ). Then, using theinverse Laplace transform, we have

Vn(S,τ) = L−1T

[

K(S,ζ ) gn(ζ )]

(13.49)

and its derivative

V(m)n (S,τ) =

∂ mVn(S,τ)∂Sm = L

−1T

[

K(m)(S,ζ ) gn(ζ )]

, (13.50)

where

K(m)(S,ζ ) =∂ mK(S,ζ )

∂Sm ,


andL−1T denotes the inverse operator of the Laplace transform.

According to (13.40), (13.42), (13.43) and the related recursion formulas,gn(τ),fn(τ) andBn(τ) depend only onV(m)

n (1,τ) for 0≤ m≤ n−1, where the superscript(m) denotes themth-order differentiation with respect toS. Therefore, it is unnec-

essary to gain the general expressionV(m)n (S,τ) for all Sbut only onS= B0(τ) = 1.

So, the key is to gainV(m)n (1,τ) efficiently. From (13.50), it holds

V(m)n (1,τ) = L

−1T

[

K(m)(1,ζ ) gn(ζ )]

. (13.51)

According to (13.48), it is convenient to gain

K(1,ζ ) = − 2

(1− γ)−√

4ζ +(1+ γ)2,

K(1)(1,ζ ) =∂ K(S,ζ )

∂S

∣

∣

∣

∣

S=1=−1,

K(2)(1,ζ ) =1+ γ

2+

12

√

4ζ +(1+ γ)2,

...

and so on. Their inverse Laplace transforms read

K(1,τ) =1√πτ

exp

[

−14(1+ γ)2τ

]

+(1− γ)

2e−γτ Erfc

[

− (1− γ)2

√τ]

,

K(1)(1,τ) = −δ (τ),

K(2)(1,τ) = − 1

2√

πτ3exp

[

−14(1+ γ)2τ

]

+12(1+ γ)δ (τ),

...

where Erfc(x) is the complementary error function andδ (τ) is the Dirac delta func-tion, respectively.

As mentioned above, we only need obtainV(m)n (1,τ) so as to gain the optimal

exercise boundaryB(τ). The procedure starts from the initial guessB0(τ) = 1. Sinceg0(τ) = 1, we have its Laplace transform

g0(ζ ) = LT [g0(τ)] =1ζ.

Then,V0(1,τ) and its derivatives are given by the inverse Laplace transform (13.51),B1(τ) is gained by (13.43),g1(τ) and f1(τ) are obtained by (13.37) with the explicitformulas (13.38) to (13.41), successively. Similarly, ˆg1(ζ ) is given by means ofthe Laplace transform, thenV1(1,τ) and its derivatives are given by the inverseLaplace transform (13.51),B2(τ) is gained by (13.43),g2(τ) and f2(τ) are obtained


by (13.37) with the explicit formulas (13.38) to (13.41), successively. In theory, wecan gainBm(τ) successively in this way, wherem= 1,2,3, · · · .

Note that the inverse Laplace transform (13.51) can be expressed by a convolu-tion integral ofK(m)(1,τ) andgn(τ), say,

V(m)n (1,τ) =

∫ τ

0K(m)(1,τ − t) gn(t) dt. (13.52)

Therefore, sinceK(1,τ) contains the complementary error function, the expressions

of V(m)n (1,τ), and thenBn(τ), gn(τ) and fn(τ), become more and more complicated

asn increases. Thus, it becomes more and more difficult to exactly calculate theLaplace transform ˆgn(ζ ) = LT [gn(τ)] and especially the inverse Laplace transform(13.51). Therefore, it becomes more and more difficult to exactly solve the high-order deformation equations.

In order to obtain the high-order approximations, Zhu [46] employed a numericalmethod to calculate the related integrals similar to (13.52). Unlike Zhu [46], Cheng,Zhu and Liao [14] used analytic approximations ofB(τ) in powers of

√τ about the

expiry dateτ = 0. Note thatK(1,τ) is expressed by the exponential function and thecomplementary error function, whose Taylor series atτ = 0 have the infinite radiusof convergence. Besides, the dimensionlessτ to expiry is often small in practice.Therefore,B(τ) may be well approximated by a polynomial of

√τ as long as many

enough terms are used, say, the ordero(τM) is high enough.Cheng, Zhu and Liao [14] used the following strategy. First,both ofK(m)(1,τ)

andgn(τ) are expanded in power series ofτ aboutτ = 0 up-to the ordero(τM),denoted byK(m)(1,τ) andgn(τ), respectively. It is found that they can be expressedby

K(m)(1,τ) =2M

∑n=0

am,n(√

τ)n

(13.53)

and

gn(τ) =2M

∑n=0

cn(√

τ)n, (13.54)

which are Holder continuous with exponent 1/2 in time and therefore agree wellwith Blanchet’s [4] proof. Then, it is easy to gain their Laplace transforms

K(m)(1,ζ ) = LT [K(m)(1,τ)], gm(ζ ) = LT [gm(τ)].

More importantly, it is rather convenient to gain their inverse Laplace transform

V(m)n (1,τ) = L

−1T

[

K(m)(1,ζ ) gm(ζ )]

,

since both ofK(m)(1,ζ ) andgm(ζ ) are expressed in polynomials ofζ−1/2.The Nth-order homotopy-approximation of the dimensionlessB(τ) in polyno-

mial of√

τ to the ordero(τM) is givenexplicitlyby


B(τ)≈N

∑m=0

Bm(τ) =2M

∑k=0

bk(√

τ)k, (13.55)

where the coefficientbk is dependent uponγ = 2r/σ2 and the convergence-controlparameterc0. To further modify the above approximation ofB(τ), we first write

z=√

τ, and then calculate the Pade approximation toB=2M∑

n=0bn zn, centered atz= 0

of degree[M,M]. Then, replacingzby√

τ , we gain the[M,M] Pade approximant ofB(τ) to the ordero(τM).

13.3 Validity of the explicit homotopy-approximations

To show the accuracy and validity of the explicit expression(13.55) of the optimalexercise boundaryB(τ) given by the above HAM approach, we compare it withsome published analytic results. As pointed out by Cheng, Zhu and Liao [14], allpublished explicit approximations ofB(τ) were valid forτ ≪ τexp. For example,

B(τ) = exp(−2√

ατ), τ ≪ τexp, (13.56)

where

α = −12

ln(

9πγ2τ)

, by Kushe and Keller [25], (13.57)

α = −12

ln

(

4eγ2τ2−B2

p

)

, by Bunch and Johnson [8], (13.58)

with the perpetual optimal exercise price

Bp =γ

1+ γ. (13.59)

Besides, Knessl [24] gave the following asymptotic formula

ln [B(τ)] =−√

2τ| ln(4πγ2τ)|

1+∣

∣ln(4πγ2τ)∣

∣

−2

, τ ≪ τexp. (13.60)

By means of two examples, these asymptotic or perturbation formulas are com-pared with the explicit formula (13.55) given by the HAM approach mentionedabove. All results reported below are gained by a Mathematica code given in the Ap-pendix 13.2, which is free available athttp://numericaltank.sjtu.edu.cn/HAM.htm.Besides, all results given below have units, say, they are not dimensionless.

13.3 Validity of the explicit homotopy-approximations 443

Fig. 13.1 Approximations ofthe optimal exercise boundaryin the case of Example 13.1:X = $100,r = 0.1, σ = 0.3andT = 1 (year). Solid line:the 12th-order homotopy-approximation in polynomialof

√τ to o(τ8) by means

of c0 = −1; Circles: resultgiven by Zhu [46] with nu-merical integral; Dashed lineA: (13.57) given by Kuskeand Keller [25]; Dashed lineB: (13.60) by Knessl [24];Dashed line C: (13.58) byBunch and Johnson [8].

time to expiry (year)

Opt

imal

exer

cise

boun

dary

($)

0 0.2 0.4 0.6 0.8 165

70

75

80

85

90

95

100A

B

C

Dashed line A: by Kuske & Keller [25]Dashed line B: by Knessl [24]Dashed line C: by Bunch & Johnson [8]Symbols: numerical integration by Zhu [46]

Example 13.1

Let us first consider the following sample case discussed by Wu and Kwok [43],Carr and Faguet [10], Zhu [46], Cheng, Zhu and Liao [14] and also Cheng [13]:

• strike priceX = $100,• risk-free interest rater = 0.1,• volatility σ = 0.3,• time to expirationT = 1 (year).

Note that the zeroth-order deformation equation contains the convergence-controlparameterc0, which is used in the frame of the HAM to guarantee the convergenceof the homotopy-series. In this example, we choosec0 =−1 for the sake of simplic-ity.

It is found that, neither of (13.57) given by Kuske and Keller[25], nor of (13.60)by Knessl [24], nor of (13.58) by Bunch and Johnson [8], is valid for more than acouple of weeks prior to expiry, as shown in Fig. 13.1.

By means of the HAM-based approach mentioned above, we obtain the corre-sponding optimal exercise boundaryB(τ) in the polynomial of

√τ to the order

o(τ8), which agrees well with the results of Zhu [46] given by the numerical inte-gral in thewholetime, as shown in Fig. 13.1. At the expiration timeT = 1 (year),the optimal exercise boundaryB(T) is $76.25 in Wu and Kwok’s [43] numericalsolution, $76.11 by Zhu’s [46] numerical integral, and $76.17 by the 20th-orderhomotopy-approximation in polynomial to the ordero(τ8) given by the HAM, asshown in Table 13.1. This indicates the validity of the analytic approach based onthe HAM described above. Therefore, unlike all of asymptotic and perturbation ap-proximations mentioned above, which are often valid for only a couple of days orweeks prior to expiry, the explicit polynomial approximation of the optimal exer-cise boundary given by the HAM-based approach is accurate and valid even up-toseveral years. Note that even the 12th-order homotopy-approximation ofB(T) in


Table 13.1 The mth-order homotopy-approximation of the optimal exercise boundaryB(τ) inpolynomial of

√τ up too(τ8) by means ofc0 =−1 at different time in case ofX = $100,r = 0.1,

σ = 0.3 andT = 1 (year)

Order of approx.m 3 month 6 month 9 month 12 month

4 84.02 79.86 77.44 75.848 82.86 79.22 77.24 75.9612 82.63 79.27 77.41 76.1516 82.60 79.35 77.49 76.1818 82.62 79.38 77.50 76.1820 82.63 79.40 77.50 76.17

polynomial of√

τ to the ordero(τ8) is accurate enough, as shown in Fig. 13.1.Mathematically speaking, all asymptotic and perturbationformulas are valid onlyfor τ ≪ T, but the homotopy-approximation in polynomials of

√τ to the order

o(τ8) is valid even atτ = T, i.e. the time to expiry!It is found that the 10th-order homotopy-approximation ofB(τ) in polynomial

of√

τ to the ordero(τ8) by means ofc0 = −1 is accurate enough up to 3 yearsprior to expiry, as shown in Fig. 13.2, which is about 60 timeslonger than those ofother asymptotic and/or perturbation formulas mentioned above! Besides, the Padetechnique can be used to further enlarge the valid time ofB(τ): writing z=

√τ and

then using Pade technique, we gain the [8,8] Pade approximant ofB(τ) centered atz= 0, say,

B(τ)≈ 100

[

1+ b1(τ)1+ b2(τ)

]

(13.61)

where

b1(τ) = −0.758595τ1/2+0.8748811τ−0.474758τ3/2+0.209634τ2

− 0.0551746τ5/2+0.01333τ3−0.00210274τ7/2+2.15592×10−4τ4,

b2(τ) = −0.228157τ1/2+0.297781τ−0.0406677τ3/2+0.0324789τ2

− 0.0018480τ5/2+0.0016651τ3+3.545×10−6τ7/2+4.245×10−5τ4.

As shown in Fig. 13.2, this [8,8] Pade approximant ofB(τ) is accurate enough evenup to 10 years prior to expiry, about 200 times longer than those of other asymptoticand/or perturbation formulas mentioned above!

For the same problem, Cheng, Zhu and Liao [14] and Cheng [13] gave homotopy-approximations ofB(τ) up too(τ6) ando(τ7) by means of the HAM, respectively.Both of them are valid even atτ = T = 1 year, i.e. the time to expiry. In this chapter,we gained the 10th-order homotopy-approximations ofB(τ) up too(τ8), which isaccurate up to 3 years prior to expiry. In theory,B(τ) can be expanded into thepolynomial of

√τ to the ordero(τM) for arbitrary positive integerM ≥ 1. Does the

maximum valid time of the homotopy-approximation ofB(τ) up too(τM) stronglydepend uponM?


Fig. 13.2 The [8,8] Padeapproximant of the optimalexercise boundaryB(τ) inthe case of Example 13.1:X = $100,r = 0.1, σ = 0.3.Dashed-line: the 10th-orderhomotopy-approximation ofB(τ) in the polynomial of

√τ

to o(τ8) by c0 = −1; Solidline: the corresponding [8,8]Pade approximat ofB(τ);Dash-dotted line: perpetualoptimal exercise price.

Time to expiry (year)

Opt

imal

exer

cise

boun

dary

($)

0 2 4 6 8 1060

65

70

75

80

85

90

95

100

Fig. 13.3 The influence of theordero(τM) to the 10th-orderhomotopy-approximations ofthe optimal exercise boundaryby means ofc0 = −1 inthe case of Example 13.1:X = $100,r = 0.1 andσ =0.3. Dashed line A:M = 8;Dashed line B:M = 16;Dashed line C:M = 24;Dashed line D:M = 32;Solid line:M = 48; Symbols:[48,48] Pade approximant;Dash-dotted line: perpetualoptimal exercise price. Time to expiry (year)

Opt

imal

exer

cise

boun

dary

($)

0 5 10 15 2060

65

70

75

80

85

90

95

100

A

B

CD

Using the explicit formulas given in the Appendix 13.1, it isconvenient nowfor us to investigate the influence of the ordero(τM) of B(τ). It is found that, thehigher the order ofo(τM), the longer the maximum valid time of the 10th-orderhomotopy-approximation ofB(τ) prior to expiry, as shown in Fig. 13.3. Note that,the 10th-order homotopy-approximation ofB(τ) in polynomial of

√τ to o(τ48) is

accurate enough even up to 20 years prior to expiry, which is about 400 times longerthan those of asymptotic and/or perturbation formulas mentioned above! As shownin Fig.13.3, the optimal exercise price given byM = 48 tends to the perpetual opti-mal exercise priceBp so closely that the combination of the 10th-order homotopy-approximation ofB(τ) in polynomial too(τ48) with the known perpetual optimalexercise priceBp = γ/(1+ γ) gives accurate enough optimal exercise price even inthewholetime (0≤ τ <+∞) prior to expiry! Using the Mathematica code given inthe Appendix 13.2, such kind of accurate analytic approximation is gained by meansof a laptop (MacBook Pro with 2.8 GHz and 4GB MHz DDR3) in 102 seconds only.

As shown in Fig. 13.4 and Table 13.2, the 10th-order homotopyapproximationof B(τ) in polynomial of

√τ to o(τ128) is valid even up toa half centuryprior to


Fig. 13.4 The influence of theordero(τM) to the maximumvalid time (year) prior toexpiry for the 10th-orderhomotopy-approximations ofthe optimal exercise boundaryby c0 = −1 in the case ofExample 13.1:X = $100,r = 0.1 andσ = 0.3. Dash-dotted line: formula 1.6008+0.3853M given by a least-squares fit; Symbols: themaximum valid time prior toexpiry for B(τ) in polynomialof

√τ to o(τM). M, Order of power seires of B(τ)

Val

idtim

e(y

ear)

toex

piry

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

Example 13.1: X=$100, r=0.1,σ=0.3

Dashed line: 1.6008 + 0.3853 M

Table 13.2 The maximum valid time (years) prior to expiry for the 10th-order homotopy-approximation of the optimal exercise boundaryB(τ) in polynomial of

√τ to different orders

o(τM) in case of Example 13.1 withX = $100,r = 0.1 andσ = 0.3 by means ofc0 =−1.

The ordero(τM) of the polynomial forB(τ) Maximum valid time (years) prior to expiry

M = 8 3M = 16 7.5M = 24 11M = 32 14.5M = 48 21M = 64 28M = 80 32M = 96 38.5M = 128 50

expiry, which is aboutone thousandtimes longer than those of asymptotic and/orperturbation formulas mentioned above!

Especially, it is found that the maximum valid time of the 10th-order homotopy-approximation ofB(τ) in polynomial of

√τ to o(τM) is directly proportional toM,

although approximately, as shown in Fig. 13.4. It suggests that, given an arbitrarytime T prior to expiry, we can always gain the corresponding accurate enough op-timal exercise price, as long as theNth-order homotopy-approximation ofB(τ) inpolynomial of

√τ to o(τM) has large enoughM with a reasonably high orderN.

This is an excellent example to illustrate that the HAM can give much better ex-plicit, analytic approximations of some nonlinear problems than asymptotic and/orperturbation methods!

Indeed,a truly new method always gives something new and/or different. Thisexample shows the originality and great potential of the HAMonce again.


Fig. 13.5 The optimal exer-cise boundaryB(τ) in caseof Example 13.2:X = $1,r = 0.08 andσ = 0.4. Dashedline A: (13.57) by Kuskeand Keller [25]; Dashed lineB: (13.60) by Knessl [24];Dashed line C: (13.58) byBunch & Johnson [8]; Solidline: 10th-order homotopy-approximation ofB(τ) byc0 = −1 in polynomialof

√τ to o(τ48); Sym-

bols: 10th-order homotopy-approximation ofB(τ) byPade method; Dash-dottedline: perpetual optimal exer-cise price $0.5.

Time (year) prior to expiry

Opt

imal

exer

cise

pric

e($

)

0 5 10 15 200

0.2

0.4

0.6

0.8

1 A

B

C

Example 13.2: X= 1, r = 0.08,σ = 0.4Dashed line A: by Kuske & Keller [25]Dashed line B: by Knessl [24]Dashed line C: by Bunch & Johnson [8]Solid line: by the HAM (10th-order)

Example 13.2

The second example is a long term option considered by Chen and Chadam et.al [11]:

• strike priceX = $1,• risk-free interest rater = 0.08,• volatility σ = 0.4,• time to expirationT = 3 (year).

As shown in Fig. 13.5, the 10th-order homotopy-approximation of B(τ) in thepolynomial of

√τ to o(τ48) by means ofc0 = −1 is valid even up to 20 years prior

to expiry, while neither of (13.57) given by Kuske and Keller[25], nor of (13.60)by Knessl [24], nor of (13.58) by Bunch & Johnson [8], is validfor more thanone month prior to expiry! Note that our homotopy-approximation fits well with its[48,48] Pade approximant in the whole time interval 0≤ τ ≤ 20 year. Such kindof accurate analytic approximation is gained by means of a laptop (MacBook Prowith 2.8 GHz and 4GB MHz DDR3) in only 57 seconds. At the expirytimeT = 20year, the 10th-order homotopy-approximation and the corresponding [48,48] Padeapproximant give the same optimal exercise price $0.5029, which is so close to theperpetual optimal exercise price $0.5 thatB(τ) = 0.5 is an accurate enough approx-imation forτ > 20. So, combining the 10th-order homotopy-approximation of B(τ)in the polynomial of

√τ to o(τ48) for 0 ≤ τ ≤ 20 (year) and the perpetual opti-

mal exercise priceB(τ) = 0.5 for τ > 20 (year), we have an analytic approximationaccurate enough in thewhole time domain 0≤ τ < +∞, i.e. for arbitrary time toexpiry!

Like Cheng, Zhu and Liao [14] and Cheng [13], we obtained all of abovehomotopy-approximations by means of the convergence-control parameterc0 =


Fig. 13.6 The optimal ex-ercise boundary in caseof Example 13.2:X = $1,r = 0.08 andσ = 0.4. Dashedline A: (13.57) by Kuskeand Keller [25]; Dashed lineB: (13.60) by Knessl [24];Dashed line C: (13.58)by Bunch & Johnson [8];Dash-dotted line: 10th-orderhomotopy-approximation ofB(τ) in polynomial of

√τ

to o(τ8) by c0 = −1; Solidline: 10th-order homotopy-approximation ofB(τ) inpolynomial of

√τ to o(τ8) by

c0 =−1/2; Long-dashed line:perpetual optimal exerciseprice $0.5.

Time to expiry (year)

Opt

imal

exer

cise

boun

dary

($)

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1A

B

C

Example 13.2: X = 1, r = 0.08,σ = 0.4Dashed line A: by Kuske & Keller [25]Dashed line B: by Knessl [24]Dashed line C: by Bunch & Johnson [8]Dash-dotted line: by HAM with c0 = -1Solind line: by HAM with c0 = -1/2

−1. However, as proved in Chapter 4 and shown in this book, the convergence-control parameterc0 provides us a convenient way to guarantee the convergence ofhomotopy-series. As shown in Fig. 13.6, the 10th-order homotopy-approximationof B(τ) in polynomial of

√τ to o(τ8) by c0 = −1 is valid about 6 years prior to

expiry. However, whenc0 = −1/2, it is valid about 10 years prior to expiry: themaximum valid time ofB(τ) increases about 66%. So, the convergence-control pa-rameterc0 also provides us an alternative way to enlarge the maximum valid timeof the homotopy-approximation ofB(τ) prior to expiry.

These two examples illustrate that, unlike other asymptotic and /or perturbationformulas which are often valid only a couple of days or weeks prior to expiry, theHAM provides us an accurate approximation of the optimal exercise boundaryB(τ)in polynomial of

√τ to o(τM), which is often valid a couple of dozen years, or

even a half century, prior to expiry, as long asM and the order of approximation arereasonably large with a properly chosen convergence-control parameterc0.

As pointed out by Kim [23] and Carr et al. [9], when the optimalexercise bound-ary B(τ) is known, it is easy to gain the priceV(S,τ) of American put option bymeans of (13.7). So, by means of the explicit analytic approximation (13.55) ofB(τ) in polynomial of

√τ given by the HAM-based approach mentioned in this

chapter, it is convenient to gain accurate approximation ofthe priceV(S,τ), whichmaybe valid even up to a half century prior to expiry!

13.4 A practical code for businessmen

For all examples considered by Cheng, Zhu and Liao [14] and Cheng [13], we gainaccurate optimal exercise price in much longer time intervals (prior to expiry) by

13.4 A practical code for businessmen 449

means of the 10th-order homotopy-approximation ofB(τ) in polynomial of√

τ too(τ48). Besides, it is found that the 10th-order homotopy-approximation ofB(τ) inpolynomial of

√τ to o(τ48) by means ofc0 =−1 is often valid to such a long time

that the known perpetual optimal exercise price becomes a good enough approx-imation thereafter. Therefore, practically speaking, we have the accurate optimalexercise priceB(τ) and the option priceV(S,τ) for arbitrarily large expiration-time0< T < +∞. This suggests that the homotopy-approximation ofB(τ) in the poly-nomial of

√τ to o(τ48) is valid in general and thus can be widely used in business

related to American put option.As mentioned above, by means of a laptop (MacBook Pro with 2.8GHz and

4GB MHz DDR3), only 102 and 57 seconds CPU times are needed to gain the 10th-order homotopy-approximations ofB(τ) in polynomial of

√τ to o(τ48) by means of

c0 =−1 for Examples 13.1 and 13.2, respectively. This is fast enough for a scholar,but not for a businessman. For convenience sake, a practicalMathematica code isprovided in Appendix 13.3, which can give accurate enough optimal exercise priceof American put option for rather large expiration-timeT in a few seconds!

Note that the dimensionless equations (13.10) to (13.14 ) contain only one dimen-sionless parameterγ = 2r/σ2. So, the dimensionlessB(τ) is only dependent uponγ. Using the Mathematica code given in Appendix 13.2 and a high-performancecomputer, we obtained the 10th-order homotopy-approximation of the dimension-lessB(τ) in polynomial of

√τ to o(τM) by means ofc0 = −1 for the unknown

parameterγ, whereM is a reasonably large integer such asM = 24,36,48 and soon. Then, these lengthy homotopy-approximations of the dimensionlessB(τ) aresaved in data files with different names such asAPO-48-10.txt , correspondingto the 10th-order homotopy-approximation to the ordero(τ48). A short Mathematicacode, namelyAPOh, is given in Appendix 13.3, which first reads all dimensionlesshomotopy-approximations ofB(τ) saved in such a data file, and then calculates thedimensional optimal exercise price according to a given strike priceX, risk-free in-terest rater, volatility σ and time to expirationT (year). Using a laptop, one canoften gain an accurate enough exercise price of American putoption for rather largeexpiration-timeT in only a few seconds! This practical Mathematica code with asimple users guide is available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

Note that such kind ofexplicit homotopy-approximations ofB(τ) in polynomialof

√τ are analytic, because they are continuous for any 0< τ < +∞ and their

derivativesB′(τ), B′′(τ) and so on exist for allτ > 0. Thus, any interpolations areunnecessary at all. It is true that such kind of analytic approximations given by theHAM are very lengthy, which might be hundreds pages long if printing out. Fromthe traditional points of view, such kind of explicit, lengthy analytic formula mightbe useless. Fortunately, we are now in the times of computer:a laptop can read andcalculate such kind of explicit, lengthy analytic formulasin a few seconds, which iseven faster than calculating a simple, half-page length analytic formula by hands!So, if we regard the keyboard of a laptop as a pen, its hard diskas papers, and itscentral processing unit (CPU) as a brain of human-being, then an explicit, analyticformula in the times of computer can be very lengthy, as shownin this section. Notethat our traditional concept “analytic” appeared several hundreds ago, when our


computational tools were rather inefficient. The practicalMathematica codeAPOhis an excellent example to verify that our traditional concept “analytic” isout of date,and thus should be greatlymodifiedin the times of computer and internet. This alsoshows, on the other hand, the originality of the HAM.

Indeed,a truly new method always gives something new and/or different.


The American put option is governed by a linear PDE with variable coefficients,subject to some linear initial and boundary conditions. However, it is in principle anonlinear problem, since the two boundary conditions are satisfied on an unknownmoving boundaryB(τ). The asymptotic and/or perturbation formulas (13.57) givenby Kuske & Keller [25], (13.58) by Bunch & Johnson [8] and (13.60) by Knessl[24], respectively, are often valid only a couple of days or weeks prior to expiry,which is too short for the practical use in business.

It was Cheng, Zhu and Liao [14] who first combined the HAM with the Laplacetransform in such a way that the homotopy-approximations ofthe optimal exerciseboundaryB(τ) in polynomial of

√τ to o(τ6) were obtained. Cheng [13] further

gave a homotopy-approximation ofB(τ) in polynomial too(τ7). Unlike the asymp-totic and/or perturbation formulas, these homotopy approximations ofB(τ) are validfor several years, and thus are much better than all asymptotic and/or perturbationformulas mentioned above.

In this chapter, we further modified the HAM-based approach of Cheng, Zhu andLiao [14] and Cheng [13] by means of deriving explicit formulas (13.35) to (13.41)related to the unknown moving boundary. Especially, we investigate, for the firsttime, the influence of the ordero(τM) of polynomials ofB(τ) to its maximum validtime prior to expiry. It is found that, the maximum valid timeof B(τ) in polynomialof

√τ to o(τM) is directly proportional toM, so that, as long asM is large enough,

the explicit homotopy-approximation ofB(τ) can be valid even up to a half-centuryprior to expiry, about 1000 times longer than those of asymptotic and/or perturbationformulas mentioned above, as shown in Example 13.1. Besides, it is found that the10th-order homotopy-approximationofB(τ) in polynomial of

√τ to o(τ48) by c0 =

−1 is often valid up to so many years prior to expiry that the theoretical perpetualoptimal exercise price

Bp =

(

γ1+ γ

)

X

is accurate enough thereafter. Therefore, the combinationof such kind of homotopy-approximation ofB(τ) in polynomial of

√τ to o(τ48) and the theoretical per-

petual optimal exercise priceBp mentioned above can be regarded as an accu-rate analytic approximation ofB(τ) valid in thewhole time domain 0≤ τ < +∞.Based on the HAM approach, a Mathematica code is given in Appendix 13.2 andfree available athttp://numericaltank.sjtu.edu.cn/HAM.htm. Using this code with ahigh-performance computer, we gained the 10th-order (dimensionless) homotopy-


approximation ofB(τ) in polynomial of√

τ to o(τ48) for the unknown dimension-less parameterγ = 2r/σ2, and saved it in a data-file by the name ofAPO-48-10.txt ,which can be directly used to gain accurate enough optimal exercise price of Amer-ican put option in a few seconds by means of the short, practical Mathematica codeAPOh. Both of the practical codeAPOhand the data-fileAPO-48-10.txt are avail-able at the same website mentioned above. Obviously, theAPOhmay provide busi-nessmen a convenient tool in business.

In addition, we also investigate, for the first time, the influence of the convergence-control parameterc0 to the maximum valid time (prior to expiry) of the homotopy-approximation ofB(τ) in polynomial of

√τ . As shown in Fig. 13.6, the maximum

valid time of the 10th-order homotopy-approximation ofB(τ) in polynomial of√

τto o(τ8) given byc0 = −1/2 is about 66% longer than that byc0 = −1. This sug-gests that the convergence-control parameterc0 also provides us a convenient wayto enlarge the maximum valid time ofB(τ) prior to expiry.

Note that Landau transform [26] is not used here. So, this analytic approachbased on the HAM and Laplace transform has quite general meanings, and thus canbe widely applied to solve similar problems in finance, such as the optimal exerciseboundary of American options on an underlying asset with dividend yield [47] andso on.

Finally, we emphasize that, unlike all asymptotic and/or perturbation formulasmentioned above, which are often valid only a couple of days or weeks prior to ex-piry, the analytic homotopy-approximations ofB(τ) in polynomials of

√τ to o(τ48)

may be valid a couple of dozen years, or even a half century! Besides, although theseaccurate homotopy-approximations ofB(τ) saved in the data-fileAPO-48-10.txt

might be several hundreds pages long when printed out, we cangain accurate opti-mal exercise price of American put option in a few seconds by means of the short,practical Mathematica codeAPOhin a laptop! This is an excellent example to showthat the traditional concept of “analytic” solutions isout of dateand thus should bemodifiedin the times of computer and internet.

Acknowledgements The formulas about the inverse Laplace transform used in Appendix 13.2 areprovided by Dr. Jun Cheng.


Appendix 13.1 Detailed derivation of fn(τ) and gn(τ)

According to (13.28), we define

[Λ(τ;q)−B0(τ)]m =

[

+∞

∑i=1

Bi(τ)qi

]m

=+∞

∑n=m

µm,n(τ) qn, (13.62)

with the definitionµ1,n(τ) = Bn(τ), n≥ 1. (13.63)

Then, it holds

[Λ(τ;q)−B0(τ)]m+1 =+∞

∑n=m+1

µm+1,n(τ) qn

=

[

+∞

∑n=m

µm,n(τ) qn

] [

+∞

∑i=1

Bi(τ)qi

]

=+∞

∑n=m+1

qn

[

n−1

∑i=m

µm,i(τ) Bn−i(τ)

]

(13.64)

which gives the recursion formula

µm+1,n(τ) =n−1

∑i=m

µm,i(τ) Bn−i(τ). (13.65)

For the sake of simplicity, define

ψn,0(τ) =Vn(B0,τ), ψn,m(τ) =1m!

∂ mVn(S,τ)∂Sm

∣

∣

∣

∣

S=B0(τ). (13.66)

Then, on the moving boundaryS=Λ(τ;q), We have by means of Taylor expansionat S= B0(τ) that

Vn(S,τ) = ψn,0(τ)++∞

∑m=1

ψn,m(τ) [Λ(τ;q)−B0(τ)]m

= ψn,0(τ)++∞

∑m=1

ψn,m(τ)

[

+∞

∑i=m

µm,i(τ) qi

]

= ψn,0(τ)++∞

∑i=1

qi

[

i

∑m=1

ψn,m(τ)µm,i(τ)

]

= Vn(B0,τ)++∞

∑i=1

αn,i(τ) qi, (13.67)

where

αn,i(τ) =i

∑m=1

ψn,m(τ) µm,i(τ), i ≥ 1. (13.68)

Appendix 13.1 Detailed derivation offn(τ) andgn(τ) 453

Therefore, on the moving boundaryS= Λ(τ;q), we have

φ(S,τ;q) =+∞

∑n=0

Vn(S,τ) qn =+∞

∑n=0

qn

[

Vn(B0,τ)++∞

∑i=1

αn,i(τ) qi

]

=+∞

∑n=0

qn

[

Vn(B0,τ)+n−1

∑j=0

α j ,n− j(τ)

]

=+∞

∑n=0

[Vn(B0,τ)+ fn(τ)] qn, (13.69)

where

fn(τ) =n−1

∑j=0

α j ,n− j(τ). (13.70)

Whenq= 0, it holds onS= Λ(τ;0) = B0(τ) that

φ(Λ ,τ;q) = φ(B0,τ;0) =V0(B0,τ).

So, we havef0(τ) = 0 and thus it holds

φ(S,τ;q) =V0(B0,τ)++∞

∑n=1

[Vn(B0,τ)+ fn(τ)] qn (13.71)

on the moving boundaryS= Λ(τ;q).Similarly, we have on the moving boundaryS= Λ(τ;q) that

∂Vn(S,τ)∂S

= ψn,1(τ)++∞

∑m=1

(m+1)ψn,m+1(τ) [Λ(τ;q)−B0(τ)]m

= V ′n(B0,τ)+

+∞

∑i=1

βn,i(τ) qi , (13.72)

where the prime denotes the differentiation with respect toS, and

βn,i(τ) =i

∑m=1

(m+1)ψn,m+1(τ) µm,i(τ), i ≥ 1. (13.73)

Furthermore, it holds on the moving boundaryS= Λ(τ;q) that

∂φ(S,τ;q)∂S

=V ′0(B0,τ)+

+∞

∑n=1

[

V ′n(B0,τ)+gn(τ)

]

qn, (13.74)

where

gn(τ) =n−1

∑j=0

β j ,n− j(τ), n≥ 1. (13.75)


Appendix 13.2 Mathematica code for American put option

The American put option equation is solved by means of the HAMwith the Laplacetransform. This code is available athttp://numericaltank.sjtu.edu.cn/HAM.htm.


code Developer. All rights reserved.Redistribution and use in source and binary forms, with or without modification,




• Redistributions in source and binary forms for profit purpose, with or withoutmodification, are not allowed without written agreement from the code developer.










Appendix 13.2 Mathematica code for American put option 455

Mathematica code for American put optionby Shijun LIAO

Shanghai Jiao Tong UniversityDecember 2010


( * Define approx[f] for Taylor expansion of f * )approx[f_] := Module[temp,temp[0] = Series[f, t, 0, OrderTaylor]//Normal;temp[1] = temp[0]/.tˆ(n_.) * Derivative[j_][DiracDelta][0]->0;temp[2] = temp[1]/.tˆ(n_.) * DiracDelta[0]->0;temp[3] = temp[2]/.DiracDelta[0]->0;temp[4] = temp[3]/.Derivative[j_][DiracDelta][0]->0;temp[5] = N[temp[4],60]//Expand;If[KeyCutOff == 1, temp[5] = temp[5]//Chop];temp[5]];

( * Define approx2[f] for Taylor expansion of f * )approx2[f_] := Module[temp,temp[0] = Expand[f];temp[1] = temp[0] /. Derivative[n_][DiracDelta][t] -> dd[ n];temp[2] = temp[1] /. DiracDelta[t] -> dd[0];temp[3] = Series[temp[2],t, 0, OrderTaylor]//Normal;temp[4] = temp[3] /. dd[0] -> DiracDelta[t];temp[5]=temp[4]/.dd[n_]->Derivative[n][DiracDelta][ t];temp[6] = N[temp[5],60]//Expand;If[KeyCutOff == 1, temp[6] = temp[6]//Chop];temp[6]];

( * Define GetLK[n] * )lamda := (1 - gamma)/2 - 1/2 * Sqrt[(4 p + (1 + gamma)ˆ2)];kernel[s_] := -sˆlamda/lamda;lK0[0] = -1/lamda;lK0[i_] := D[kernel[s], s, i] /. s -> 1 // Expand;

GetLK[m0_,m1_,Nappr_]:= Module[temp,K1,K2,lK1,lK2,For[i = Max[m0,0], i <= m1, i++,

K[i] = invl[lK0[i]];K1[i] = K[i]/.Derivative[_][DiracDelta][t_]->0,

DiracDelta[t_]->0;K2[i] = Collect[K[i]-K1[i],DiracDelta[t],

Derivative[Blank[]][DiracDelta][t]];temp = Series[K1[i],t,0,Nappr]//Normal;lK1[i] = LaplaceTransform[temp, t, p];lK2[i] = LaplaceTransform[K2[i],t, p];LK[i] = Collect[lK1[i] + lK2[i], p];


];];

( * Define Getf[n] and Getg[n] * )mu[m_,n_]:=If[m ==1,b[n],Sum[mu[m-1,i] * b[n-i],i,m-1,n-1]];psi[n_,m_] := dV[n,m]/m!;alpha[n_,i_] := Sum[psi[n,m] * mu[m,i],m,1,i];beta[n_,i_] := Sum[(m+1) * psi[n,m+1] * mu[m,i],m,1,i];f[0] := 0;g[0] := 1;Getf[n_] := Sum[alpha[j,n-j],j,0,n-1];Getg[n_] := Sum[beta[j,n-j] ,j,0,n-1];

( * Define Getb[n] * )b[0] := 1;BB[0] := 1;B[0] := X;Getb[n_] := Module[temp,If[n == 1,

b[1] = c0 * dV[0, 0] // Expand,temp = b[n - 1] + c0 * (b[n -1]+dV[n-1,0]+f[n-1])//Expand;b[n] = approx[temp];];

];

( * Define GetDV[m,n] * )GetDV[m_, n_] := Module[temp,If[n == 1, DV[m, 1] = -g[m],

temp[1] = Expand[LK[n] * Lg[m]];DV[m, n] = invl[temp[1]];];

DV[m, n] = approx[DV[m, n]];];

( * Define dV[m,n] * )dV[m_,n_] := Module[temp,If[NumberQ[flag[m,n]],

Goto[100],GetDV[m,n];flag[m,n] = 1];

Label[100];DV[m,n]];

( * Define hp[f_,m_,n_] * )hp[f_,m_,n_]:= Module[k,i,df,res,q,df[0] = f[0];For[k = 1, k <= m+n, k++, df[k] = f[k] - f[k-1]//Expand ];res = df[0] + Sum[df[i] * qˆi,i,1,m+n];Pade[res,q,0,m,n]/.q->1];


( * Get [m,n] Pade approximant of B * )pade[order_]:= Module[temp,s,i,j,temp[0] = BB[order] /. tˆi_. -> sˆ(2 * i);temp[1] = Pade[temp[0],s,0,OrderTaylor,OrderTaylor] ;If[KeyCutOff == 1, temp[1] = temp[1]//Chop];BBpade[order] = temp[1] /. sˆj_. -> tˆ(j/2);Bpade[order] = X * BBpade[order]/. t -> (sigmaˆ2 * t/2);];

( * Define inverse Laplace transformation * )invl[Sqrt[p]] := -1/(2 * Sqrt[Pi] * tˆ(3/2));

invl[pˆn_] := Module[temp, nInt,nInt = IntegerPart[n];If[n > 1/2 && n > nInt,

Goto[100],temp[2] = InverseLaplaceTransform[pˆn, p, t];Goto[200];];

Label[100];temp[1] = -1/2/Sqrt[Pi]/tˆ(3/2);temp[2] = D[temp[1], t, nInt];Label[200];temp[2]//Expand];

invl[d_./(c_. + a_. * Sqrt[4p + b_.])] := Module[temp,temp[1] = d/(4a) * Exp[-b * t/4];temp[2] = 2/Sqrt[Pi * t];temp[3] = c/a * Exp[cˆ2 * t/(4aˆ2)] * Erfc[c * Sqrt[t]/(2a)];temp[1] * (temp[2]-temp[3])//Expand];

invl[d_./(p * (c_. + a_. * Sqrt[4p + b_.]))]:= Module[temp,temp[1] = Sqrt[b] * Erf[Sqrt[b * t]/2];temp[2] = c/a * Exp[-(b-(c/a)ˆ2) * t/4] * Erfc[c * Sqrt[t]/(2a)];temp[3] = -1/(b - (c/a)ˆ2) * d/a * (c/a-temp[1]-temp[2]);temp[3]//Expand];

invl[pˆi_. * Sqrt[c_. * p + a_.]] := Module[temp,temp = D[-Exp[-a * t/c]/(2 * c* Sqrt[Pi] * (t/c)ˆ(3/2)),t, i];temp//Expand];

invl[Sqrt[c_. * p+a_.]]:=-Exp[-a * t/c]/(2 * c* Sqrt[Pi] * (t/c)ˆ(3/2));invl[f_] := InverseLaplaceTransform[f, p, t] // Expand;invl[p_Plus] := Map[invl, p];invl[c_ * f_] := c * invl[f] /; FreeQ[c, p];


( * Main code * )ham[m0_, m1_] := Module[temp, k, n,If[m0 == 1,Print[" Strike price = ?"];temp[0] = Input[];If[!NumberQ[temp[0]],Goto[100]];X = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;Print[" Risk-free interest rate = ?"];temp[0] = Input[];If[!NumberQ[temp[0]],Goto[100]];r = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;Print[" Volatility = ?"];temp[0] = Input[];If[!NumberQ[temp[0]],Goto[100]];sigma = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;Print[" Time to expiry = ?"];temp[0] = Input[];If[!NumberQ[temp[0]],Goto[100]];T = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;gamma = 2* r/sigmaˆ2;texp = sigmaˆ2 * T/2;Bp = X* gamma/(1 + gamma);Label[100];If[!NumberQ[gamma],

X = .;r = .;sigma = .;gamma = .;T = .;];

Print["-------------------------------------------- ---------"];Print[" INPUT PARAMETERS: "];Print[" Strike price (X) = ",X," ($) "];Print[" Risk-free interest rate (r) = ",r];Print[" Volatility (sigma) = ",sigma];Print[" Time to expiry (T) = ",T," (year)"];Print["-------------------------------------------- ---------"];Print[" CORRESPONDING PARAMETERS: "];Print[" gamma = ",gamma];Print[" dimensionless time to expiry (texp)=",texp//N];Print[" perpetual optimal exercise price (Bp)=",Bp//N,"( $)"];Print["-------------------------------------------- ---------"];Print[" CONTROL PARAMETERS: "];Print[" OrderTaylor = ",OrderTaylor];Print[" c0 = ",c0];Print["-------------------------------------------- ---------"];KeyCutOff = If[OrderTaylor < 80 && NumberQ[gamma], 1, 0];If[KeyCutOff == 1,

Print["Command Chop is used to simplify the result"],Print["Command Chop is NOT used "]];

If[NumberQ[gamma],Print["Pade technique is used"],Print["Pade technique is NOT used"]


];Clear[flag,DV];];For[k = Max[1, m0], k <= m1, k++,

Print[" k = ", k];If[k == 1, GetKK[]; GetBJ[]; GetKn[]];If[k == 1, Lg[0] = LaplaceTransform[g[0], t, p]];If[k == 1, GetLK[0,2,OrderTaylor],

GetLK[k+1,k+1,OrderTaylor]];Getb[k];BB[k] = Collect[BB[k - 1] + b[k], t];temp[0] = X * BB[k] /. t-> (sigmaˆ2 * t/2)//Expand;B[k] = Collect[temp[0],t];If[NumberQ[gamma],pade[k]];temp[1] = Getg[k];temp[2] = Getf[k];g[k] = approx2[temp[1]];f[k] = approx2[temp[2]];Lg[k] = LaplaceTransform[g[k], t, p];If[NumberQ[gamma] && NumberQ[sigma] && NumberQ[X],

Print[" Optimal exercise price at the timeto expiration = ", B[k]/.t->T//N];

Print[" Modified result givenby Pade technique = ",Bpade[k]/.t->T//N];

];];

Print[" Well done !"];If[NumberQ[gamma] && NumberQ[sigma] && NumberQ[X],

Plot[Bp,B[m1],Bpade[m1],t,0,1.25 * T,PlotRange -> 0.8 * Bp,X, PlotStyle ->RGBColor[1,0,0],RGBColor[0,1,0],RGBColor[0,0,1]];

Print[" Order of homotopy-approximation : ",m1];Print[" Green line : optimal exercise boundary B

in polynomial "];Print[" Blue line : optimal exercise boundary B

by Pade method "];Print[" Red line : perpetual optimal exercise price "];];

];

( * Dimensionless analytic formula given by Kuske and Keller * )GetKK[] := Module[alpha,alpha = -Log[9 * Pi * gammaˆ2* t]/2;KK0 = Exp[-2 * Sqrt[alpha * t]];KK = X* KK0 /. t->sigmaˆ2/2 * t;];

( * Dimensionless formula given by Bunch and Johnson * )GetBJ[] := Module[alpha,Bp0 = gamma/(1+gamma);alpha = -Log[4 * E* gammaˆ2* t/(2 - Bp0ˆ2)]/2;BJ0 = Exp[-2 * Sqrt[alpha * t]];BJ = X* BJ0 /. t->sigmaˆ2/2 * t;];


( * Dimensionless formula given by Knessl * )GetKn[] := Module[z,z = Abs[Log[4 * Pi * gammaˆ2* t]];Kn0 = Exp[-Sqrt[2 * t * z] * (1+1/zˆ2)];Kn = X* Kn0 /. t->sigmaˆ2/2 * t;];

( * Define the order of Taylor’s series expansion * )OrderTaylor = 8;

( * Assign the convergence-control parameter * )c0 = -1;

( * Get 8th-order homotopy-approximation of B * )ham[1,8];

( * Get 10th-order homotopy-approximation of B * )ham[8,10]

Appendix 13.3 Mathematica codeAPOhfor businessmen 461

Appendix 13.3 Mathematica codeAPOh for businessmen

The Mathematica codeAPOhgives the accurate optimal exercise price at a giventime prior to expiry in a few second.

This practical code first reads the dimensionless results from the data-file by thename ofAPO-48-10.txt , which were gained by Shijun Liao using the Mathe-matica code in Appendix 13.2 and a high-performance computer for the unknowndimensionless parameterγ = 2r/σ2 in general, and then calculate the dimensionaloptimal exercise price up to the time to expiry. Unlike otherasymptotic and/or per-turbation formulas that are often valid a couple of days or weeks prior to expiry,this code often gives an accurate approximation that is often valid a couple of dozenyears. So, it is especially useful for businessmen.

The codeAPOhis available athttp://numericaltank.sjtu.edu.cn/HAM.htm.


AOPhDeveloper. All rights reserved.Redistribution and use in source and binary forms, with or without modification,




• Redistributions in source and binary forms for profit purpose, with or withoutmodification, are not allowed without written agreement from theAOPhdevel-oper.











A Simple Users Guide

APOh[order ] This module gives the homotopy-approximationof dimensionaloptimal exercise priceB(τ), with order denoting the order of homotopy-approximation. The code first reads the data-fileAPO-48-10.txt , then asksthe user to input strike priceX, risk-free interest rater, volatility σ and timeto expirationT (year), and finally lists the results ofB(τ) at different order ofapproximations, their modified approximations by the Pademethod, and plots acurve ofB(τ) in the interval 0≤ τ ≤ 1.25T with the theoretical perpetual optimalexercise priceBp = Xγ/(1+ γ). To begin a new case, simply run the codeAPOhonce again and input new parameters.

B[n] Thenth-order homotopy-approximation of the optimal exercise price withdimension ($), a polynomial of

√τ to o(τM), whereM = OrderTaylor .

Bpade[n] The [M,M] Pade approximant of thenth-order homotopy approxi-mation ofB(τ) in polynomial of

√τ to o(τM), whereM = OrderTaylor .

OrderHAM The highest order of homotopy-approximations of the dimension-less results in the data fileAPO-48-10.txt . Its defaults is 10 in the data fileAPO-48-10.txt .

OrderTaylor The order ofB(τ) in polynomial of√

τ. Its defaults is 48 in thedata fileAPO-48-10.txt .

Bp The perpetual optimal exercise priceBp = Xγ/(1+ γ).

Appendix 13.3 Mathematica codeAPOhfor businessmen 463

Mathematica codeAPOhfor businessmenby Shijun LIAO

Shanghai Jiao Tong UniversityDecember 2010


( * Input dimensionless results given by means of the HAM * )<<APO-48-10.txt;Print["-------------------------------------------- ---------"];Print["OrderTaylor = ",OrderTaylor];Print["OrderHAM = ",OrderHAM];Print["-------------------------------------------- ---------"];

( * Define APOh[Order_] * )APOh[Order_]:=Module[temp,n,i,j,s,Print[" OrderTaylor = ",OrderTaylor];Print[" strike price = ?"];temp[0] = Input[];X = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;Print[" risk-free interest rate = ?"];temp[0] = Input[];r = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;Print[" volatility = ?"];temp[0] = Input[];sigma = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;Print[" time to expiry (year) = ?"];temp[0] = Input[];T = IntegerPart[temp[0] * 10ˆ10]/10ˆ10;gamma = 2* r/sigmaˆ2;texp = sigmaˆ2 * T/2;Bp = X* gamma/(1 + gamma);Print["-------------------------------------------- ---------"];Print[" INPUT PARAMETERS: "];Print[" Strike price (X) = ",X," ($) "];Print[" Risk-free interest rate (r) = ",r];Print[" Volatility (sigma) = ",sigma];Print[" Time to expiry (T) = ",T," (year)"];Print["-------------------------------------------- ---------"];Print[" CORRESPONDING PARAMETERS: "];Print[" gamma = ",gamma];Print[" dimensionless time to expiry (texp)=",texp//N];Print[" perpetual optimal exercise price (Bp)=",Bp//N,"( $)"];Print["-------------------------------------------- ---------"];Print[" CONTROL PARAMETERS: "];Print[" OrderTaylor = ",OrderTaylor];Print[" c0 = ",c0];Print["-------------------------------------------- ---------"];For[n = 1,n <= Min[Order,OrderHAM], n++,

Print[" n = ",n];


temp[0] = X * BB[n] /. t-> (sigmaˆ2 * t/2)//Expand;B[n] = Collect[temp[0],t];If[NumberQ[gamma],

temp[0] = BB[n] /. tˆi_. -> sˆ(2 * i);temp[1] = Pade[temp[0],s,0,OrderTaylor,OrderTaylor] ;BBpade[n] = temp[1] /. sˆj_. -> tˆ(j/2);Bpade[n] = X * BBpade[n]/. t -> (sigmaˆ2 * t/2);];

If[NumberQ[gamma] && NumberQ[sigma] && NumberQ[X],Print["Optimal exercise price at the time to expiration

= ", B[n]/.t->T//N];Print["Modified result given by Pade technique = ",

Bpade[n]/.t->T//N];];

];Print["Well done"];If[NumberQ[gamma] && NumberQ[sigma] && NumberQ[X],

n = Min[Order,OrderHAM];Plot[Bp, B[n], Bpade[n], t, 0, 1.25 * T,

PlotRange -> 0.8 * Bp, X, PlotStyle ->RGBColor[1,0,0],RGBColor[0,1,0],RGBColor[0,0,1]];

Print[" Order of homotopy-approximation : ",n];Print[" Green line : optimal exercise boundary B

in polynomial "];Print[" Blue line : optimal exercise boundary B

by Pade method "];Print[" Red line : perpetual optimal exercise price "];];

];

References 465

References

1. Allegretto, W., Lin, Y., Yang, H.: Simulation and the early-exercise option problem. Discr.Contin. Dyn. Syst. B.8, 127-136 (2001)

2. Allobaidi, G., Mallier, R.: On the optimal exercise boundary for an American put option.Journal of Applied Mathematics.1, No. 1, 39-45 (2001)

3. Barles, G., Burdeau, J., Romano, M., Samsoen, N.: Critical stock price near expiration. Math-ematical Finance.5, No. 2, 77-95 (1995)

4. Blanchet, A.: On the regularity of the free boundary in theparabolic obstacle problem - Ap-plication to American options. Nonlinear Analysis.65, 1362-1378 (2006)

5. Brennan, M., Schwartz, E.: The valuation of American put options. Journal of Finance.32,449–462 (1977)

6. Broadie, M., Detemple, J.: American option valuation: new bounds, approximations, and acomparison of existing methods. Review of Financial Studies.9, No. 4, 1211-1250 (1996)

7. Broadie, M., Detemple, J.: Recent advances in numerical methods for pricing derivative se-curities. In Numerical Methods in Finance, 43-66, edited byRogers, L. C. G. and Talay, D.,Cambridge University Press, England (1997)

8. Bunch, D.S., Johnson, H.: The American put option and its critical stock price. Journal ofFinance.5, 2333-2356 (2000)

9. Carr, P., Jarrow, R., Myneni, R.: Alternative characterizations of American put options. Math-ematical Finance.2, 87-106 (1992)

10. Carr, P., Faguet, D.: Fast accurate valuation of American options. Working paper, CornellUniversity (1994)

11. Chen, X.F., Chadam, J., Stamicar R.: The optimal exercise boundary for American put op-tions: analytic and numerical approximations. Working paper (http://www.math.pitt.edu/-xfc/Option/CCSFinal.ps), University of Pittsburgh (2000)

12. Chen, X.F., Chadam, J.: A mathematical analysis for the optimal exercise boundary Americanput option. Working paper (http://www.pitt.edu/-chadam/papers/2CC9-30-05.pdf), Universityof Pittsburgh (2005)

13. Cheng, J.: Application of the Homotopy Analysis Method in the Nonlinear Mechanics andFinance (in Chinese). PhD Dissertation, Shanghai JiaotongUniversity (2008)

14. Cheng, J., Zhu, S.P., Liao, S.J.: An explicit series approximation to the optimal exerciseboundary of American put options. Communications in Nonlinear Science and NumericalSimulation.15 1148 – 1158 (2010),

15. Cox, J., Ross, S., Rubinstein, M.: Option pricing: a simplified approach. Journal of FinancialEconomics.7, 229-263 (1979)

16. Dempster, M.: Fast numerical valation of American, exotic and complex options. Departmentof Mathematics Research Report, University of Essex, Colchester, England (1994)

17. Evans, J.D., Kuske, R., Keller, J.B.: American options on asserts with dividends near expiry.Mathematical Finance.12, No. 3, 219-237 (2002)

18. Geske, R., Johnson, H.E.: The American put option valuedanalytically. Journal of Finance.5, 1511-1523 (1984)

19. Grant, D., Vora, G., Weeks, D.: Simulation and the early-exercise option problem. Journal ofFinancial Engineering.5, 211-227 (1996)

20. Hon, Y.C., Mao, X.Z.: A radial basis function method for solving options pricing model.Journal of Financial Engineering.8, 31-49 (1997)

21. Huang, J.Z., Marti, G.S., Yu, G.G.: Pricing and Hedging American Options: A RecursiveIntegration Method. Review of Financial Studies.9, 277-300 (1996)

22. Jaillet, P., Lamberton, D., Lapeyre, B.: Variational inequalities and the pricing of Americanoptions. Acta Applicandae Math.21, 263-289 (1990)

23. Kim, I.J.: The analytic valuation of American options. Review of Financial Studies.3, 547-572 (1990)

24. Knessl, C.: A note on a moving boundary problem arising inthe American put option. Studiesin Applied Mathematics.107, 157-183 (2001)


25. Kuske, R.A., Keller, J.B.: Optional exercise boundary for an American put option. AppliedMathematical Finance.5, 107-116 (1998)

26. Landau, H.G.: Heat conduction in melting solid. Quarterly of Applied Mathematics.8, 81-94(1950)

27. Longstaff, F., Schwartz, E.S.: A radial basis function method for solving options pricingmodel. Review of Finanial Studies.14, 113-147 (2001)
















43. Wu, L., Kwok, Y.K.: A front-fixing finite difference method for the valuation of Americanoptions. Journal of Financial Engineering.6, 83-97 (1997)


45. Zhang, J.E., Li, T.C.: Pricing and hedging American options analytically: a perturbationmethod. Working paper, University of Hong Kong (2006)



Chapter 14Two and three dimensional Gelfand equation

Abstract Using the 2D and 3D Gelfand equation as an example, we illustrate thatthe homotopy analysis method (HAM) can be used to solve a 2nd-order nonlinearpartial differential equation (PDE) in a rather easy way by transforming it into aninfinite number of the 4th or 6th-order linear PDEs. This is mainly because theHAM provides us extremely large freedom to choose auxiliarylinear operator andbesides a convenient way to guarantee the convergence of solution series. To thebest of our knowledge, such kind of transformation has neverbeen used by otheranalytic/numerical methods. This illustrates the originality and great flexibility ofthe HAM for strongly nonlinear problems. It also suggests that we must keep anopen mind, since we might have much larger freedom to solve nonlinear problemsthan we thought traditionally.

14.1 Introduction

In Chapter 2, we illustrate that the homotopy analysis method (HAM) [12–25, 32]provides us extremely large freedom to choose the auxiliarylinear operator: the 2nd-order nonlinear ordinary differential equation (ODE) describing a periodic oscilla-tion can be transferred into an infinite number of linear 2κ-order linear ODEs, whereκ ≥ 1 is any a positive integer, and besides the convergence of the series solution isguaranteed by means of the so-called convergence-controlparameterc0. In addition,it is due to this extremely large freedom that the nonlinear PDEs describing the non-similarity boundary-layer flows (in Chapter 11) and the unsteady boundary-layerflows (in Chapter 12) can be transferred into an infinite number of linear ODEs, andthus can be solved by means of the Mathematica packageBVPh1.0. In this chapter,we further illustrate that, such kind of freedom on the choice of the auxiliary linearoperatorL can greatly simplify solving some high dimensional nonlinear PDEs.

For example, let us consider the high dimensional Gelfand equation [5,26]

467

468 14 Two and three dimensional Gelfand equation

∆u+λ eu = 0, x ∈ Ω ⊂ RN,u= 0, x ∈ ∂Ω ,

(14.1)

where∆ denotes the Laplace operator,λ is an eigenvalue,u is the eigenfunction,x isa vector for the spatial variables,N= 1,2,3 corresponds to the number of the dimen-sion,Ω is the domain,∂Ω denotes the boundary, respectively. Physically, Gelfandequation arises in several contexts, such as chemical reactor theory, the steady-stateequation for a nonlinear heat conduction problem, questions on geometry and rel-ativity about the expansion of universe, and so on. Mathematically, the governingequation has strong nonlinearity, since it contains the exponential term exp(u).

Generally speaking, it is difficult to gain accurate analytic approximations ofeigenvalues and eigenfunctions of a high-dimensional PDE with strong nonlinear-ity. The investigation on Gelfand problem [5, 26] has a long history. Liouville [26]gave a closed-form expression of eigenvalue for one-dimensional (1D) Gelfandequation. Based on Chebyshev functions, Boyd [5] proposed an analytic approachand a numerical method for two-dimensional (2D) Gelfand equation on the square[−1,1]× [−1,1], and suggested a one-point analytic approximation

λ = 3.2 A e−0.64A, (14.2)

and a three-point analytic approximation

λ = (2.667A+4.830B+0.127C) e−0.381A−0.254B−0.018C, (14.3)

whereA= u(0,0) and

B = A(

0.829−0.566e0.463A−0.0787e−0.209A)

/G,

C = A(

−1.934+0.514e0.463A+1.975e−0.209A)

/G,

G = 0.2763+e0.463A+0.0483e−0.209A.

This problem still attracts the attention of current researchers [8,27].In this chapter, the HAM is employed to solve the 2nd-order 2D(or 3D) Gelfand

equation with the high nonlinearity in a rather easy way by transferring it into aninfinite number of the 4th (or 6th) order linear 2D (or 3D) PDEs.

14.2 Homotopy-approximations of 2D Gelfand equation


Following Boyd [5], let us consider the 2D Gelfand equation

∆u+λ eu = 0, −1< x< 1, −1< y< 1, (14.4)

14.2 Homotopy-approximations of 2D Gelfand equation 469

subject to the boundary condition on the four walls

u(x,±1) = 0, u(±1,y) = 0. (14.5)

The above nonlinear eigenvalue equation has an infinite number of eigenvaluesand eigenfunctions. Obviously, different eigenfunctionshave different values at theorigin (0,0). So, we can use the different values of

A= u(0,0) (14.6)

to distinguish different eigenfunctions and the corresponding eigenvalues. Writing

u(x,y) = A+w(x,y), (14.7)

whereA is a given constant, the original 2D Gelfand equation becomes

∆w+λ eA ew = 0, −1< x< 1, −1< y< 1, (14.8)

subject to the boundary conditions on the four walls

w(x,±1) =−A, w(±1,y) =−A, (14.9)

with the restrictionw(0,0) = 0. (14.10)

Let w0(x,y) andλ0 denote the initial guesses of the eigenfunctionw(x,y) and theeigenvalueλ , respectively. Note that the initial guessw0(x,y) is unnecessary to sat-isfy the boundary conditions (14.9) and the restriction (14.10). Letq∈ [0,1] denotean embedding parameter. In the frame of the HAM, we should first of all constructsuch two continuous variations (or deformations)φ(x,y;q) andΛ(q) that, asq in-creases from 0 to 1,φ(x,y;q) varies continuously from the initial guessw0(x,y) tothe eigenfunctionw(x,y), and at the same time,Λ(q) varies continuously from theinitial guessλ0 to the eigenvalueλ , respectively. Such two continuous variations aregoverned by the zeroth-order deformation equation

(1−q)L [φ(x,y;q)−w0(x,y)] = c0 q N [φ(x,y;q),Λ(q)] (14.11)

on the square(x,y) ∈ [−1,1]× [−1,1], subject to the boundary conditions on thefour walls

(1−q) [φ(±1,y;q)−w0(±1,y)] = c0 q [φ(±1,y;q)+A] , (14.12)

(1−q) [φ(x,±1;q)−w0(x,±1)] = c0 q [φ(x,±1;q)+A] , (14.13)

with the additional restriction at the origin

(1−q) [φ(0,0;q)−w0(0,0)] = c0 q φ(0,0;q), (14.14)

where


N [φ(x,y;q),Λ(q)]

=∂ 2φ(x,y;q)

∂x2 +∂ 2φ(x,y;q)

∂y2 +eA Λ(q) exp[φ(x,y;q)] (14.15)

is a nonlinear operator corresponding to the 2D Gelfand equation (14.8),L is anauxiliary linear operator with the propertyL [0] = 0, andc0 6= 0 is the convergence-control parameter, respectively. It should be emphasized here that we have extremelylarge freedom to choose the auxiliary linear operatorL and the convergence-controlparameterc0, as shown later.

Whenq= 0, sinceL [0] = 0, the zeroth-order deformation equations (14.11) to(14.14) have the solution

φ(x,y;0) = w0(x,y). (14.16)

Whenq= 1, sincec0 6= 0, the zeroth-order deformation equations (14.11) to (14.14)are equivalent to the original PDEs (14.8) to (14.10), provided

φ(x,y;1) = w(x,y), Λ(1) = λ . (14.17)

Thus, as the embedding parameterq∈ [0,1] increases from 0 to 1,φ(x,y;q) indeedvaries continuously from the initial guessw0(x,y) to the eigenfunctionw(x,y), sodoesΛ(q) from the initial guessλ0 to the eigenvalueλ . So, mathematically, thezeroth-order deformation equations (14.11) to (14.14) construct two continuous ho-motopies

φ(x,y;q) : w0(x,y)∼ w(x,y), Λ(q) : λ0 ∼ λ .

Expandingφ(x,y;q) andΛ(q) into Maclaurin series with respect to the embed-ding parameterq and using (14.16), we have the homotopy-Maclaurin series

φ(x,y;q) = w0(x,y)++∞

∑n=1

wn(x,y) qn, (14.18)

Λ(q) = λ0++∞

∑n=1

λn qn, (14.19)

where

wn(x,y) =1n!

∂ nφ(x,y;q)∂qn

∣

∣

∣

∣

q=0= Dn [φ(x,y;q)] , (14.20)

λn =1n!

∂ nΛ(q)∂qn

∣

∣

∣

∣

q=0= Dn [Λ(q)] , (14.21)

are the so-called homotopy-derivatives ofφ(x,y;q) andΛ(q), andDn is thenth-order homotopy-derivative operator, respectively. It should be emphasized that, inthe frame of the HAM, we have great freedom to choose the auxiliary linear operatorL , the initial guessw0(x,y), and especially the convergence-control parameterc0:all of them influence the convergence of the series (14.18) and (14.19). Assume thatall of them are chosen so properly that the homotopy-Maclaurin series (14.18) and


(14.19) absolutely converge atq= 1. Then, due to (14.17), we have the homotopy-series solution

w(x,y) = w0(x,y)++∞

∑n=1

wn(x,y), (14.22)

λ = λ0++∞

∑n=1

λn. (14.23)

The differential equations for the unknownwn(x,y) and λn−1 (n ≥ 1) can bededuced directly from the zeroth-order deformation equations (14.11) to (14.14):taking thenth-order homotopy-derivative on both sides of the zeroth-order defor-mation equations (14.11) to (14.14), we have the so-callednth-order deformationequation

L [wn(x,y)− χn wn−1(x,y)] = c0 δn−1(x,y), (14.24)

subject to the boundary conditions on the four walls

wn(x,±1) = µn(x,±1), wn(±1,y) = µn(±1,y), (14.25)

and the additional restriction at the origin

wn(0,0) = (χn+ c0) wn−1(0,0), (14.26)

where

χn =

1, n> 1,0, n= 1,

(14.27)

and

δk(x,y) = DkN [φ(x,y;q)]

= ∆wk(x,y)+eAk

∑j=0

λk− j D j

[

eφ(x,y;q)]

, (14.28)

µn(x,y) = (χn+ c0) wn−1(x,y)+ c0 (1− χn) A, (14.29)

are gained by Theorem 4.1. According to Theorem 4.7, we have the recursion for-mula

D0

[

eφ(x,y;q)]

= ew0(x,y), Dk

[

eφ(x,y;q)]

=k−1

∑j=0

(

1− jk

)

wk− j D j

[

eφ(x,y;q)]

.

(14.30)Thus, it is easy to calculate the termδn−1(x,y) of (14.24) by means of computeralgebra system like Mathematica. Note that (14.24) can be gained by Theorem 4.15directly. For details, please refer to Chapter 4.

As mentioned before, in the frame of the HAM, we have extremely large freedomto choose the auxiliary linear operatorL and the initial guessw0(x,y). Note that the


Gelfand equation contains a linear operator, i.e. the Laplace operator∆ . However,even the 2D linear PDE

∆u(x,y) = 0, −1≤ x≤ 1, −1≤ y≤ 1,

has a complicated common solution

u(x,y) =+∞

∑k=0

(

Bk,1 e−αkx+Bk,2 eαkx)

[

Bk,3 sin(αky)+Bk,4 cos(αky)]

++∞

∑k=1

(

Bk,5 e−β ky+Bk,6 eβ ky)

[

Bk,7 sin(βkx)+Bk,8 cos(βkx)]

,

where the coefficientsα,β andBk,i are determined by the boundary conditions. Forexample, on the boundaryx= 1, the above expression reads

u(1,y) =+∞

∑k=0

(

Bk,1 e−αk+Bk,2 eαk)

[

Bk,3 sin(αky)+Bk,4 cos(αky)]

++∞

∑k=1

(

Bk,5 e−β ky+Bk,6 eβ ky)

[

Bk,7 sin(βk)+Bk,8 cos(βk)]

.

So, it is not easy to satisfy the boundary condition onx= 1 by means of the aboveexpression. Therefore, if we choose the Laplace operator∆ as the auxiliary linearoperatorL , it is not easy to solve the high-order deformation equations (14.24)to (14.26). Thus, we should choose an auxiliary linear operator L better than theLaplace operator∆ . Fortunately, in the frame of the HAM, we have extremely largefreedom to choose the auxiliary linear operatorL . Using this kind of freedom, weindeed could choose such an auxiliary linear operatorL that it is rather easy tosolve the high-order deformation equation, as shown below.

Note that the boundary conditions (14.5) and the boundary

∂Ω : (x,y) ∈ [−1,1]× [−1,1]

itself are symmetric aboutx andy axes. Besides, it is easy to prove that, ifw(x,y)is a solution of the 2D Gelfand equation, thenw(±x,±y) is also its solution. So,w(x,y) is symmetric aboutx andy axis, and therefore can be expressed by the basefunctions

x2m y2n | m= 1,2,3, · · · , n= 1,2,3, · · ·

(14.31)

in the form

w(x,y) =+∞

∑m=1

+∞

∑n=1

bm,n x2m y2n, (14.32)

wherebm,n is a coefficient to be determined. It provides us the so-called solutionexpressionof w(x,y). Our aim is to find a convergent series solution of the eigen-functionw(x,y) in the form (14.32) and the corresponding convergent seriesof theeigenvalueλ , for a given value ofA.


To satisfy the additional restriction (14.10) and the solution-expression (14.32),we choose the simplest initial guess

w0(x,y) = 0. (14.33)

Note that this initial guess satisfies the restriction (14.10), but not the boundaryconditions (14.9) on the four walls.

Next, we should choose the auxiliary linear operatorL . To obey the solutionexpression (14.32) ofw(x,y), it should hold

L [C1] = 0 (14.34)

for any a non-zero constantC1. Besides, sincew0(x,y) = 0, it holdsδ0(x,y) = λ0 eA,therefore,δn−1(x,y) may contain a non-zero constant. So, to obey the solution ex-pression (14.32), the inverse operatorL −1 of the auxiliary linear operatorL shouldhave the property

L−1[1] =C2 x2 y2, (14.35)

whereC2 is a non-zero constant. Especially, the linear auxiliary operatorL shouldbe chosen properly so that it is rather easy to solve the high-order deformation equa-tion (14.24) with the boundary conditions (14.25) at the four walls. Let w∗

m(x,y)denote a special solution of (14.24). Obviously,

w∗n(x,y)−w∗

n(x,±1)−w∗n(±1,y)+w∗

n(±1,±1)

vanishes on the four walls, and besides

µn(x,±1)+ µn(±1,y)− µn(±1,±1)

satisfies the boundary conditions (14.25) on the four walls,whereµn(x,y) is definedby (14.29). Therefore,

wn(x,y) = w∗n(x,y)−w∗

n(x,±1)−w∗n(±1,y)+w∗

n(±1,±1)

+ µn(x,±1)+ µn(±1,y)− µn(±1,±1) (14.36)

satisfies thenth-order deformation equation (14.24) and the boundary conditions(14.25), as long as the auxiliary linear operatorL has the property

L [ f (x)] = L [g(y)] = 0 (14.37)

for arbitrary smooth functionsf (x) andg(y).There exist an infinite number of linear differential operators satisfying the

above-mentioned properties (14.34), (14.35) and (14.37),such as the 2nd-order lin-ear operator

L u = c2

(

1xy

)

∂ 2u∂x∂y

, (14.38)


and the 4th-order linear operator

L u = c4∂ 4u

∂x2∂y2 , (14.39)

wherec2 andc4 are constants. These two linear operators are special casesof themore general linear operator

L u =

(

c2

xy

)

∂ 2u∂x∂y

+ c4∂ 4u

∂x2∂y2 , (14.40)

whose inverse operator is

L−1[xk yn] =

xk+2 yn+2

(k+2)(n+2)[c2+ c4(k+1)(n+1)]. (14.41)

Using the above inverse operatorL −1, it is very easy to gain a special solution

w∗n(x,y) = c0 L

−1 [δn−1(x,y)]+ χn wn−1(x,y) (14.42)

of the nth-order deformation equation (14.24). Thereafter, the solution wn(x,y) ofthe high-order deformation equations (14.24) to (14.25) isobtained by means of(14.36). Then,λn−1 is determined by the linear algebraic equation (14.26). Formoredetails, please refer to Liao and Tan [25].

Note that the above approach needs only algebraic calculations. Thus, it is rathereasy to obtain high-order approximations of the eigenfunctionw(x,y) and the eigen-value λ , especially by means of algebra computer system such as Mathematica,Maple, and so on. In this way, we greatly simplifies solving the 2D Gelfand equa-tion, as shown later.

The corresponding Mathematica code for the 2D Gelfand equation is given in theAppendix 14.1 and free available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

14.2.2 Homotopy-approximations

In the frame of the HAM, the convergence-controlparameterc0 provides us a conve-nient way to guarantee the convergence of solution series. It should be emphasizedthat the auxiliary linear operator (14.40) contains two parametersc2 andc4, whichcan be also regarded as the convergence-controlparameterslike c0. So, we have nowthree convergence-control parametersc0,c2 andc4, which are used to guarantee theconvergence of (14.22) for the eigenfunction and (14.23) for the eigenvalue.

It is found that, for the arbitrary values ofc2 andc4, thenth-order approximationof the eigenfunctionu(x,y) on the four walls reads

A(1+ c0)n,


Table 14.1 The minimum of the averaged squared residual of the 2D Gelfand equation (14.8)whenA= 1 by means ofc0 =−1 and the 2nd-order auxiliary linear operator (14.38)

Order of approximationm Minimum of squared residual Optimal value ofc2

10 1.23×10−2 -5.354515 1.57×10−2 -7.934920 1.89×10−2 -10.5021

which vanishes asn→+∞ only if

|1+ c0|< 1. (14.43)

The above expression restricts the choice of the convergence-control parameterc0.Especially, whenc0 = −1, the boundary conditions on the four walls are exactlysatisfied at every order of approximation. Due to this reason, we choosec0 =−1 forthe sake of simplicity. Then, there are two unknown convergence-controlparametersc2 andc4 left.

Define the averaged squared residual of the 2D Gelfand equation

Em =1

100

9

∑i=0

9

∑j=0

[

∆u(i∆x, j∆y)+λ eu(i∆x, j∆y)]2, ∆x= ∆y=

110

, (14.44)

whereu andλ are themth-order homotopy-approximation of the eigenfunction andthe eigenvalue, respectively. Obviously,Em is dependent upon the two convergence-control parametersc2 andc4.

Since the original Gelfand equation (14.8) is2nd-order, we first use the2nd-order linear operator (14.38) as the auxiliary linear operator, corresponding toc4 =0of the linear operator (14.40). Due toc0 =−1 andc4 = 0, there exists only one non-zero convergence-control parameterc2 now. Without loss of generality, let us con-sider the case ofA= 1. It is found that the minimum of the averaged squared resid-ualEm of the governing equation (14.8) doesnotdecreases asm increases, as shownin Table 14.1. Besides, there does not exist an interval ofc2, where the averagedsquared residualEm of the governing equation (14.8) decreases asm increases, asshown in Fig. 14.1. For example, in case ofA= 1 andc2 =−20, the correspondinghomotopy-series is divergent. Therefore, we can not gain convergent series solutionof the 2D Gelfand equation (14.8) by using the 2nd-order linear operator (14.38) asthe auxiliary linear operator.

Then, we use the4th-order linear operator (14.39) as the auxiliary linear op-erator, corresponding toc2 = 0 of the linear operator (14.40). The minimum ofthe averaged squared residual of (14.8) decreases to 2.85×10−4 at the 20th-orderhomotopy-approximation by means of the optimal convergence-control parameterc∗4 = −0.2494, as listed in Table 14.2. Besides, for arbitrary convergence-controlparameterc4 ≤ −1/4, the averaged squared residualEm of (14.8) decreases asm,the order of approximation, increases, as shown in Fig. 14.2. This suggests that we


Table 14.2 The minimum of the averaged squared residual of the 2D Gelfand equation whenA= 1by means ofc0 =−1 and the 4th-order auxiliary linear operator (14.39)


3 2.65×10−2 -0.44435 9.07×10−3 -0.31448 3.17×10−3 -0.286610 1.83×10−3 -0.276315 6.31×10−4 -0.255520 2.85×10−4 -0.2494

Fig. 14.1 Averaged squaredresidual of (14.8) versusc2when A = 1 by means ofc0 = −1 and using the 2nd-order linear operator (14.38)as the auxiliary linear opera-tor. Dashed line: 10th-orderhomotopy-approximation;Dash-dotted line: 15th-orderhomotopy-approximation;Solid line: 20th-orderhomotopy-approximation.

c2

Ave

rage

dre

sidu

alsq

uare

-20 -15 -10 -5 010-2

10-1

100

101

A = 1, c0 = -1, c4 = 0

Fig. 14.2 Averaged squaredresidual of (14.8) versusc4when A = 1 by means ofc0 = −1 and using the 4th-order linear operator (14.39)as the auxiliary linear opera-tor. Dashed line: 10th-orderhomotopy-approximation;Dash-dotted line: 15th-orderhomotopy-approximation;Solid line: 20th-orderhomotopy-approximation.

c4

Ave

rage

dre

sidu

alsq

uare

-2 -1.5 -1 -0.5 010-4

10-3

10-2

A = 1, c0 = -1, c2 = 0

can gain convergent series solution of the eigenfunction and the eigenvalue by us-ing the4th-order linear operator (14.39) as the auxiliary linear operator with theconvergence-control parameterc4 ≤−1/4. This is indeed true: whenA= 1, the av-eraged squared residual of the 2D Gelfand equation (14.8) monotonously decreasesto 1.5×10−7 at the 100th-order homotopy-approximationby means of the4th-orderauxiliary linear operator (14.39) withc4 = −1/4, and besides the corresponding


Table 14.3 The eigenvalueλ and averaged squared residualEm of (14.8) whenA= 1, c0 =−1 bymeans of the 4th-order linear operator (14.39) withc4 =−1/4 as the auxiliary linear operator.

Order of approximationm Minimum of squared residual Eigenvalueλ

3 0.574 1.57655 9.4×10−2 1.605910 4.9×10−3 1.621215 6.7×10−4 1.623720 2.8×10−4 1.623325 1.4×10−4 1.623130 7.6×10−5 1.623140 2.5×10−5 1.6231150 8.9×10−6 1.623115860 3.3×10−6 1.623115870 1.3×10−6 1.623115880 5.5×10−7 1.6231158490 2.6×10−7 1.62311584100 1.5×10−7 1.62311584

Fig. 14.3 Profiles ofu(x,y)of 2D Gelfand equation(14.8) whenA = 1, c0 = −1by means of the 4th-orderlinear operator (14.39)with c4 = −1/4 as theauxiliary linear operator.Lines: 20th-order homotopy-approximation; Symbols:10th-order homotopy-approximation; Solid line:y= 0; Dashed line:y= 1/2;Dash-dotted line:y= 1/4.

x

u

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

A = 1

eigenvalueλ converges to 1.62311584, as shown in Table 14.3. Note that even the10th-order homotopy-approximationof the eigenfunctionu(x,y) is accurate enough,as shown in Fig. 14.3.

It is found that, for other values ofA, we can gain convergent eigenfunctionand eigenvalue by means of the4th-order auxiliary linear operator (14.39) witha proper convergence-control parameterc4 in a similar way. For example, whenA= 5, we gain the convergent eigenfunction and eigenvalue by means of the optimalconvergence-control parameterc4 = −2/5: as shown in Table 14.4, the averagedsquared residua of (14.8) decreases monotonously and the corresponding eigenvalueconverges toλ = 0.516. Besides, the 30th-order homotopy-approximation of theeigenfunction agrees well with the 50th-order ones, as shown in Fig. 14.4.

RegardingA as an unknown parameter, we obtain the 30th-order homotopy-approximation of the eigenvalue by means ofc0 = −1 and the 4th-order auxil-


Table 14.4 The eigenvalueλ and the averaged squared residualEm of (14.8) whenA= 5, c0 =−1by means of the 4th-order linear operator (14.39) withc4 =−2/5 as the auxiliary linear operator.

Order of approximationm Averaged squared residual Eigenvalueλ

5 21.1 0.47110 3.31 0.49015 1.11 0.50120 0.56 0.50730 0.18 0.51240 7.3×10−2 0.51450 3.3×10−2 0.51560 1.6×10−2 0.51670 8.3×10−3 0.51680 4.6×10−3 0.516

Fig. 14.4 Profiles ofu(x,y)of 2D Gelfand equation(14.8) whenA = 5, c0 = −1by means of the 4th-orderlinear operator (14.39)with c4 = −2/5 as theauxiliary linear operator.Lines: 50th-order homotopy-approximation; Symbols:30th-order homotopy-approximation; Solid line:y= 0; Dashed line:y= 1/2;Dash-dotted line:y= 1/4.

x

u

-1 -0.5 0 0.5 10

1

2

3

4

5A = 5

iary linear operator (14.39) withc4 = −1. It is found that the eigenvalue of the2D Gelfand equation has the maximum value 1.702 atA= 1.391, which agrees wellwith Boyd’s numerical resultλmax= 1.702 atA = 1.39, as shown in Table 14.5.Besides, the simplified formula

λ ≈ e−A (3.39403927A+0.85308129A2+0.14514688A3

+1.83020402×10−2 A4+1.65401606×10−3 A5

+1.03116169×10−4 A6+5.35313091×10−6 A7)

(14.45)

given by the first seven terms of the 30th-order homotopy-approximation of theeigenvalue agrees well with the 20th-order homotopy-approximation and Boyd’snumerical results [5] in 0≤ A≤ 10, as shown in Fig. 14.5.

All of these verify that, in the frame of the HAM, the2nd-ordernonlinear PDE(14.4) can be transferred into an infinite number of the4th-orderlinear PDEs (14.24)governed by the4th-orderauxiliary linear operator


Table 14.5 Comparison of the maximum eigenvalueλmaxof the 2D Gelfand equation with Boyd’sanalytic and numerical results. The homotopy-approximations are obtained by means ofc0 = −1and the 4th-order auxiliary linear operator (14.39) withc4 =−1.

λmax The corresponding value ofA

5th-order HAM approx. 1.701 1.38310th-order HAM approx. 1.702 1.38915th-order HAM approx. 1.702 1.39120th-order HAM approx. 1.702 1.39125th-order HAM approx. 1.702 1.39130th-order HAM approx. 1.702 1.391Boyd’s 1-point formula (14.2) 1.84 1.56Boyd’s 3-point formula (14.3) 1.735 1.465Boyd’s numerical result 1.702 1.39

Fig. 14.5 Comparison of theeigenvalue of 2D Gelfandequation given by differ-ent methods. Solid line: thesimplified formula (14.45)given by the first 7 terms ofthe 30th-order homotopy-approximation; Open cir-cles: 20th-order homotopy-approximation; Filled circles:Numerical results given byBoyd [5]; Dashed line: Boyd’s1-point approximation (14.2);Dash-dotted line: Boyd’s 3-points approximation (14.3). A

λ

0 2 4 6 8 100

0.5

1

1.5

2

eigenvalue of 2D Gelfand equation

L u= c4∂ 4u

∂x2∂y2 .

Note that, using above4th-order auxiliary linear operator, there exist an infinitenumber of smooth functions, such as

sin(x), exp(y), x f(y), y g(x)

and so on, which satisfy

L [sin(x)] = L [exp(y)] = L [x f(y)] = L [y g(x)] = 0,

wheref (y) andg(x) arearbitrary smooth functions except the polynomials ofy andx, respectively. However, all of them arenot allowed in the solution of the high-order deformation equation (14.24), because they disobeythe solution-expression(14.32). In other words, ifw∗

n(x,y) is a special solution of (14.24), then


w∗n(x,y)+B1 sin(x)+B2 exp(y)+B3 x f(y)+B4 y g(x)+ · · ·

also satisfies (14.24), whereB1,B2,B3 andB4 are coefficients, andf (y) andg(x) arenot polynomials ofy andx, respectively. However, to obey the solution-expression(14.32), wehad toenforce

B1 = B2 = B3 = B4 = · · ·= 0

in the frame of the HAM. This illustrates that the so-calledsolution-expressionindeed plays an important role in the frame of the HAM, which however can greatlysimplify solving some nonlinear problems, if properly used.

It should be emphasized that the convergence of (14.22) for the eigenfunction and(14.23) for the eigenvalue is guaranteed by two convergence-control parametersc0

andc4. In fact, it is these two convergence-control parameters that provide a strongsupport for the extremely large freedom of the HAM on the choice of the auxiliarylinear operatorL , because a divergent series has no meanings at all.

Note that, by means of perturbation methods [4, 6, 7, 10, 28–31], it is possible totransfer a nonlinear differential equation into an infinitenumber oflower-orderlin-ear differential equations, when the highest derivative ismultiplied by a perturbationquantity. However, to the best of the author’s knowledge, a2nd-ordernonlinear PDEhasneverbeen transferred into an infinite number of the4th-order linear PDEs insuch a way by any other analytic and numerical methods! It suggests that we mighthave much larger freedom to solve nonlinear problems than wethought and be-lieved traditionally! This shows the originality and greatflexibility of the HAM fornonlinear problems.

Indeed,a truly new method always gives something new and/or different.

14.3 Homotopy-approximations of 3D Gelfand equation

Let us further consider the 3D Gelfand equation

∆u+λ eu = 0, −1≤ x,y,z≤ 1 (14.46)


u(±1,y,z) = 0, u(x,±1,z) = 0, u(x,y,±1) = 0. (14.47)

The above 2nd-order nonlinear PDE can be easily solved by means of the HAM in arather similar way as mentioned above. A corresponding Mathematica code is givenin the Appendix 14.2 and available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

WriteA= u(0,0,0) (14.48)

andu(x,y,z) = A+w(x,y,z). (14.49)


The original 3D Gelfand equation become

∆w+λ eA ew = 0, −1< x,y,z< 1, (14.50)

subject to the boundary conditions on the six walls

w(±1,y,z) = w(x,±1,z) = w(x,y,±1) =−A, (14.51)

with the restrictionw(0,0,0) = 0. (14.52)

Let w0(x,y,z) andλ0 denote the initial guesses of the eigenfunction and eigen-value,q∈ [0,1] the embedding parameter, respectively. In the frame of the HAM, wefirst construct such two continuous mappingsφ(x,y,z;q) andΛ(q) that, asq∈ [0,1]increases from 0 to 1,φ(x,y,z;q) varies from the initial guessw0(x,y,z) to the eigen-functionw(x,y,z), and at the same time,Λ(q) varies from the initial guessλ0 to theeigenvalueλ , respectively. Such kind of two continuous mappingsφ(x,y,z;q) andΛ(q) are governed by the zeroth-order deformation equation

(1−q) [φ(x,y,z;q)−w0(x,y,z)] = q c0 N [φ(x,y,z;q),Λ(q)] (14.53)

on the cubic−1< x,y,z<+1, subject to the boundary conditions on the six walls

(1−q) [φ(±1,y,z;q)−w0(±1,y,z)] = c0 q [φ(±1,y,z;q)+A] , (14.54)

(1−q) [φ(x,±1,z;q)−w0(x,±1,z)] = c0 q [φ(x,±1,z;q)+A] , (14.55)

(1−q) [φ(x,y,±1;q)−w0(x,y,±1)] = c0 q [φ(x,y,±1;q)+A] , (14.56)

with the additional restriction at the origin

(1−q) [φ(0,0,0;q)−w0(0,0,0)] = c0 q φ(0,0,0;q), (14.57)

where

N [φ(x,y,z;q),Λ(q)]

=∂ 2φ(x,y,z;q)

∂x2 +∂ 2φ(x,y,z;q)

∂y2 +∂ 2φ(x,y,z;q)

∂z2

+eA Λ(q) exp[φ(x,y,z;q)], (14.58)

is a nonlinear operator corresponding to (14.50),L is an auxiliary linear operatorwith the propertyL [0] = 0, andc0 6= 0 is the convergence-control parameter, re-spectively. Note that we have great freedom to choose the auxiliary linear operatorL and the convergence-control parameterc0 in the frame of the HAM.

Similarly, we have the homotopy-series solution

w(x,y,z) = w0(x,y,z)++∞

∑n=1

wn(x,y,z), (14.59)


λ = λ0++∞

∑n=1

λn. (14.60)

According to Theorem 4.15,wn(x,y,z) is governed by thenth-order deformationequation

L [wn(x,y,z)− χn wn−1(x,y,z)] = c0 δn−1(x,y,z), −1< x,y,z< 1, (14.61)

subject to the boundary conditions on the six walls

wn(±1,y,z) = µn(±1,y,z), wn(x,±1,z) = µn(x,±1,z), wn(x,y,±1) = µn(x,y,±1),

where

µn(x,y,z) = (χn+ c0) wn−1(x,y,z)+ c0 (1− χn) A, (14.62)

δk(x,y,z) = ∆wk(x,y,z)+eAk

∑j=0

λk− j D j(

eφ) , (14.63)

are gained by Theorem 4.1, and the termD j(

eφ) is given by the recursion formula(14.30).

Similarly, the solution of the above linear PDE reads

wn(x,y,z) = w∗n(x,y,z)−w∗

n(±1,y,z)−w∗n(x,±1,z)−w∗

n(x,y,±1)

+ w∗n(x,±1,±1)+w∗

n(±1,y,±1)+w∗n(±1,±1,z)

− w∗n(±1,±1,±1)

+ µn(±1,y,z)+ µn(x,±1,z)+ µn(x,y,±1)

− µn(x,±1,±1)− µn(±1,y,±1)− µn(±1,±1,z)

+ µn(±1,±1,±1), (14.64)

where

w∗n(x,y,z) = c0 L

−1[δn−1(x,y,z)]+ χn wn−1(x,y,z) (14.65)

is a special solution of (14.61). Besides, the eigenvalueλn−1 is determined by thelinear algebraic equation

wn(0,0,0) = (χn+ c0) wn−1(0,0,0). (14.66)

We also choose the same initial guess

w0(x,y,z) = 0

for the 3D Gelfand equation. Quite similarly, we choose an auxiliary linear operatorL in the form


Fig. 14.6 Averaged squaredresidual of the 3D Gelfandequation versusc6 whenA= 1, c0 = −1 by means ofthe 6th-order auxiliary lin-ear operator (14.69). Dashedline: 5th-order homotopy-approximation; Dash-dottedline: 10th-order homotopy-approximation; Solid line:15th-order homotopy-approximation.

c6

Ave

rage

dre

sidu

alsq

uare

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210-2

10-1

100

A = 1, c3 = 0

L w=

(

c3

xyz

)

∂ 3w∂x∂y∂z

+ c6∂ 6w

∂x2∂y2∂z2 , (14.67)

wherec3 andc6 are constants. Its inverse operator reads

L−1[xm yn zk] =

xm+2 yn+2 zk+2

(m+2)(n+2)(k+2)[c3+ c6(m+1)(n+1)(k+1)](14.68)

for arbitrary positive integersm,n,k. Especially, whenc3 = 0, we have the6th-orderauxiliary linear operator

L w= c6∂ 6w

∂x2∂y2∂z2 . (14.69)

For more details, please refer to Liao and Tan [25].Note that there exist three convergence-control parameters c0,c3 andc6. All of

them have no physical meanings, but provide a convenient wayto guarantee theconvergence of the homotopy-series (14.59) and (14.60). Without loss of generality,let us first consider a special caseA = 1. Similarly, it is found that thenth-orderapproximation ofu(x,y,z) on the six walls reads

A(1+ c0)n,

which vanishes asn → +∞ when |1+ c0| < 1, i.e.−2 < c0 < 0. Similarly, it isalso found that we can not obtain convergent series of eigenvalue and eigenfunctionwhenc3 6= 0 and−2 < c0 < 0. Therefore, we choosec0 = −1 andc3 = 0 so thatthe boundary conditions on the six walls are exactly satisfied. Then, we focus ourattention on the influence of the convergence-control parameterc6 in (14.69) to theconvergence of the homotopy-series (14.59) and (14.60).

Similarly, we define the averaged squared residualEm of the 3D Gelfand equationin a similar way to (14.44), which is dependent only uponc6 whenA= 1, c0 = −1by means of the 6th-order auxiliary linear operator (14.69), as shown in Fig. 14.6.The minimum of the averaged squared residualEm of the 3D Gelfand equation and


Table 14.6 The minimum of the averaged squared residual of the 3D Gelfand equation whenA= 1, c0 =−1 by means of the 6th-order auxiliary linear operator (14.69)


3 0.127 0.16145 7.7×10−2 0.147810 2.2×10−2 0.118515 1.2×10−2 0.108820 7.9×10−3 0.099425 5.8×10−3 0.0982

Table 14.7 The eigenvalueλ and the averaged squared residualEm of the 3D Gelfand equationwhenA= 1, c0 =−1 by means of the 6th-order auxiliary linear operator (14.69) with c6 =+1/8

Order of approximationm Averaged squared residual Eigenvalueλ

1 0.77 1.68783 0.33 2.17575 9.6×10−2 2.293510 2.2×10−2 2.266815 1.3×10−2 2.263520 9.5×10−3 2.263625 7.3×10−3 2.263630 5.9×10−3 2.2636

the corresponding optimal values of the convergence-control parameterc6 are listedin Table 14.6. It is found that, as the order of approximationincreases, the averagedsquared residualEm decreases for an arbitrary convergence-control parameterc6 inthe interval 0.1≤ c6 <+∞, and besides the optimal value ofc6 is close to 0.1. Thisis indeed true: whenA = 1, we gain the convergent eigenfunction and its corre-sponding eigenvalueλ = 2.2636 by means ofc0 = −1 and the 6th-order auxiliarylinear operator (14.69) withc6 =+1/8, as shown in Table 14.7.

RegardingA as a unknown parameter, by means ofc0 = −1 and the 6th-orderauxiliary linear operator (14.69) withc6 = +1, we gain the 25th-order approxima-tion of the eigenvalue using the Mathematica code in the Appendix 14.2. It is foundthat the eigenvalue of the 3D Gelfand equation has the maximum valueA= 2.476 atA≈ 1.61, as shown in Table 14.8. The 25th-order homotopy-approximation of theeigenvalue is valid in the interval 0≤ A≤ 12, and the simplified formula given byits first eight terms, i.e.

λ ≈ e−A(4.48514605A+1.30348867A2

+ 0.31378876A3+0.056269253A4

+ 7.77343016×10−3A5+8.58885688×10−4A6

+ 6.90477890×10−5A7+2.20710623×10−6A8)

, (14.70)


Fig. 14.7 Eigenvalue of the3D Gelfand equation bymeans ofc0 = −1 and the6th-order auxiliary linear op-erator (14.69) withc6 = +1.Solid line: 25th-order HAMapproximation; Circles: 20th-order HAM approximation;Squares: simplified formula(14.70) by means of the first8 terms of the 25th-order ap-proximation.

A

λ

0 3 6 9 120

1

2

3

eigenvalue of 3D Gelfand equation

Table 14.8 The maximum eigenvalueλmax of the 3D Gelfand equation by means ofc0 =−1 andthe 6th-order auxiliary linear operator (14.69) withc6 = 1.

λmax The corresponding value ofA

4th-order HAM approx. 2.477 1.6038th-order HAM approx. 2.476 1.60012th-order HAM approx. 2.476 1.60216th-order HAM approx. 2.476 1.60520th-order HAM approx. 2.476 1.60725th-order HAM approx. 2.476 1.610

is a good approximation ofλ , as shown in Fig. 14.7.All of these verify that, in the frame of the HAM, the2nd-order3D nonlinear

PDE (14.46) can be transferred into an infinite number of the6th-orderlinear PDEswhose solutions can be obtained very easily by means of algebra calculations only.In this way, the original 3D nonlinear Gelfand equation is solved in a rather easyway. It is mainly because the HAM provides us extremely largefreedom to choosethe auxiliary linear operator, and at the same time, it also provides us a convenientway to guarantee the convergence of series solution by meansof the convergence-control parameters. To the best of our knowledge, such kind of transform hasneverbeen reported by means of other analytic and numerical methods. This suggests thatwe might have much larger freedom to solve nonlinear problems than we thoughand believed traditionally!


In this chapter, a simple but rather efficient analytic approach is proposed to solvethe high-dimensional Gelfand equation with strong nonlinearity. In the frame ofthe HAM, the2nd-ordernonlinear PDE is transferred into an infinite number of


4th-order2D or 6th-order3D linear PDEs, which are very easy to solve under therestriction of the so-called solution-expression.

By means of the HAM, the 3rd-order nonlinear PDE describing anon-similarityboundary-layer flow can be transferred into an infinite number of the 3rd-order lin-ear ODEs so that the non-similarity boundary-layer flow can be solved in a rathersimilar way likesimilarity boundary-layer ones, as shown in Chapter 11. Besides,the 3rd-order nonlinear PDE describing theunsteadyboundary-layer flow can betransferred into an infinite number of the 3rd-order linear ODEs so that theun-steadyboundary-layer flow can be solved in a very similar way likesteady-stateboundary-layer ones, as shown in Chapter 12. Here, we further illustrate that, in theframe of the HAM, the2nd-ordernonlinear PDE (Gelfand equation) can be trans-ferred into an infinite number of the4th-order2D or6th-order3D linear PDEs. Allof these verify that the HAM indeed provides us extremely large freedom to choosethe auxiliary linear operatorL . Using this kind of extremely large freedom on thechoice of the auxiliary linear operatorL , some nonlinear differential equations maybe solved in a much easier way, as mentioned above.

It should be emphasized that the convergence-control parametersc0,c4 andc6

play a very important role in guaranteeing the convergence of the series of the eigen-function and eigenvalue of the Gelfand equation. For the 2D Gelfand equation, theconvergent results are gained by means ofc0 = −1 and the 4th-order auxiliary lin-ear operator (14.39) withnegativeconvergence-control parameterc4. However, forthe 3D Gelfand equation, the convergent results are obtained by means ofc0 = −1and the 6th-order auxiliary linear operator (14.69) withpositiveconvergence-controlparameterc6. Therefore, without these properly choosing convergence-control pa-rameters, it is very difficult to gain convergent results. Note that a divergent serieshas no meanings at all. Thus, in the frame of the HAM, the freedom on the choiceof the auxiliary linear operatorL is in essence based on this kind of guarantee ofthe convergence of homotopy-series solution, otherwise such kind of freedom hasno meanings at all. This indicates once again the importanceof the convergence-control parameters [11]: it is the convergence-control parameter that essentially dis-tinguishes the HAM fromall other analytic methods.

It should be emphasized that, the transform used in this chapter hasneverbeen re-ported by any other analytic and numerical methods, to the best of our knowledge. Itreveals the originality and great flexibility of the HAM. Besides, it also suggests thatwe might have much larger freedom to solve nonlinear problems than we thoughtand believed traditionally. Indeed, it is a good example to keep us an open mind fornonlinear problems: it is some of our “traditional” thoughts that might be the largestrestriction to our mind.

It is a pity that many things are still unclear now. For example, how to find thebestor theoptimalauxiliary linear operator among an infinite number of possibleones? Can we give some rigorous mathematical proofs in general? The freedomon the choice of the auxiliary linear operator might bring forward some new andinteresting problems in applied and pure mathematics, and might, I wish, finallygive us the “true” freedom on solving highly nonlinear differential equations, ifsuch kind of freedom really exists.


Appendix 14.1 Mathematica code of 2D Gelfand equation

The 2D Gelfand equation

∆u+λ eu = 0, −1< x< 1, −1< y< 1,

subject to the boundary condition on the four walls

u(x,±1) = 0, u(±1,y) = 0,

is solved by means of the HAM. This Mathematica code is free available athttp://numericaltank.sjtu.edu.cn/HAM.htm.

Mathematica code of 2D Gelfand equationby Shijun LIAO


( *************************************************** ********** )( * * )( * For given A, we find such an eigenvalue lambda and * )( * a normalized eigenfunction w(x,y) satisfying: * )( * w_xx + w_yy + lambda * Exp[w(x)] = 0 * )( * subject to the boundary conditions: * )( * w(1,y)=w(-1,y)=w(x,1)=w(x,-1)=-A,w(0,0)=0, * )( * * )( *************************************************** ********** )<<Calculus‘Pade‘;<<Graphics‘Graphics‘;

( *************************************************** ********** )( * Define initial guess of w(x) * )( *************************************************** ********** )w[0] = 0;U[0] = A + w[0];

( *************************************************** ********** )( * Define the function chi[k] * )( *************************************************** ********** )chi[k_]:=If[ k <= 1, 0, 1];

( *************************************************** ********** )( * Define the the auxiliary linear operator L * )( *************************************************** ********** )L[f_] := Module[temp,Expand[c2 * D[f,x,1,y,1]/x/y + c4 * D[f,x,2,y,2]]];


( *************************************************** ********** )( * Define the inverse operator of L * )( *************************************************** ********** )Linv[xˆm_. * yˆn_.] := xˆ(m+2) * yˆ(n+2)/(m+2)/(n+2)

/(c2+c4 * (n+1) * (m+1));Linv[xˆm_.] := xˆ(m+2) * yˆ2/(m+2)/2/(c2+c4 * (m+1));Linv[yˆn_.] := xˆ2 * yˆ(n+2)/(n+2)/2/(c2+c4 * (n+1));Linv[c_] := c * xˆ2 * yˆ2/4/(c2+c4)/;FreeQ[c,x] && FreeQ[c,y];

( *************************************************** ********** )( * The linear property of the inverse operator of L * )( * Linv[f_+g_] := Linv[f]+Linv[g]; * )( *************************************************** ********** )Linv[p_Plus] := Map[Linv,p];Linv[c_ * f_] := c * Linv[f] /; FreeQ[c,x] && FreeQ[c,y];

( *************************************************** ********** )( * Define Getdelta[k] * )( *************************************************** ********** )Getdelta[k_]:=Module[temp,n,temp[1] = D[w[k],x,2] + D[w[k],y,2];temp[2] = Sum[lambda[n] * Dexp[k-n],n,0,k];delta[k] = Expand[ temp[1] + temp[2] ];];

( *************************************************** ********** )( * Define GetDexp[n] * )( *************************************************** ********** )GetDexp[n_]:=Module[j,If[n==0,Dexp[0]=Exp[w[0]]];If[n >0,Dexp[n]=Sum[(1-j/n) * Dexp[j] * w[n-j],j,0,n-1]//Expand];];

( *************************************************** ********** )( * Define Getlambda * )( * This module gets lambda[k-1] * )( *************************************************** ********** )Getlambda[k_]:=Module[eq,temp,eq = w[k]-(c0+chi[k]) * w[k-1] /. x -> 0,y->0;temp = Solve[eq == 0, lambda[k-1]];lambda[k-1] = temp[[1,1,2]]//Expand;];

( *************************************************** ********** )( * Define GetwSpecial * )( * This module gets a special solution of (14.24) and (14.25) * )( *************************************************** ********** )GetwSpecial[k_]:=Module[temp,temp[0] = Expand[RHS[k]];temp[1] = Linv[temp[0]];temp[2] = temp[1] + chi[k] * w[k-1]//Simplify;wSpecial = temp[2]//Expand;];


( *************************************************** ********** )( * Define Getw * )( * This module gets a solution of (14.24) and (14.25) * )( *************************************************** ********** )Getw[k_]:=Module[temp,alpha,alpha = (c0+chi[k]) * w[k-1]+c0 * (1-chi[k]) * A;temp[1] = wSpecial /. y->1;temp[2] = wSpecial /. x->1;temp[3] = wSpecial /. x->1, y->1;temp[4] = alpha /. x->1;temp[5] = alpha /. y->1;temp[6] = alpha /. x->1,y->1;temp[7] = wSpecial - temp[1] - temp[2] + temp[3]

+ temp[4] + temp[5] - temp[6];w[k] = Simplify[temp[7]]//Expand;];

( *************************************************** ********** )( * Define GetErr[k] * )( *************************************************** ********** )GetErr[k_]:=Module[temp,sum,dx,dy,Num,i,j,X,Y,err[k] = D[U[k],x,2] + D[U[k],y,2] + LAMBDA[k] * Exp[U[k]];Nx = 10;Ny = 10;dx = N[1/Nx,100];dy = N[1/Ny,100];sum = 0;Num = 0;For[i = 0, i <= Nx-1, i++,

X = i * dx;For[j = 0, j <= Ny - 1, j++,

Y = j * dy;temp = err[k]ˆ2 /. x->X, y->Y;sum = sum + temp;Num = Num + 1;];

];Err[k] = sum/Num;If[NumberQ[Err[k]], Print["Squared Residual of G.E. = ",

Err[k]//N]]];

( *************************************************** ********** )( * Define Body1[A] * )( * Boyd’s one-point formula * )( *************************************************** ********** )Boyd1[A_] := 3.2 * A* Exp[-0.64 * A];

( *************************************************** ********** )( * Define Body3[A] * )( * Boyd’s three-point formula * )( *************************************************** ********** )Boyd3[A_] := Module[temp,B,C,G,G = 0.2763 + Exp[0.463 * A] + 0.0483 * Exp[-0.209 * A];


B = A* ( 0.829-0.566 * Exp[0.463 * A]-0.0787 * Exp[-0.209 * A])/G;C = A* (-1.934+0.514 * Exp[0.463 * A]+1.9750 * Exp[-0.209 * A])/G;temp[1] = 2.667 * A + 4.830 * B + 0.127 * C;temp[2] = 0.381 * A + 0.254 * B + 0.018 * C;temp[1] * Exp[-temp[2]]];

( *************************************************** ********** )( * Main Code * )( *************************************************** ********** )ham[m0_,m1_]:=Module[temp,k,j,For[k=Max[1,m0], k<=m1, k=k+1,Print[" k = ",k];GetDexp[k-1];

Getdelta[k-1];RHS[k] = c0 * delta[k-1]//Expand;

GetwSpecial[k];Getw[k];U[k] = U[k-1] + w[k]//Simplify;Getlambda[k];Lambda[k-1] = Sum[lambda[j],j,0,k-1]//Expand;

LAMBDA[k-1] = Lambda[k-1] * Exp[-A];Print[k-1,"th approximation of LAMBDA = ",N[LAMBDA[k-1], 10] ];

];Print["Successful !"];];

( *************************************************** ********** )( * Define the parameters * )( *************************************************** ********** )c0 = -1;A = 1;c2 = 0;c4 = -1/4;

( *************************************************** ********** )( * Print the parameters * )( *************************************************** ********** )Print[" A = ",A];Print[" c0 = ",c0];Print[" c2 = ",c2];Print[" c4 = ",c4];

( * Get the 20th-order approximation * )ham[1,21];

( * Get squared residual of up to 20th-order approximation * )For[k=5, k<=20, k=k+5, Print["k = ",k]; GetErr[k]]


Appendix 14.2 Mathematica code of 3D Gelfand equation

The 3D Gelfand equation

∆u+λ eu = 0, −1≤ x,y,z≤ 1,


u(±1,y,z) = 0, u(x,±1,z) = 0, u(x,y,±1) = 0,

is solved by means of the HAM. This Mathematica code is free available athttp://numericaltank.sjtu.edu.cn/HAM.htm

Mathematica code of 3D Gelfand equationby Shijun LIAO


( *************************************************** ********** )( * * )( * For given A, we find such an eigenvalue lambda and * )( * a normalized eigenfunction w(x,y) that: * )( * w_xx+w_yy+w_zz+lambda * Exp[w(x)] = 0 * )( * subject to the boundary conditions: * )( * w(1,y)=w(-1,y)=w(x,1)=w(x,-1)=-A,w(0,0)=0, * )( *************************************************** ********** )<<Calculus‘Pade‘;<<Graphics‘Graphics‘;<<tool2010.nb;

( *************************************************** ********** )( * Define initial guess * )( *************************************************** ********** )w[0] = 0;U[0] = A + w[0];

( *************************************************** ********** )( * Define chi[k] * )( *************************************************** ********** )chi[k_]:=If[k<=1,0,1];

( *************************************************** ********** )( * Define inverse operator of auxiliary linear operator L * )( *************************************************** ********** )Linv[xˆm_ * yˆn_ * zˆk_] := xˆ(m+2) * yˆ(n+2) * zˆ(k+2)/(m+2)

/(n+2)/(k+2)/(c3+c6 * (n+1) * (m+1) * (k+1));Linv[yˆn_ * zˆk_] := xˆ2 * yˆ(n+2) * zˆ(k+2)/2/(n+2)/(k+2)

/(c3+c6 * (n+1) * (k+1));Linv[xˆm_ * zˆk_] := xˆ(m+2) * yˆ2 * zˆ(k+2)/(m+2)/2/(k+2)


/(c3+c6 * (m+1) * (k+1));Linv[xˆm_ * yˆn_] := xˆ(m+2) * yˆ(n+2) * zˆ2/(m+2)/(n+2)/2

/(c3+c6 * (n+1) * (m+1));Linv[xˆm_] := xˆ(m+2) * yˆ2 * zˆ2/(m+2)/2/2/(c3+c6 * (m+1));Linv[yˆn_] := xˆ2 * yˆ(n+2) * zˆ2/2/(n+2)/2/(c3+c6 * (n+1));Linv[zˆk_] := xˆ2 * yˆ2 * zˆ(k+2)/2/2/(k+2)/(c3+c6 * (k+1));Linv[c_] := c * xˆ2 * yˆ2 * zˆ2/8/(c3+c6) /; FreeQ[c,x]

&& FreeQ[c,y] && FreeQ[c,z];

( *************************************************** ********** )( * The linear property of the inverse operator of L * )( * Linv[f_+g_] := Linv[f]+Linv[g]; * )( *************************************************** ********** )Linv[p_Plus] := Map[Linv,p];Linv[c_ * f_] := c * Linv[f]/;FreeQ[c,x]&&FreeQ[c,y]&&FreeQ[c,z];

( *************************************************** ********** )( * Define GetR[k] * )( *************************************************** ********** )Getdelta[k_]:=Module[temp,n,temp[1] = D[w[k],x,2] + D[w[k],y,2] + D[w[k],z,2];temp[2] = Sum[lambda[n] * Dexp[k-n],n,0,k];delta[k] = Expand[ temp[1] + temp[2] ];];

( *************************************************** ********** )( * Define GetDexp[n] * )( *************************************************** ********** )GetDexp[n_]:=Module[,If[n == 0, Dexp[0] = 1 ];If[n >0,Dexp[n]=Sum[(1-j/n) * Dexp[j] * w[n-j],j,0,n-1]//Expand];];

( *************************************************** ********** )( * Define Getlambda * )( * This module gets lambda[k-1] * )( *************************************************** ********** )Getlambda[k_]:=Module[eq,temp,eq = w[k]-(c0+chi[k]) * w[k-1] /. x->0, y->0, z->0;temp = Solve[eq == 0, lambda[k-1]];lambda[k-1] = temp[[1,1,2]]//Expand;];

( *************************************************** ********** )( * Define GetwSpecial * )( * This module gets a special solution of (14.61) * )( *************************************************** ********** )GetwSpecial[k_]:=Module[temp,temp[0] = Expand[RHS[k]];temp[1] = Linv[temp[0]];temp[2] = temp[1] + chi[k] * w[k-1];wSpecial = temp[2]//Expand;];


( *************************************************** ********** )( * Define Getw * )( * This module gets a solution of the high-order EQ * )( *************************************************** ********** )Getw[k_]:=Module[temp,mu,mu = (c0+chi[k]) * w[k-1] + c0 * (1-chi[k]) * A;temp[1] = wSpecial /. x->1;temp[2] = wSpecial /. y->1;temp[3] = wSpecial /. z->1;temp[4] = wSpecial /. x->1, y->1;temp[5] = wSpecial /. x->1, z->1;temp[6] = wSpecial /. y->1, z->1;temp[7] = wSpecial /. x->1, y->1, z->1;temp[8] = wSpecial-temp[1]-temp[2]-temp[3]+temp[4]

+temp[5]+temp[6]-temp[7];temp[1] = mu /. x->1;temp[2] = mu /. y->1;temp[3] = mu /. z->1;temp[4] = mu /. x->1, y->1;temp[5] = mu /. x->1, z->1;temp[6] = mu /. y->1, z->1;temp[7] = mu /. x->1, y->1, z->1;temp[9] = temp[8]+temp[1]+temp[2]+temp[3]-temp[4]

-temp[5]-temp[6]+temp[7];w[k] = Expand[temp[9]];];

( *************************************************** ********** )( * Define GetErr[k] * )( *************************************************** ********** )GetErr[k_]:=Module[temp,sum,dx,dy,dz,Num,i,j,m,X,Y,Z,Nx,Ny,Nz,err[k] = D[U[k],x,2] + D[U[k],y,2] + D[U[k],z,2]

+ LAMBDA[k] * Exp[U[k]];Nx = 10;Ny = 10;Nz = 10;dx = N[1/Nx,100];dy = N[1/Ny,100];dz = N[1/Nz,100];sum = 0;Num = 0;For[i = 0, i <= Nx-1, i++,

X = i * dx;For[j = 0, j <= Ny - 1, j++,

Y = j * dy;For[m = 1, m <= Nz-1, m++,

Z = m* dz;temp = err[k]ˆ2 /. x->X, y->Y,z->Z;sum = sum + temp;Num = Num + 1;

];];

];


Err[k] = sum/Num;If[NumberQ[Err[k]], Print["Squared Residual of G.E. = ",

Err[k]//N]]];

( *************************************************** ********** )( * Define HP[F,m,n] * )( * This module gives [m,n] homotopy-Pade approximation * )( * of the series : F[k] = sum[f[i],i,0,k] * )( *************************************************** ********** )hp[F_,m_,n_]:=Block[i,k,dF,temp,q,dF[0] = F[0];For[k = 1, k <= m+n, k = k+1, dF[k] = Expand[F[k]-F[k-1]]];temp = dF[0]+Sum[dF[i] * qˆi,i,1,m+n];Pade[temp,q,0,m,n]/.q->1];

( *************************************************** ********** )( * Main Code * )( *************************************************** ********** )ham[m0_,m1_]:=Module[temp,k,j,For[k=Max[1,m0], k<=m1, k=k+1,

Print[" k = ",k];GetDexp[k-1];Getdelta[k-1];RHS[k] = c0 * delta[k-1]//Expand;GetwSpecial[k];Getw[k];U[k] = U[k-1] + w[k]//Expand;Getlambda[k];Lambda[k-1] = Expand[Sum[lambda[j],j,0,k-1]];LAMBDA[k-1] = Lambda[k-1] * Exp[-A];Print[k-1,"th approximation of lambda = ",

N[LAMBDA[k-1],10] ];];

Print["Successful !"];];

( *************************************************** ********** )( * Physical and control parameters * )( *************************************************** ********** )A = 1;c0 = -1;c3 = 0;c6 = 1/8;

( *************************************************** ********** )( * Print physical and control parameters * )( *************************************************** ********** )Print[" A = ",A];Print[" c0 = ",c0];Print[" c3 = ",c3];Print[" c6 = ",c6];


( * Get the 10th-order approximation of LAMBDA * )ham[1,11];

( * Get squared residual of up to10th-order approximation * )For[k=2, k<=10, k=k+2, Print["k = ",k]; GetErr[k]]


References




4. Awrejcewicz, J., Andrianov, I.V., Manevitch, L.I.: Asymptotic Approaches in Nonlinear Dy-namics. Springer-Verlag, Berlin (1998)

5. Boyd, J.P.: An analytical and Numerical Study of the Two-dimensional Bratu Equation. Jour-nal of Scientific Computing.1, 183-206 (1986)


7. Hinch, E.J.: Perturbation Methods. In seres of CambridgeTexts in Applied Mathematics ,Cambridge University Press, Cambridge (1991)

8. Jacobsen, J., Schmitt, K.: The Liouville-Bratu-Gelfandproblem for radial operators. Journalof Differential Equations.184, 283-298 (2002)

9. Karmishin, A.V., Zhukov, A.T., Kolosov, V.G.: Methods ofDynamics Calculation and Testingfor Thin-walled Structures (in Russian). Mashinostroyenie, Moscow (1990)

10. Lagerstrom, P.A.: Matched Asymptotic Expansions: Ideas and Techniques. In series of Ap-plied Mathematical Sciences,76, Springer-Verlag , New York (1988)
















References 497

26. Liouville, J.: Sur lequation aux derivees partielles (∂ lnλ )/(∂u∂v)± 2λa2 = 0. J. de Math.18, 71 – 72 (1853)

27. McGough, J.S.: Numerical continuation and the Gelfand problem. Applied Mathematics andComputation.89, 225-239 (1998)

28. Murdock, J.A.: Perturbations: - Theory and Methods. John Wiley & Sons, New York (1991)29. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (1973)30. Nayfeh, A.H.: Perturbation Methods. John Wiley & Sons, New York (2000)31. Von Dyke, M.: Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford

(1975)32. Xu, H., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy-based solutions of the Navier-


Chapter 15Interaction of nonlinear water wave andnonuniform currents

Abstract In this chapter, we illustrate the validity of the homotopy analysis method(HAM) for a complicated nonlinear PDE describing the nonlinear interaction of aperiodic traveling wave on a non-uniform current with exponential distribution ofvorticity. In the frame of the HAM, the original highly nonlinear PDE with vari-able coefficient is transferred into an infinite number of much simpler linear PDEs,which are rather easy to solve. Physically, it is found that Stokes’ criterion of wavebreaking is still correct for traveling waves on non-uniform currents. It verifies thatthe HAM can be used to solve some complicated nonlinear PDEs so as to deepenand enrich our physical understanding about some interesting nonlinear phenomena.

15.1 Introduction

In science and engineering, there are many complicated nonlinear partial differentialequations (PDEs). In this chapter, the homotopy analysis method (HAM) [11, 13–27,40] is used to solve such a kind of nonlinear problem.

Consider a train of permanent deep-water waves propagatingon a steady, non-uniform, shear current with a vertical distribution of vorticity. Assume that the wateris inviscid and incompressible, and the flow is two dimensional, so that the streamfunctionψ exists. Besides, the magnitude of the vorticity varies exponentially withthe magnitude of the stream functionψ , but remains constant along the same stream-line. This mainly because the profile of the currents is mostly exponential over thedepth in practice.

Mathematically, this problem is described by a nonlinear PDE with a nonlin-ear boundary condition on an unknown wave surface, as shown below. Due to itsmathematical difficulty, earlier works either assume that the current is uniform andhence ignore the effects of the vorticity distribution, or that the waves are of smallamplitude and the current is weak and hence simply neglect the nonlinear interac-tion between them. In the former case, the wave is regarded topropagate on stillwater in a reference frame moving with the current at the samespeed, so that the

499

500 15 Interaction of nonlinear water wave and nonuniform currents

uniform current can be linearly added to the pure waves. Thiskind of problem wasinvestigated by Fenton [9] who derived Stokes’ fifth-order solution for deep-waterwaves. For shallow water waves with a uniform current, Clamond [2, 3] and Cla-mond & Fructus [4] used the renormalization method to obtainaccurate approxi-mations. Unfortunately, in practice, currents are mostly nonuniform and hence thevorticity appears. In this case, linear wave theory is commonly used, if the wavesare of small amplitude or the current is weak. Contributionswere made by Thomp-son [36], Kishida & Sobey [10] who investigated the linear shear current, Eastwood& Watson [8] who examined the bilinear current. For details,please refer to Pere-grine [30], Toumazis [37] and Wang [38, 39]. But if neither the wave amplitudeis small nor the current is weak, the above mentioned methodsare too crude: it isfound that they overestimate the waves in an aiding current but underestimate themin an opposing current.

The nonlinear theory of the wave-current interaction begins from Dalrymple [5],who numerically studied the interaction of wave with a current in 1/7 power-law.Thomas [35] numerically investigated the interaction of nonlinear waves and a cur-rent with exponential distribution of vorticity. Wang et al. [38] applied the pertur-bation method to obtain a fifth-order approximation. However, it is well-known thatperturbation approximations are often valid for weakly nonlinear problems in gen-eral.

In 2003, Liao and Cheung [26] successfully applied the HAM [11, 13–27, 40]to gain accurate analytic approximations of two-dimensional (2D) nonlinear pro-gressive deep-water waves. In 2009, Cheng, Cang and Liao [1]further applied theHAM to investigate the interaction between the 2D nonlineardeep-water wave andexponentially shear currents.

15.2 Mathematical modeling

15.2.1 Original boundary-value equation

Let (x,y) denote the Cartesian coordinates, with the horizontalx-axis pointing in thedirection of wave propagation, they-axis pointing vertically upwards, and the originlying the mean water level, respectively. A moving reference frame(x′− ct,y′) 7→(x,y) is used, wherec is the wave phase speed,(x′,y′) is the coordinate in the stillreference frame, respectively, so that the problem is steady. Let y= η(x) denote thefree surface,H the wave height,λ the wave length, andu = u(x,y),v(x,y) thetotal velocity of the wave on the current, respectively.

Since the fluid is impressible and the flow is two dimensional,there exists thestream functionψ(x,y) defined by

ψy(x,y) = u(x,y)− c, ψx(x,y) =−v(x,y). (15.1)

15.2 Mathematical modeling 501

Let Ω(ψ) denote the given vorticity distribution of the current. Dueto the definitionof vorticity, we have the governing equation

ψxx+ψyy =−Ω(ψ), (15.2)

where the subscript denotes the partial derivatives.Here, we investigate a special case

Ω(ψ) = ε exp(−ψ) (15.3)

indicated by Phillips [31], corresponding to a current withexponentially decayingvorticity in the direction of water depth, whereε is a physical parameter determiningthe strength of the vorticity. Whenε < 0, the current moves in the same direction ofwave propagation, called the aiding current. Whenε > 0, the current moves in theopposing direction of wave propagation, called opposing current. Note that the vor-ticity of all uniform currents is zero. So, a given distribution of vorticity correspondsto an infinite number of currents. To avoid this, we define thatε = 0 corresponds tothe still water.

On the free surface, we have the kinematic condition

ψ = 0, on y= η(x), (15.4)

and the dynamic condition

12(ψ2

x +ψ2y )+g y= Q, on y= η(x), (15.5)

whereQ is the so-called Bernoulli constant, which is unknown and will be deter-mined later. Besides, in the infinite depth, we have the boundary condition

ψx → 0 as y→−∞. (15.6)

Given the vorticity distributionΩ(ψ) = ε exp(−ψ), the above-mentioned non-linear boundary-value problem contains four unknowns: thestream functionψ(x,y),the free surfaceη(x), the wave phase speedc, and the Bernoulli constantQ.Note that the partial differential equation (15.2) contains the exponentialtermΩ(ψ) = ε exp(−ψ), which possesses a rather strong nonlinearity for the unknownstream-functionψ . Besides, thenonlineardynamic condition (15.5) is satisfied ontheunknownfree surfacey = η(x). Mathematically, the strong nonlinearity of thegoverning equation (15.2) and the boundary condition (15.5), and especially theunknown free surfacey = η(x), bring us great difficulties to solve this nonlinearboundary-value problem.


15.2.2 Dubreil-Jacotin transformation

To handle the unknown free surface, the Dubreil-Jacotin transformation [6, 7, 35]is employed to reformulate the problem in aknowndomain. The strategy is to re-gard the stream functionψ as an independent variable (likex), andy as a unknownfunction dependent on bothx andψ , expressed byy(x,ψ). Then, according to thefundamental theorems in calculus, we have from (15.1) the local velocity compo-nents

u− c=1yψ

, v=yx

yψ. (15.7)

For simplicity, the following dimensionless variables

x= k x∗, y= k y∗, ψ =kc

ψ∗, Ω(ψ) =1

k cΩ ∗(ψ), (15.8)

are introduced, where the variables with asterisk are dimensional, andk = 2π/λ isthe wave number, respectively. Thus, the problem governed by the original PDEs(15.2) to (15.5) in theunknowndomain

−∞ < y≤ η(x), −∞ < x<+∞

is reformulated in theknowndomain

0≤ ψ <+∞, −∞ < x<+∞

by a complicated nonlinear PDE

yxx y2ψ −2 yx yψ yxψ +(1+ y2

x) yψψ = y3ψ Ω(ψ), (15.9)


µ (1+ y2x)+2 (y− γ) y2

ψ = 0 on ψ = 0, (15.10)

yx → 0 as ψ →+∞, (15.11)

where bothµ = k c2/g andγ = k Q/g are unknown constants.Note that, by means of the Dubreil-Jacotin transformation [6, 7, 35], the original

nonlinear PDE (15.2) with the exponentially nonlinear termΩ(ψ) = ε exp(−ψ)becomes the nonlinear PDE (15.9). Although (15.9) looks much more compli-cated than (15.2), its nonlinearity is actually weaker, because the exponential termΩ(ψ) = ε exp(−ψ) in (15.9) is now regarded as aknownvariable coefficient. Moreimportantly, the free surface boundary conditions are now satisfied on theknownboundaryψ = 0. In addition, the above PDEs contain only three unknowns: oneunknown functiony(x,ψ) and two unknown constantsµ andγ. So, compared to theoriginal equations, the number of the unknowns is reduced byone, i.e. from fourto three. Therefore, the use of the Dubreil-Jacotin transformation [6, 7, 35] in fact


simplifies the original problem, although (15.9) looks morecomplicated than theoriginal PDE (15.2).

The free surfaceη(x) corresponds to the stream lineψ = 0. Since the origin isdefined in the mean water level, the free surface is given by

η(x) = y(x,0)− 1π

∫ π

0y(x,0) dx. (15.12)

The dimensionless wave heightH (scaled byk−1) is given by

H = y(0,0)− y(π ,0). (15.13)

Thus, it is straightforward to gain the free surfacey= η(x) and the wave height, aslong asy(x,ψ) is obtained.


The HAM is employed to gain the accurate analytic approximations of the compli-cated nonlinear PDE (15.9) with the variable coefficientΩ(ψ) = ε exp(−ψ), sub-ject to the boundary conditions (15.10) and (15.11).

15.3.1 Solution expression

As mentioned in previous chapters, the so-called solution-expression plays an im-portant role in the frame of the HAM. Mathematically, from the highly nonlinearPDE (15.9), it is hard to guess the expression form ofy(x,ψ). Fortunately, from thephysical background of the problem, it is easy to find a propersolution expression ofit. Physically speaking, in the moving reference frame, thesolution is composed ofthree parts: a train of deep-water propagation wave, a uniform current caused by themotion of the reference frame, and a nonuniform current possessing an exponentialdistribution of vorticity. In case of the pure deep-water wave (i.e.Ω = 0), the waveelevation is periodic, expressed by

y=+∞

∑m=0

am cos(mx),

wheream is a constant coefficient. Even if there exists a non-uniformcurrent, thewave elevation is still periodic in thex direction, and therefore should be always ex-pressed by the above expression. Besides, since (15.9) contains the term exp(−ψ),it is straightforward that the solutiony(x,ψ) should contain the terms exp(−nψ),wheren ≥ 1 is an integer. Such kind of terms exp(−nψ) (n ≥ 1) are also neces-sary for the boundary condition (15.11) asψ →+∞. Furthermore, it is well-known


that the uniform current due to the motion of the reference frame won’t give riseto the wave-current interaction. Therefore, from the physical viewpoints, even with-out solving the nonlinear PDEs (15.9) to (15.13), it is clearthaty(x,ψ) should beexpressed in the form

y=−ψ ++∞

∑n=1

+∞

∑m=0

αn,m exp(−nψ) cos(mx), (15.14)

where the first part results from the uniform current due to the motion of the ref-erence frame, and the second part from the combined effect ofthe nonlinear waveand the non-uniform current. The above formula provides us the so-calledsolution-expressionof y(x,ψ), which plays an important role in the frame of the HAM,as shown below. Note that (15.14) automatically satisfies the boundary condition(15.11).

15.3.2 Zeroth-order deformation equation

Let q∈ [0,1] denote the embedding parameter, andy0(x,ψ), µ0, γ0 denote the initialguesses ofy(x,ψ), µ andγ, respectively. In the frame of the HAM, we should firstof all construct such three continuous mappingsY(x,ψ ;q) 7→ y(x,ψ), ∆(q) 7→ µandΓ (q) 7→ γ such that, asq increases from 0 to 1,Y(x,ψ ;q) varies continuouslyfrom the initial guessy0(x,ψ) to the wave elevationy(x;ψ), so does∆(q) from theinitial guessµ0 to µ , andΓ (q) from γ0 to γ, respectively. Such kind of continuousvariations (or deformations) are defined by the zeroth-order deformation equation

(1−q)L [Y(x,ψ ;q)− y0(x,ψ) ] = c0 q N [Y(x,ψ ;q) ], (15.15)


(1−q)Lb [Y(x,ψ ;q)− y0(x,ψ) ]

= c0 q Nb[Y(x,ψ ;q), ∆(q), Γ (q) ], on ψ = 0, (15.16)

and∂Y(x,ψ ;q)

∂x= 0, asψ →+∞, (15.17)

with the definition of the wave height

Y(0,0;q)−Y(π ,0;q) = H, (15.18)

wherec0 6= 0 is the convergence-control parameter,L andLb are two auxiliarylinear operators,N andNb are two nonlinear operators defined by

N [Y(x,ψ ;q) ] =Yxx Y2ψ −2Yx YxψYψ +(1+Y2

x )Yψψ −Ω Y3ψ , (15.19)

Nb [Y(x,ψ ;q), ∆(q), Γ (q) ] = ∆(

1+Y2x

)

+2 (Y−Γ )Y2ψ , (15.20)


corresponding to (15.9) and (15.10), respectively. The auxiliary linear operatorsLandLb have the propertyL [0] = 0 andLb[0] = 0, and will be chosen later.

Whenq= 0, sinceL [0] = 0 andLb[0] = 0, the zeroth-order deformation equa-tions (15.15) to (15.18) have the solution

Y(x,ψ ;0) = y0(x,ψ), ∆(0) = µ0, Γ (0) = γ0. (15.21)

Whenq= 1, sincec0 6= 0, the zeroth-order deformation equations (15.15) to (15.18)are equivalent to the original PDEs (15.9) to (15.13), provided

Y(x,ψ ;1) = y(x,ψ), ∆(1) = µ , Γ (1) = γ. (15.22)

Hence, as the embedding parameterq increases from 0 to 1,Y(x,ψ ;q),∆(q) andΓ (q) indeed vary from the initial guessesy0(x,ψ), µ0 andγ0 to the exact solutionsy(x,ψ), µ andγ, respectively. Mathematically speaking, the zeroth-order deforma-tion equations (15.15) to (15.18) construct three homotopies:

Y(x,ψ ;q) : y0(x,ψ)∼ y(x,ψ), ∆(q) : µ0 ∼ µ , Γ (q) : γ0 ∼ γ. (15.23)

Using (15.21) and expandingY(x,y;q), ∆(q) andΓ (q) into the Maclaurin serieswith respect to the embedding parameterq, we have the homotopy-Maclaurin series

Y(x,ψ ;q) = y0(x,ψ)++∞

∑n=1

yn(x,ψ) qn, (15.24)

∆(q) = µ0++∞

∑n=1

µn qn, (15.25)

Γ (q) = γ0++∞

∑n=1

γn qn, (15.26)

where

yn(x,ψ) = Dn [Y(x,ψ ;q)] =1n!

∂ nY(x,ψ ;q)∂qn

∣

∣

∣

∣

q=0,

andµn = Dn [∆(q)] , γn = Dn [Γ (q)]

are thenth-order homotopy-derivatives ofY(x,ψ ;q), ∆(q) andΓ (q), respectively,andDn is thenth-order homotopy-derivative operator. It is well-known that a powerseries has often a finite convergence radius. Fortunately, the zeroth-order deforma-tion equations (15.15) and (15.16) contains the convergence-control parameterc0,which can be used to adjust and control the convergence radius of the series. Be-sides, we have extremely large freedom to choose the auxiliary linear operatorsLandLb. Assuming that all of them are so properly chosen that the above homotopy-Maclaurin series are absolutely convergent atq= 1, we have, according to (15.22),the homotopy-series


y(x,ψ) = y0(x,ψ)++∞

∑n=1

yn(x,ψ), (15.27)

µ = µ0++∞

∑n=1

µn, (15.28)

γ = γ0++∞

∑n=1

γn. (15.29)

15.3.3 High-order deformation equation

The governing equation and boundary conditions of the unknown yn(x,ψ) can bededuced uniquely from the zeroth-order deformation equations (15.15) to (15.18).Directly substituting the series (15.24) to (15.26) into the zeroth-order deformationequations (15.15) to (15.18), then equating the coefficients of the like-power ofq,we have thenth-order deformation equations

L [ yn(x,ψ)− χn yn−1(x,ψ) ] = c0 δn−1(x,ψ) , (15.30)

subject to the boundary condition

Lb ( yn− χn yn−1 ) = c0 δ bn−1 (x) on ψ = 0, (15.31)

and

∂yn

∂x= 0 asψ →+∞, (15.32)

with the restrictionyn(0,0)− yn(π ,0) = 0, (15.33)

where

χk =

0, k≤ 1,1, k> 1,

(15.34)

and

δk =k

∑i=0

∂ 2yk−i

∂x2

i

∑j=0

∂yi− j

∂ψ∂y j

∂ψ−2

k

∑i=0

∂yk−i

∂x

i

∑j=0

∂ 2yi− j

∂x∂ψ∂y j

∂ψ+

∂ 2yk

∂ψ2

+k

∑i=0

∂ 2yk−i

∂ψ2

i

∑j=0

∂y j−i

∂x∂yi

∂x−Ω(ψ)

k

∑i=0

∂yk−i

∂ψ

i

∑j=0

∂y j−i

∂ψ∂yi

∂ψ, (15.35)

δ bk = µk+

k

∑i=0

µk−i

i

∑j=0

∂yi− j

∂x∂y j

∂x+2

k

∑i=0

(yk−i − γk−i)i

∑j=0

∂yi− j

∂ψ∂y j

∂ψ. (15.36)


are gained by Theorem 4.1. Equations (15.30) and (15.31) canbe gained directly byTheorem 4.15. For details, please refer to Chapter 4 and Cheng et al. [1].

Note that the high-order deformation equation contains thetwo auxiliary linearoperatorsL andLb. Fortunately, in the frame of the HAM, we have extremelylarge freedom to choose the auxiliary linear operators, as illustrated in Chapter 14.Note that the PDE (15.9) contains only one linear termyψψ . However, if we choose

L [y(x,ψ)] =∂ 2y(x,ψ)

∂ψ2

as the auxiliary linear operator, we obtain a solutiony(x,ψ) expressed in powerseries ofψ , which disobeysthe solution-expression (15.14). Since a power seriesoften has a finite radius of convergence, it is difficult for such kind of solution tosatisfy the boundary condition (15.11) asψ →+∞. So, if one follows the traditionalideas of perturbation methods, which have a high opinion of the linear terms of anonlinear governing equation, this linear termyψψ of the original PDE (15.9) mightgreatly mislead us. Fortunately, the HAM provides us extremely large freedom tochoose the auxiliary linear operators, so that we cancompletelyforget the lineartermyψψ in (15.9) and choose a proper auxiliary linear operatorL mainly based onthe solution-expression (15.14), which is gained under thephysical considerations.Note thatu= exp(−ψ)cos(x) satisfies

∂ 2u∂ψ2 +

∂ 2u∂x2 = 0.

So, to obey the solution expression (15.14), we choose the following auxiliary linearoperator

L u=∂ 2u∂ψ2 +

∂ 2u∂x2 . (15.37)

Note that the boundary condition (15.10) does not contain any linear terms, too.Similarly, to obey the solution-expression (15.14), we choose the following auxil-iary linear operator

Lbu= u+∂u∂ψ

(15.38)

for the nonlinear boundary condition (15.10).It should be emphasized that the above two auxiliary linear operators (15.37)

and (15.38) have nearly no relationships with the governingequation (15.9) and theboundary condition (15.10), respectively! This is mainly because the HAM providesus extremely large freedom to choose the auxiliary linear operatorsL andLb sothat the solution-expression (15.14) is satisfied. As mentioned in previous chapters,this kind of freedom is an obvious advantage of the HAM over other analytic tech-niques, and can greatly simplify resolving some nonlinear problems.


Besides, the HAM also provides us great freedom to choose theinitial guesses.The initial guessy0(x,ψ) should also satisfy the solution expression (15.14) andin addition the definition of wave height (15.13). So, it is straightforward for us tochoose such an initial guess

y0(x,ψ) =−ψ +H2

exp(−ψ) cosx. (15.39)

Note that the initial solutionsµ0 andγ0 are unknown up to now.In the aboventh-order deformation equations (15.30) to (15.33), the right hand-

side termsδn−1(x,ψ) and δ bn−1(x) are only dependent uponyk(x,ψ),µk and γk,

where 0≤ k ≤ n− 1, so that all of them are regarded as the known terms. Thus,according to the definitions (15.37) and (15.38) of the two auxiliary linear opera-torsL andLb, the high-order deformation equation (15.30) islinear, subject to thetwo linear boundary conditions (15.31) and (15.32) on theknownboundaryψ = 0andψ → +∞, respectively. Such kind of linear boundary-value problemin a fixeddomain is much simpler, and thus easier to solve, than the original highly nonlinearPDE (15.9) with variable coefficient exp(−ψ).

15.3.4 Successive solution procedure

Note that thenth-order deformation equations (15.30) to (15.33) containthree un-knowns,yn(x,ψ), µn−1 andγn−1. Due to the definition (15.37) of the auxiliary linearoperatorL , it holds for any positive integerM that

L

[

M

∑m=1

Cn,m exp(−mψ)cos(mx)

]

= 0,

whereCn,m is a constant. Thus, the general solution of the high-order deformationequation (15.30) reads

yn(x,ψ) = χn yn−1(x,ψ) + y∗n(x,ψ)+M

∑m=1

Cn,m exp(−mψ)cos(mx), (15.40)

whereM is a positive integer to be determined later, and

y∗n(x,ψ) = c0 L−1 [δn−1(x,ψ) ] (15.41)

is a special solution of (15.30). Here,L −1 is an inverse operator ofL , defined by

L−1 [exp(−mψ)cos(nx)] =

exp(−mψ)cos(nx)(m2−n2)

, (m 6= n) (15.42)

L−1 [exp(−mψ)sin(nx)] =

exp(−mψ)sin(nx)(m2−n2)

, (m 6= n). (15.43)


Using the above formulas, it is easy to gain the special solution y∗n(x,ψ) of (15.30),especially by means of the computer algebra system such as Mathematica, Mapleand so on.

To determine the unknownµn−1, γn−1 andCn,m, we substitute the expression(15.40) into the boundary condition (15.31), i.e.

Lb

[

y∗n+M

∑m=1

Cn,m exp(−mψ)cos(mx)

]

= c0 δ bn−1(x) , onψ = 0, (15.44)

whereM is undetermined. The above equation can be rewritten in the form

M

∑m=2

(1−m)Cn,m cos(mx) =2n+1

∑m=0

Bn,m(γn−1,µn−1) cos(mx), (15.45)

where the coefficientBn,m(γn−1,µn−1) is determined by the right-hand side termc0 δ b

n−1 (x) of (15.44). Balancing both sides of the above equation, we have

M = 2n+1, Cn,m =Bn,m(γn−1,µn−1)

(1−m), 1< m≤ 2n+1,

and

Bn,0(γn−1,µn−1) = 0, Bn,1(γn−1,µn−1) = 0, (15.46)

which exactly provide us the two algebraic equations to determine the unknownγn−1 andµn−1.

Up to now, only the coefficientCn,1 is unknown. Substituting (15.40) into (15.33)gives the algebraic equation

y∗n(0,0)− y∗n(π ,0)+2n+1

∑m=1

Cn,m−2n+1

∑m=1

Cn,m cos(mπ) = 0 (15.47)

for Cn,1, whose solution reads

Cn,1 =−12

y∗n(0,0)− y∗n(π ,0)+2n+1

∑m=2

[1− (−1)m]Cn,m

. (15.48)

Hence, all unknowns are determined and thus the problem is closed.In this way, we obtainyn(x,ψ), µn−1 andγn−1 successively in ordern= 1,2,3, · · ·

by means of only algebra computations. For example, solvingthe 1st-order defor-mation equations, we have

γ0 =1

320

[

160+25H2+ ε(

80+3 H2)] , (15.49)

µ0 =1

8+H2

[

8− 34

H2+532

H4+ ε(

4− 160

H2+3

160H4) ]

(15.50)


and

y1(x,ψ) = c0 [A1,0(ψ)+A1,1(ψ)cosx+A1,2(ψ)cos2x+A1,3(ψ)cos3x] , (15.51)

where

A1,0(ψ) = −H2

8exp(−2ψ)+ ε exp(−ψ)+

εH2

24exp(−3ψ),

A1,1(ψ) =3H3

64exp(−ψ)− H3

64exp(−3ψ)− εH

2exp(−ψ)

−9εH3

2240exp(−ψ)+

εH2

exp(−2ψ)+εH3

160exp(−4ψ),

A1,2(ψ) = −H2(256+40H2−5H4)

128(8+H2)exp(−2ψ)

−εH2(384+136H2−9H4)

1920(8+H2)exp(−2ψ)+

3εH2

40exp(−3ψ),

A1,3(ψ) = −H3

32exp(−3ψ)− 3εH3

448exp(−3ψ)+

εH3

224exp(−4ψ).

Similarly, we can obtainγ1,µ1,y2(x,ψ), and so on. It is found that all of our so-lutions yn(x,ψ) contain the term exp(−mψ), wherem≥ 1, so that the boundarycondition (15.32) is automatically satisfied. As long as theconvergent series solu-tion of y(x,ψ) is obtained, one can get the velocity field by (15.7) and the waveelevation by (15.12), respectively. Note that the dimensionless wave phase speedk c2/g is given byµ , and the energy on the free surface is given byγ, respectively.

It is found thaty(x,ψ) can be expressed by

y(x,ψ) = yc(ψ)+ yw(x,ψ)+ yi(x,ψ),

whereyc(ψ) = y(x,ψ)|H=0

is related to the pure current without waves,

yw(x,ψ) = y(x,ψ)|ε=0

is related to the pure waves without the current, andyi(x,ψ) is related to the wave-current interaction, respectively.

For more details, please refer to Cheng, Cang and Liao [1].

15.4 Homotopy approximations 511

Fig. 15.1 Averaged squaredresidual of the governingequation (15.9) versusc0whenH = 3/10 andε = 1/5.Dashed line: 5th-order ho-motopy approximation;Dash-dotted line: 8th-orderhomotopy-approximation;Solid line: 10th-orderhomotopy-approximation

c0

Em

-1 -0.8 -0.6 -0.4 -0.2 010-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

H = 0.3, ε =0.2

Fig. 15.2 Averaged squaredresidual of boundary-condition (15.10) versusc0whenH = 3/10 andε = 1/5.Dashed line: 5th-order ho-motopy approximation;Dash-dotted line: 8th-orderhomotopy-approximation;Solid line: 10th-orderhomotopy-approximation

c0

Emb

-1 -0.8 -0.6 -0.4 -0.2 010-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

H = 0.3, ε =0.2

15.4 Homotopy approximations

Based on the formulas mentioned above, a corresponding Mathematica code is de-veloped, which is given in the Appendix 15.1 and free available as open resource athttp://numericaltank.sjtu.edu.cn/HAM.htm

Note that the homotopy-series ofy(x,ψ) andµ = kc2/g, γ = k Q/g contain theso-called convergence-control parameterc0, which provides us a convenient way toguarantee the convergence of the homotopy-series solution, as shown below.

The accuracy of themth-order approximation is indicated by means of the aver-aged squared residual of the governing equation

Em =1

(1+Nx)(1+Nψ)

Nx

∑m=0

Nψ

∑j=0

N

[

m

∑n=0

yn(xi ,ψ j)

]2

(15.52)

and the averaged squared residual of the nonlinear boundarycondition


Table 15.1 Minimum of squared residual of the governing equation (15.9) and the correspondingoptimal convergence-control parameterc0 whenH = 3/10 andε = 1/5.

Order of approximationm Minimum of squared residualEm Optimalc0

5 2.3×10−7 -0.60498 5.5×10−9 -0.577510 4.4×10−10 -0.568615 1.3×10−10 -0.5569

Ebm =

11+Nx

Nx

∑i=0

Nb

[

m

∑n=0

yn(xi ,0),m

∑n=0

µn,m

∑n=0

γn

]2

, (15.53)

where

xi = i

(

πNx

)

, ψ j = j

(

2πNψ

)

with Nx = Nψ = 10. From the physical point of view, the residual error of thegov-erning equation below one wave-length (2π) in water depth is rather small and thusis neglected, because it is well-known that the wave velocity field decays exponen-tially as the water depth increases.

Without of loss of generality, let us first consider the caseH = 3/10 andε = 1/5.The averaged squared residualEm andEb

m of the governing equation and the nonlin-ear boundary condition versus the convergence-control parameterc0 are as shownin Fig. 15.1 and Fig. 15.2, respectively. Note that both ofEm and Eb

m decreasesin the interval−0.6 ≤ c0 < 0 as m increases. Besides, the optimal value ofc0

given by the minimum ofEm andEbm is about -0.55, as shown in Table 15.1 and

Table 15.2, respectively. This is indeed true: the homotopy-approximations whenH = 3/10 andε = 1/5 given by means ofc0 =−11/20 converges quickly, as shownin Table 15.3. Notice that the averaged squared residuals ofthe governing equationand the nonlinear boundary condition decrease monotonously to 1.1× 10−21 and5.2×10−22 at the 40th-order of approximation, respectively. Besides, we gain theconvergent valuesµ = 0.81442133 andγ = 0.40556503, respectively. By means ofthe homotopy-Pade acceleration technique, we obtain the more accurate convergentvalues ofµ = 0.814421334285 andγ = 0.405565029876, as shown in Table 15.4.Furthermore, it is found that even the 3rd-order homotopy approximation of thewave elevation is accurate enough, as shown in Fig. 15.3. Allof these verify thatwe can indeed gain accurate analytic approximations of the complicated nonlinearPDE (15.9) subject to the nonlinear boundary condition (15.10) by means of theHAM as mentioned above. Thus, given physically reasonable values ofH andε,we can gain convergent homotopy-approximations in a similar way by means of theMathematica code given in the Appendix 15.1.

Before discussing the wave-current interaction, let us consider the two limitingcases: the pure water wave, corresponding to the zero corticity Ω = 0, and the purecurrent, corresponding to the zero wave-heightH = 0. WhenΩ = 0, it is the Stokes


Table 15.2 Minimum of squared residual of the boundary condition (15.10) and the correspondingoptimal convergence-control parameterc0 whenH = 3/10 andε = 1/5.

Order of approximationm Minimum of squared residualEbm Optimalc0

5 2.2×10−6 -0.59768 8.1×10−8 -0.574910 7.9×10−9 -0.572315 5.7×10−11 -0.5707

Table 15.3 Themth-order homotopy-approximations ofµ , γ and the corresponding squared resid-ual Em, Eb

m whenH = 3/10, ε = 1/5 by means ofc0 =−11/20.

m Em Ebm µ = k c2/g γ = k Q/g

3 7.2×10−6 3.6×10−5 0.8167 0.40785 4.4×10−7 3.2×10−6 0.8131 0.404210 6.0×10−10 1.0×10−8 0.8142 0.405315 1.4×10−12 8.3×10−11 0.8144 0.405420 7.2×10−15 7.4×10−13 0.81442 0.4055625 9.4×10−17 5.6×10−15 0.814421 0.40556530 2.3×10−18 3.5×10−17 0.81442133 0.405565035 5.3×10−20 1.6×10−19 0.81442133 0.4055650340 1.1×10−21 5.2×10−22 0.81442133 0.40556503

Table 15.4 The[m,m] homotopy-Pade approximation ofµ andγ whenH = 3/10 andε = 1/5

m µ = k c2/g γ = k Q/g

2 0.8029 0.40354 0.8154 0.40566 0.814421 0.405568 0.814421 0.40556510 0.81442133 0.4055650212 0.814421334 0.40556502914 0.81442133428 0.4055650298716 0.814421334285 0.40556502987618 0.814421334285 0.40556502987620 0.814421334285 0.405565029876

deep-water wave model [32]. According to Stokes’ theory [29], a train of prop-agating deep-water waves breaks when the wave height arrives at its maximumvalueH = 0.886, corresponding to the maximum crest angleα = 120 degree. Itis found that, even for the waves with large wave-amplitude close to the limitingwave, our[m,m] homotopy-Pade approximations ofµ = k c2/g and the crest ele-vationηc = η(0) converge, as shown in Table 15.5 and Table 15.6, respectively. Acomparative error ofηc with the linear estimated crest elevationηc ≈ H/2 is givenat the last column in Table 15.6. It is found that the comparative error is around25% for large amplitude waves. Besides, the homotopy-Padeapproximations of the


Fig. 15.3 Wave elevationwhenH = 3/10 andε = 1/5given byc0 =−11/20. Solidline: 15th-order homotopy-approximation; Sym-bols: 3rd-order homotopy-approximation

x

η(x

)

-3 -2 -1 0 1 2 3-0.16

-0.08

0

0.08

0.16

0.24H = 3/10, ε = 1/5

Table 15.5 The [m,m] homotopy-Pade approximation ofµ = k c2/g for the pure deep-water waveswithout current, corresponding toΩ = 0.

H m= 15 m= 17 m= 19 m= 20 m= 21

0.65 1.1113 1.1113 1.1113 1.1113 1.11130.70 1.1300 1.1300 1.1300 1.1300 1.13000.75 1.1502 1.1502 1.1502 1.1502 1.15020.80 1.1712 1.1712 1.1712 1.1712 1.17120.85 1.1905 1.1904 1.1905 1.1904 1.19040.86 1.1926 1.1929 1.1930 1.1930 1.19300.87 1.1939 1.1940 1.1941 1.1941 1.19410.875 1.1933 1.1932 1.1936 1.1936 1.19360.88 1.1915 1.1909 1.1917 1.1916 1.19160.882 1.1902 1.1892 1.1902 1.1902 1.19020.8825 1.1898 1.1887 1.1898 1.1898 1.1897

wave phase speed agrees well with the analytic results givenby Schwartz [34] andLonguet-Higgins [28], as shown in Fig. 15.4. Note that, as shown in Table 15.5 andTable 15.6, the maximum wave height given by the HAM is aboutH = 0.8825,which is a little less thanH = 0.886 given by Stokes theory [29].

Note thatH = 0 corresponds to the pure current without waves. Whenc0 = −1,the fifth-order homotopy-approximation of the pure currentreads

y(ψ) = −ψ − ε exp(−ψ)+34

ε2 exp(−2ψ)− 56

ε3 exp(−3ψ)

+3532

ε4 exp(−4ψ)− 6340

ε5 exp(−5ψ)+ const. (15.54)

Using (15.2), we have the velocity profile of the pure currentin the still coordinates

uc(ψ) = −ε exp(−ψ)+12

ε2 exp(−2ψ)− 12

ε3 exp(−3ψ)+58

ε4 exp(−4ψ)


Table 15.6 The [m,m] homotopy-Pade approximation of the crest elevationηc for the pure deep-water waves without current, corresponding toΩ = 0, compared withH/2.

H m= 15 m= 17 m= 19 m= 20 m= 21

0.65 0.3875 0.3875 0.3875 0.3875 16.13%0.7 0.4250 0.4250 0.4250 0.4250 17.65%0.75 0.4649 0.4649 0.4649 0.4649 19.34%0.8 0.5080 0.5079 0.5079 0.5079 21.24%0.85 0.5566 0.5566 0.5565 0.5565 23.63%0.86 0.5675 0.5672 0.5672 0.5671 24.18%0.87 0.5793 0.5805 0.5798 0.5800 25.00%0.875 0.5855 0.5858 0.5856 0.5858 25.32%0.88 0.5923 0.5922 0.5922 0.5922 25.70%0.882 0.5952 0.5953 0.5949 0.5949 25.87%0.8825 0.5956 0.5956 0.5956 0.5956 25.93%

Fig. 15.4 Comparison of thedispersion relationship ofthe pure deep-water waves.Solid line: [23,23] homotopy-Pade approximation; Filledcircles: Longuet-Higgins’sresult [28]; Open circles:Schwartz’s result [34].

H

gc2 /k

0.7 0.75 0.8 0.85 0.91.12

1.14

1.16

1.18

1.2

ε = 0

− 78

ε5 exp(−5ψ)+2116

ε6 exp(−6ψ), (15.55)

which is exactly the same as the perturbation solution [38, 39]. The above two lim-iting cases indicate the validity of the HAM for this kind of complicated nonlinearPDE.

The general cases ofε 6= 0 andH 6= 0 are of most interest. The[2,2] homotopy-Pade approximation of the wave phase speed reads

c2/c20 =

8∑

i=0αi(H)ε i

1+8∑j=0

β j(H)ε j

, (15.56)


where

α0 = 1−18.3545H2−19.0783H4−9.5071H6−1.6706H8

− 0.1686H10−0.3944H12+0.1497H14,

α1 = −19.8095H−2+94.9902+79.8629H2+49.2679H4+16.4719H6

+ 1.3127H8+0.7249H10−0.5257H12,

α2 = −0.8127H−4−153.8798H−2−17.4780+8.4334H2−5.1748H4

− 4.1199H6−2.9438H10+0.7047H10,

α3 = 6.6032H−4−161.7258H−2−95.2033−20.2378H2+3.0971H4

+ 2.7794H6−1.2655H8+0.8404H10,

α4 = 5.2780H−4−79.0493H−2−48.4005−17.5988H2−5.1302H4

+ 0.4472H6+0.4421H8+0.0306H10,

α5 = 5.9728H−4−28.6492H−2−22.03866−3.7271H2+3.03746H4

+ 2.7544H6+0.9785H8−0.1480H10,

α6 = 0.4561H−4−4.9730H−2−6.5921−4.9442H2−2.6846H4

− 0.9577H6−0.2226H8+0.0393H10,

α7 = −1.0606H−4−2.26127H−2−2.5945−1.6613H2−0.6098H4

− 0.1298H6−0.01597H8+0.0014H10,

α8 = 0.4849H−4+1.2882H−2+1.1771+0.5258H2+0.1341H4

+ 0.0232H6+0.0034H8+0.0004H10,

and

β0 = −18.6152H2−14.3027H4−4.7494H6+0.3693H8

− 0.1756H10−0.6505H12−0.1882H14,

β1 = −19.8095H−2+101.3902+30.9370H2+5.9367H4−1.9934H6

+ 0.5200H8+3.2108H10+1.2336H12,

β2 = −0.8127H−4−180.3940H−2+152.2704+118.3418H2+26.6539H4

− 5.0904H6−9.6998H8−3.3617H10,

β3 = 5.5196H−4−364.0214H−2−98.3501+54.1467H2+29.9964H4

+ 5.5684H6−1.2260H8−0.4686H10,

β4 = 14.4435H−4−241.5237H−2−201.5196−46.7589H2+12.6025H4

+ 11.9062H6+2.9939H8+0.1049H10,

β5 = 10.2959H−4−54.8513H−2−80.1939−46.4665H2−13.8617H4

− 1.5235H6+0.18215H8+0.0065H10,

β6 = 6.7965H−4+6.2383H−2−5.5078−9.7100H2−5.4381H4

− 1.4762H6−0.17256H8−0.0018H10,

β7 = 1.28H−4+6.4458H−2+7.2395+3.6241H2+0.9258H4


Fig. 15.5 The influence of thevorticity parameterε on thephase velocityk c2/g of deep-water waves whenH = 0.1.Solid line: [2,2] homotopy-Pade approximation; Filledcircles:[8,8] homotopy-Padeapproximation; Open circle:the 5th-order perturbationsolution [38].

ε

gc2 /k

-0.2 -0.1 0 0.1 0.20.5

1

1.5

--

H = 0.1


ε

gc2 /k

-0.2 -0.1 0 0.1 0.20.5

1

1.5

2

H = 0.3

+ 0.1129H6+0.0041H8−0.0002H10,

β8 = −1.2567H−4−1.6932H−2−0.9140−0.2487H2−0.0366H4

− 0.0037H6−0.0006H8.

As shown in Fig. 15.5 to Fig. 15.7, the above[2,2] homotpy-Pade approximationis accurate enough even for large wave-heightH, which indicates not only the ac-curacy of the general expression (15.56) but also the convergence of the homotopy-series solution for waves with large amplitude. Note that the fifth-order perturbationresults [38] are valid only for waves with small amplitude (such asH = 0.1) andbecome more and more inaccurate when the wave-heightH increases, as shownin Fig. 15.5 to Fig. 15.7. Thus, for large wave-height (H ≥ 0.3), perturbation ap-proximations often overestimate the wave phase speed for both aiding and opposingcurrents.

Given ε, one can get the dispersion relationship in a similar way by means ofhomotpy-Pade technique. The curvesµ = k c2/g versus the wave-heightH in caseof ε =−0.25, −0.15, 0, 0.25 and 0.5 are as shown in Fig. 15.8. Note that the[8,8]



ε

gc2 /k

-0.2 -0.1 0 0.1 0.20.5

1

1.5

2

2.5

H = 0.5

Fig. 15.8 The dispersionrelationship of waves ona current with an expo-nential distribution of vor-ticity Ω = ε exp(−ψ)in case ofε = −0.25,−0.15, 0, 0.25, 0.5. Solidline: [12,12] homotopy-Padeapproximation; Symbols:[8,8] homotopy-Pade approx-imation.

H

gc2 /k

0 0.4 0.8 1.20.4

0.8

1.2

1.6

2

ε = -0.25

ε = -0.15

ε = 0

ε = 0.25

ε = 0.5

Table 15.7 Maximum wave-height of propagating waves on a current with the exponential distri-bution of vorticityΩ = ε exp(−ψ).

ε Maximum wave-height

-0.25 0.670 0.88250.25 1.040.50 1.17

and[12,12] homotopy-Pade approximations agree quite well, except ata few pointsvery close to the highest wave. In comparison with pure waves(ε = 0), the aidingexponential shear currents(ε < 0) tend to enlarge the wave phase velocity for givenwave heightH, but the opposing exponential shear currents(ε > 0) tend to shortenit. Especially, as shown in Table 15.7 and Fig. 15.9, the maximum wave-heightsare strongly dependent upon the vorticity of the currents: the aiding exponentialshear currents (ε < 0) tend to shorten the maximum wave-height, but the oppos-ing exponential shear currents(ε > 0) tend to enlarge it. Note that, the maximum


Fig. 15.9 Maximum wave-height versusε in caseof shear currents withthe exponential vorticityΩ = ε exp(−ψ).

ε

max

imum

wav

ehe

ight

-0.5 -0.25 0 0.25 0.5 0.750.6

0.8

1

1.2

Fig. 15.10 Approximationsof wave elevation by meansof c0 = −1/2. Solid line:25th-order approxima-tion whenH = 1.02 andε = 0.25; Dashed line: 45th-order approximation whenH = 0.8825 andε = 0; Dot-dashed line: 25th-order ap-proximation whenH = 0.64and ε = −0.25; Circles:the 20th-order homotopy-approximations.

x

η(x)

-3 -2 -1 0 1 2 3-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

wave-height of the pure propagating waves without currentsis aboutH = 0.8825.However, in case of the opposing exponential shear currents(ε > 0), the wave-height can be greater than the limiting value for pure waves,as shown in Table15.7 and Fig. 15.9. For example, in case of the opposing exponential currents with0.25≤ ε ≤ 0.5, we obtain convergent series solutions whenH = 0.92, which iseven higher thanH = 0.886 given by Stokes theory for pure waves on still water(see [29]). It should be emphasized that, according to Stokes theory, the propagat-ing wave withH = 0.92 cannotexist for pure waves on still water (ε = 0). However,in case of the opposing shear current withε = 0.25, we obtain the convergent seriessolution of waves even withH = 1.02, as shown in Fig. 15.10. Thus, based on thehomotopy-approximations, the maximum wave-height of propagating water waveson an opposing shear current (ε > 0) can be larger than that of pure waves on stillwater. Note that the wave shape in case ofH = 1.02 andε = 0.25 is steeper eventhan that of waves on still water withH = 0.8825, as shown in Fig. 15.10. For morephysical discussions, please refer to Cheng et al. [1].


Fig. 15.11 The kinetic en-ergy KEc of flow particle atcrest versus the wave heightH. Filled circles:[11,11]homotopy-Pade approxi-mation; Solid line:[12,12]homotopy-Pade approxima-tion; Open circles:[15,15]homotopy-Pade approxima-tion; Dashed line:[20,20]homotopy-Pade approxima-tion.

HK

Ec

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.05

0.1

0.15

0.2

0.25

Kinetic energy of flow particle under crest

ε = 0.25

ε = -0.25

ε = 0

As suggested by Stokes [32,33], a train of propagating waveson still water breakswhen the fluid velocity at crest is equal to the wave phase speed. To check thiscriterion for waves on a shear current, we also calculate thekinetic energy of theflow particle at crest in the reference frame moving with speed c, defined by

KEc =12

u2 =1µ[γ − y(0,0)]. (15.57)

The curves ofKEc versus the wave-heightH in case of the aiding current (ε =−0.25), still water (ε = 0) and the opposing current (ε = 0.25) are as shown inFig. 15.11. Note thatKEc ≈ 0 whenH ≈ 0.8825 for the pure waves on still waterwithout currents (ε = 0), which exactly corresponds to the highest wave given bySchwartz [34] and Longuet-Higgins [28]. However, we haveKEc ≈ 0 whenH ≈0.67 for waves on an aiding current (ε = −0.25), and whenH ≈ 1.04 for waves onan opposing current (ε = −0.25), respectively. So, it seems that the highest-wavecorresponds toKEc = 0, i.e. the fluid velocity at crest equals the wave phase speed.Note that the three curves ofKEc ∼ H in case ofε = ±0.25 andε = 0 are parallelapproximately. So, qualitatively speaking, an aiding shear current tends to shortenthe maximum wave height, but an opposing shear current has anopposite effect.Therefore, according to the approximations given by the HAM, Stokes’ criterion ofwave breaking has general meanings and is still correct evenfor waves on a non-uniform current, i.e. a train of propagating waves on a shearcurrent breaks as thefluid velocity at crest equals the wave phase speed. For more details, please refer toCheng, Cang and Liao [1].

This is a good example to show that the HAM can be used as a helpful tool tosolve some complicated nonlinear PDEs, so as to deepen our physical understand-ings about some complicated nonlinear phenomenon.



The HAM is applied to investigate the nonlinear interactionof the periodic travelingwaves on a non-uniform current with exponential distribution of vorticity. By prop-erly choosing a simple auxiliary linear operator, the original highly nonlinear PDEwith variable coefficient is transferred into an infinite number of much simpler lin-ear PDEs, which are rather easy to solve. The convergent series solutions are gainedby means of optimal convergence-control parameterc0 and homotopy-Pade tech-nique so that the highest wave can be obtained for the given non-uniform currents.These analytic approximations reveal the basic characteristic of the nonlinear in-teraction between the periodic traveling waves and the non-uniform currents. Theyillustrate that the HAM can be used as an useful analytic toolto deepen our physicalunderstanding about some complicated nonlinear phenomena.

Mathematically, it should be emphasized that the two auxiliary linear operators(15.37) and (15.38) have nearly no relationships with the original PDE (15.9) andthe nonlinear boundary condition (15.10). Note that other analytic techniques, es-pecially perturbation techniques, are strongly dependentupon linear operators inoriginal governing equations. Unlike other analytic techniques, the HAM providesus extremely large freedom to choose the auxiliary linear operators according to thephysical background of a given nonlinear problem. So, in theframe of the HAM, weneed not spend too much time in the details of governing equations and boundaryconditions of a given nonlinear problem, but should pay moreattention on its phys-ical background, so as to find out a proper solution expression. This is in essencequite different from other analytic techniques. Note that,it is the HAM that providesus such kind of extremely large freedom to greatly simplify resolving some nonlin-ear problems, as shown in this and previous chapters. Besides, as mentioned in pre-vious chapters, such kind of extremely large freedom is based on the guarantee ofthe convergent series solution by means of the optimal convergence-control param-eterc0: the freedom on the choice of the auxiliary linear operator has no meaningsat all, if the corresponding series solution is divergent.

Physically, it is found that, for given wave amplitude, waves propagate faster onan aiding shear current but more slowly on an opposing one, compared with waves instill water. Besides, different from the uniform current, an aiding shear current tendsto sharpen the crest but smoothen the trough, while an opposing shear current hasthe opposite effects. So, it is not the magnitude but the non-uniformity of the currentthat influences the wave shape. Especially, it is found that Stokes’ criterion of wavebreaking is still correct even for propagating waves on a non-uniform current, andthe highest wave on an opposing shear current is even higher and steeper than thatof waves on still water.

All of these verify the validity of the HAM for some complicated nonlinearPDEs. This example suggests that the HAM can be used as a useful tool to deepenour physical understandings about some complicated nonlinear phenomena.


Appendix 15.1 Mathematica code of wave-current interaction

The interaction of 2D nonlinear progress waves and non-uniform currents are solvedby means of the HAM. This Mathematica code is free available at

http://numericaltank.sjtu.edu.cn/HAM.htm.

Mathematica code of wave-current interactionby Jun CHENG and Shijun LIAO


( *************************************************** ********** )( * Interaction of wave & nonuniform currents * )( * Governing equation: * )( * yxx(yz)ˆ2-2yxyxz(yz)+(1+yxˆ2)yzz==omega,for z > 0 * )( * Boundary condition * )( * mu(1+yxˆ2)+2y(yz)ˆ2-2kk(yz)ˆ2 == 0 , for z = 0 * )( * y(x,z)->0 , as z->Infinity * )( * H = y(0,0)-y(pi,0) * )( *************************************************** ********** )

( *************************************************** ********** )( * Define chi_[k] * )( *************************************************** ********** )chi[k_]:= If[k<=1,0,1];

( *************************************************** ********** )( * Define initial guess * )( *************************************************** ********** )y[0] = -z + H/2 Exp[-z] * (Exp[I * x]+Exp[-I * x])/2;

( *************************************************** ********** )( * Define GetfAll[k] * )( *************************************************** ********** )GetAll[k_]:=Module[,yz[k] = Expand[D[y[k],z]];yzz[k] = Expand[D[yz[k],z]];yx[k] = Expand[D[y[k],x]];yxx[k] = Expand[D[yx[k],x]];ygamma[k] = 2(y[k]-gamma[k])//Expand;yxz[k] = Expand[D[yx[k],z]];yxyz[k] = Expand[Sum[yx[i] * yz[k-i],i,0,k]];yzyzyz[k] = Expand[Sum[yzyz[i] * yz[k-i],i,0,k]];yzyz[k] = Expand[Sum[yz[i] * yz[k-i],i,0,k]];yxyxz[k] = Expand[Sum[yx[i] * yxz[k-i],i,0,k]];yxyx[k] = Expand[Sum[yx[i] * yx[k-i],i,0,k]];];


( *************************************************** ********** )( * Define GetRHS[k] * )( *************************************************** ********** )GetRHS[k_]:=Module[temp,i,j,n,temp[1] = Sum[yxx[k-1-i] * yzyz[i],i,0,k-1];temp[2] = Sum[yxyxz[k-1-i] * yz[i],i,0,k-1];temp[3] = Sum[yzz[k-1-i] * yxyx[i],i,0,k-1];temp[4] = yzyzyz[k-1] * Omega;delta[k] = Expand[temp[1]-2 * temp[2]+yzz[k-1]+temp[3]-temp[4]];RHS[k] = Expand[c0 * delta[k]];];

( *************************************************** ********** )( * Define GetRHSb[k] * )( *************************************************** ********** )GetRHSb[k_]:=Module[temp,i,j,n,temp[1] = Sum[mu[k-1-i] * yxyx[i],i,0,k-1];temp[2] = Sum[(ygamma[k-1-i]) * yzyz[i],i,0,k-1];temp[3] = mu[k-1];deltaB[k] = Expand[temp[1] + temp[2] + temp[3]];RHSb[k] = Expand[c0 * deltaB[k]];];

( *************************************************** ********** )( * Define the the auxiliary linear operator L and Lb * )( *************************************************** ********** )L[f_] := Expand[D[f,z,2] + D[f,x,2]];Lb[f_] := Expand[D[f,z] + f];

( *************************************************** ********** )( * Define inverse operator of auxiliary linear operator Linv * )( *************************************************** ********** )Linv[Exp[n_. * z] * Exp[m_. * x]]:= 1/(nˆ2+mˆ2) * Exp[n * z] * Exp[m * x];Linv[Exp[m_. * x]]:= Exp[m * x]/mˆ2;Linv[Exp[n_. * z]]:= Exp[n * z]/nˆ2;

( *************************************************** ********** )( * The property of the inverse operator Linv * )( *************************************************** ********** )Linv[p_Plus] := Map[Linv,p];Linv[c_ * f_] := c * Linv[f] /; FreeQ[c,z] && FreeQ[c,x];

( *************************************************** ********** )( * Define GetySpecial[k] * )( *************************************************** ********** )GetySpecial[k_]:=Module[temp,temp[0] = Expand[RHS[k]];ySpecial = Linv[temp[0]] //Expand;];

( *************************************************** ********** )( * Define two functions for common solution * )( *************************************************** ********** )therest[f_,k_]:=(-Lb[f] + RHSb[k])/.z->0//ComplexExpa nd;


ccc[f_] := Select[f, FreeQ[#, x] &];

( *************************************************** ********** )( * Define Gety[k] * )( *************************************************** ********** )Gety[k_] := Module[temp,sol,p0,p1,plist,clist,dlist,c1,property,temp[0] = therest[ySpecial,k];temp[1] = temp[0]//Expand;p0 = ccc[temp[1]];p1 = Coefficient[temp[1],Cos[x]];temp[2] = temp[1] - p0 - p1 * Cos[x];sol = Solve[p0 == 0,p1 == 0,gamma[k-1],mu[k-1]];gamma[k-1] = gamma[k-1]/.sol[[1]]//Expand;mu[k-1] = mu[k-1]/.sol[[1]]//Expand;plist = Table[Coefficient[temp[2],Cos[i * x]],i,2,2k+1]

//Simplify;clist = Table[plist[[i-1]]/(1-i),i,2,2k+1];dlist = Table[clist[[i-1]] * (1-(-1)ˆi),i,2,2k+1];temp[3] = Apply[Plus,dlist];temp[4] = ySpecial /.z->0,x->0 ;temp[5] = ySpecial /.z->0,x->Pi;temp[6] = Table[Exp[-i * z] * (Exp[i * I * x]+Exp[-i * I * x])/2,

i,2,2k+1];c1 = -1/2 * (temp[4]-temp[5]+temp[3]);property = Dot[temp[6],clist]

+ c1 * Exp[-z] * (Exp[I * x] + Exp[-I * x])/2;y[k] = Expand[ySpecial + chi[k] * y[k-1] + property];];

( *************************************************** ********** )( * Define GetYs[k] * )( *************************************************** ********** )GetYs[k_]:=Module[temp,temp[0] = Y[k] /. z->0;temp[1] = Integrate[temp[0], x,-Pi,Pi]/2/Pi;Ys[k] = temp[0] - temp[1]//Expand;];

( *************************************************** ********** )( * Define GetErrB[k] * )( * Gain squared residual of boundary condition * )( *************************************************** ********** )GetErrB[k_]:=Module[temp,i,Yx,Yz,Nx,Np,dx,xx,Yx = D[Y[k],x];Yz = D[Y[k],z];temp[1] = MU[k] * (1+Yxˆ2) + 2 * (Y[k]-GAMMA[k]) * Yzˆ2 /. z->0;temp[2] = temp[1]ˆ2;Nx = 20;dx = N[Pi/Nx,100];sum = 0;Np = 0;For[i = 0, i <= Nx, i++,

xx = i * dx;


sum = sum + temp[2] /. x->xx;Np = Np + 1;

];ErrB[k] = sum/Np;If[NumberQ[ErrB[k]],

ErrB[k] = Re[ErrB[k]];Print["k = ",k," Squared Residual of B.C. = ",ErrB[k]//N]];

];

( *************************************************** ********** )( * Define GetErr[k] * )( * Gain squared residual of governing equation * )( *************************************************** ********** )GetErr[k_]:=Module[temp,i,j,sum,xx,zz,Yx,Yz,Yxx,Yxz,Yzz,Nx,Nz,Np,Yx = D[Y[k],x];Yz = D[Y[k],z];Yxx = D[Yx,x];Yxz = D[Yx,z];Yzz = D[Yz,z];temp[1] = Yxx * Yzˆ2 - 2 * Yx* Yz* Yxz + (1+Yxˆ2) * Yzz - Yzˆ3 * Omega;temp[2] = temp[1]ˆ2;Nx = 10;Nz = 10;dx = N[ Pi/Nx,100];dz = N[2 * Pi/Nz,100];sum = 0;Np = 0;For[i = 0, i <= Nx, i++,

For[j = 0, j <= Nz,j++,xx = i * dx;zz = j * dz;sum = sum + temp[2]/.x->xx,z->zz;Np = Np + 1;

];];

Err[k] = sum/Np;If[NumberQ[Err[k]],

Err[k] = Re[Err[k]];Print["k = ",k," Squared Residual of G.E. = ",Err[k]//N]];

];

( *************************************************** ********** )( * Define hp[f_,m_,n_] * )( * Gain [m,n] Homotopy-Pade approximation * )( *************************************************** ********** )hp[f_,m_,n_]:=Block[k,i,df,res,q,df[0] = f[0];For[k = 1, k <= m+n, k++, df[k] = f[k] - f[k-1]//Expand ];res = df[0] + Sum[df[i] * qˆi,i,1,m+n];Pade[res,q,0,m,n]/.q->1];


( *************************************************** ********** )( * Main Code * )( *************************************************** ********** )ham[m0_,m1_]:=Module[temp,k,n,For[k=Max[1,m0],k<=m1,k=k+1,Print[" k = ",k];

GetAll[k-1];GetRHS[k];GetRHSb[k];GetySpecial[k];Gety[k];Y[k] = Y[k-1] + y[k]//ComplexExpand;If[k>1,

MU[k-1] = MU[k-2] + mu[k-1]//Expand;GAMMA[k-1] = GAMMA[k-2] + gamma[k-1]//Expand;

];Print[" mu = ",MU[k-1]//N," variation = ",mu[k-1]//N];Print[" gamma = ",GAMMA[k-1]//N," variation = ",

gamma[k-1]//N];];

Print[" Sucessful ! "];];

( *************************************************** ********** )( * Define initial guess u[0] and related functions * )( *************************************************** ********** )Y[0] = y[0];GAMMA[0] = gamma[0];MU[0] = mu[0];ERR[0] = ComplexExpand[(Y[0] - GAMMA[0]) /.x->0/.z->0];Omega = epsilon * Exp[-z];

( * Physical and control parameters * )epsilon = 1/5;H = 3/10;c0 =-11/20;

( * Print input data * )Print["epsilon = ",epsilon];Print[" H = ",H];Print[" c0 = ",c0];

( * Gain the 10th-order approximation * )ham[1,11];

( * Gain squared residual of governing equation * )For[k=2, k<=10, k=k+2, GetErr[k]];

( * Gain squared residual of boundary condition * )For[k=2, k<=10, k=k+2, GetErrB[k]];

References 527

References

1. Cheng, J., Cang, J., Liao, S.J.: On the interaction of deepwater waves and exponential shearcurrents. ZAMP.60, 450 – 478 (2009)

2. Clamond, D.: Steady finite-amplitude waves on a horizontal seabed of arbitrary depth. J. FluidMech.398, 45-60 (1999)

3. Clamond, D.: Cnoidal-type surface waves in deep water. J.Fluid Mech.489, 101-120 (2003)4. Clamond, D., Fructus, D.: Accurate simple approximationfor the solitary wave. C. R.

Mecanique.331, 727-732 (2003)5. Dalrymple, R.A.: A numerical model for periodic finite amplitude waves on a rotational fluid.

J. Comput. Phys.24, 29-42 (1977)6. Drennan, W.M, Hui, W.H., Tenti, G.: Accurate calculations of Stokes water waves of large

amplitude. J. App. Math. Phys. (ZAMP).43, 367-384 (1992)7. Dubreil-Jacotin, M.L.: Sur la determination rigoureuse des ondes permanentes periodiques

d’ampleur finie. J. Math. Pures et Appl.13, 217-291 (1934)8. Eastwood, J.W., Watson, C.J.H.: Implication of wave-current interaction for offshore design.

E. and P. Forum,Paris (1989)9. Fenton, J.D.: A fifth-order Stokes theory for steady waves. J. Waterw. Port Coastal and Ocean

Eng.111, 216-234 (1985)10. Kishida, N., Sobey, R.J.: Stokes theory for waves on linear shear current. J. Waterw. Port

Coastal and Ocean Eng.114, 1317-1334 (1988)11. Li, Y.J., Nohara, B.T., Liao, S.J.: Series solutions of coupled Van der Pol equation

by means of homotopy analysis method. J. Mathematical Physics 51, 063517 (2010).doi:10.1063/1.3445770
















26. Liao, S.J., Cheung, K.F.: Homotopy analysis of nonlinear progressive waves in deep water.Journal of Engineering Mathematics.45, No.2, 105-116 (2003)


28. Longuet-Higgins, M., Tanaka, M.: On the crest instabilities of steep surface waves. J. FluidMech.336, 51-68 (1997)

29. Marshall, P., Tulin, Jiyue, J.Li: On the breaking of energetic waves. Int. J. Offshore Polar Eng.2, No.1, 46-53 (1992)

30. Peregrine, D.H.: Interaction of water waves and current. Adv. Appl. Mech.16, 9-117 (1977)31. Phillips, O.M.: The dynamics of the upper ocean. Cambridge Press, Cambridge (1977)32. Stokes, G.G.: Supplement to a paper on the theory of oscillatory waves. Math. Phys. Papers.

1, 314-326 (1880)33. Stokes, G.G.: On the theory of oscillation waves. Trans.Cambridge Phil. Soc.8, 441-455

(1894)34. Schwartz, L.W.: Computer extension and analytic continuation of Stokes’ expansion for grav-

ity waves. J. Fluid Mech.62, 553-578 (1974)35. Thomas, G.P.: Wave-current interactions: an experimental and numerical study. Part2. Non-

linear waves. J. Fluid Mech.216, 505-536 (1990)36. Thompson, P.D.: The propagation of small surface disturbances through rotational flow. Ann.

N. Y. Acad. Sci.51, 463-474 (1949)37. Toumazis, A.D., Ahilan, R.V.: Review of recent analytical and approximate solutions of wave-

current interaction. Environmental forces on offshore structure and their prediction.26, 61-79(1990)

38. Wang, T. et al. : Effect of nonlinear wave-current interaction on flow fields and hydrodynamicforces. Science in China - A.40, No. 6, 622-632 (1997)

39. Wang, T. et al. : On wave-current interaction. Adv. in Mech. 29, No. 3, 331-343 (1999)40. Xu, H., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy-based solutions of the Navier-


Chapter 16Resonance of arbitrary number of periodictraveling water waves

Abstract In this chapter, we verify the validity of the homotopy analysis method(HAM) for a rather complicated nonlinear PDE describing thenonlinear interactionof arbitrary number of traveling water waves. In the frame ofthe HAM, the wave-resonance criterion for arbitrary number of waves is gained, for the first time, whichlogically contains the famous Phillips’ criterion for foursmall amplitude waves.Besides, it is found for the first time that, when the wave-resonance criterion issatisfied and the wave system is fully developed, there existmultiple steady-stateresonant waves, whose amplitude might be much smaller than primary waves sothat a resonant wave may contain much small percentage of thetotal wave energy.This example illustrates that the HAM can be used as a tool to deepen and enrichour understandings about some rather complicated nonlinear phenomena.

16.1 Introduction

In this chapter, we further illustrate that the homotopy analysis method (HAM)[7–21] can be used to investigate some rather complicated nonlinear partial differ-ential equations (PDEs) so as to deepen and enrich our understandings about someinteresting nonlinear phenomena.

Let us consider here the nonlinear interaction ofarbitrary number of periodictraveling gravity waves in deep water [20]. Letz denote the vertical co-ordinatepointing upwards,x,y the horizontal co-ordinates,t the time,z= ζ (x,y, t) the freesurface, respectively. The three axises ofx,y,z are perpendicular to each other, withthe unit vectori, j ,k, respectively, i.e.i · j = i · k = j · k = 0, where· is the mul-tiplication dot. Assume that the vorticity is negligible and there exists a potentialϕ(x,y,z, t) thatu = ∇ϕ and

∇2ϕ = 0, z≤ ζ (x,y, t), (16.1)

where

529

530 16 Resonance of arbitrary number of periodic traveling water waves

∇ = i∂∂x

+ j∂∂y

+ k∂∂z

.

On the free surfacez= ζ (x,y, t), the pressure is constant, which gives from Bernoulli’sequation the dynamic boundary condition

g ζ +∂ϕ∂ t

+12

u2 = 0, onz= ζ (x,y, t), (16.2)

whereg is the acceleration due to gravity. Besides,z−ζ vanishes following a parti-cle, which gives the kinematic boundary condition

∂ζ∂ t

− ∂ϕ∂z

+

(

∂ϕ∂x

∂ζ∂x

+∂ϕ∂y

∂ζ∂y

)

= 0, onz= ζ (x,y, t). (16.3)

Combining the above two equations gives the boundary condition

∂ 2ϕ∂ t2 +g

∂ϕ∂z

+∂ (u2)

∂ t+u ·∇

(

12

u2)

= 0, onz= ζ (x,y, t), (16.4)

whereu = ∇ϕ andu2 = ∇ϕ ·∇ϕ . On the bottom, it holds

∂ϕ∂z

= 0, asz→−∞. (16.5)

For details, please refer to Phillips [25] and Longuet-Higgins [22] .In his pioneering work about nonlinear interaction offour gravity waves in deep

water, Phillips [25] gave the criterion of wave resonance

k1± k2± k3± k4 = 0, σ1±σ2±σ3±σ4 = 0, (16.6)

whereσi =√

gki with ki = |k i | (i = 1,2,3,4) is the wave angular frequency, andk i

denotes the wave number. It should be emphasized that Phillips’ resonance criterion(16.6) works only for weakly nonlinear waves with small amplitudes, becauseσi =√

gki is the angular frequency of linear theory for a single gravity wave with smallamplitude.

In a special case ofk4 = k1, Phillips [25] showed by means of perturbation meth-ods that, ifk3 = 2k1 − k2 andσ3 = 2σ1−σ2, then a steady-state solution for thetriad did not exist, and the amplitude of the third wave, if initially zero, would growlinearly in time. This conclusion was confirmed by Longuet-Higgins [22] via per-turbation theory and supported by some experiments [23, 24]. Besides, Benney [3]solved the equations governing the time dependence of the resonant modes andstudied the energy-sharing mechanism involved.

Although half century passed since Phillips’ pioneering work [25], there existstill some open questions about nonlinear interaction between gravity waves. First,Bretherton [4] pointed out that the perturbation scheme used by Phillips breaksdown for large time, and then suggested by investigating a one-dimensional dis-persive wave model that the amplitude of each wave componentshould be bounded.

16.2 Resonance criterion of two small-amplitude primary waves 531

Besides, Phillips’ wave resonance criterion (16.6) is valid for only four waves withsmallwave-amplitudes. What is the wave-resonance criterion forarbitrary numberof waves withlarge amplitude? It seems difficult to apply the perturbation schemeused by Phillips [25] and Longuet-Higgins [22] to answer these questions, becausethe related algebra was daunting and “extraordinarily tedious”, as mentioned byPhillips [26].

In 2011, Liao [20] successfully applied the HAM to give a wave-resonance cri-terion forarbitrary number of traveling waves by means of the so-called homotopymultiple-variable method, which keeps the clear physical meaning of the multiple-scales of perturbation techniques but completely abandonsthe small physical pa-rameters. In this chapter, the nonlinear interaction of periodic traveling water wavesis used as an example to describe the basic ideas of the homotopy multiple-variablemethod. By means of this method, the wave resonance criterion for arbitrary numberof traveling waves with large amplitude is gained, which logically contains Phillips’criterion for four small-amplitude waves. The detailed mathematical derivation forthe high-order deformation equations is given in the Appendix 16.1. For more de-tails, please refer to Liao [20].

16.2 Resonance criterion of two small-amplitude primary waves

16.2.1 Brief Mathematical formulas

Without loss of generality, let us first consider a fully-developed gravity wavesystem basically composed of two primary traveling waves indeep water withwave numbersk1,k2 and the corresponding angular frequenciesσ1,σ2, respectively,wherek1× k2 6= 0 (i.e. the two traveling waves are not collinear). Due to nonlinearinteraction, this wave system contains an infinite number ofwave components withthe corresponding wave numbermk1+nk2, wherem,n are integers. Since the wavesystem is assumed to be fully developed, each wave amplitudeis independent ofthe time. Letα1,α2 denote the angles between the positivex-axis i and the wavenumber vectorsk1 andk2, respectively, wherek1 · k = k2 · k = 0, i.e. thez-axis isperpendicular to the wave numbersk1,k2. Then,

k1 = k1 (cosα1 i + sinα1 j) , k2 = k2 (cosα2 i + sinα2 j) , (16.7)

wherek1 = |k1| andk2 = |k2|.Write r = xi + yj . According to the linear gravity wave theory, the two primary

waves traveling with the wave numbersk1 andk2 are given bya1cosξ1, a2cosξ2,respectively, where

ξ1 = k1 · r −σ1 t, ξ2 = k2 · r −σ2 t. (16.8)


So, the above two variables have very clear physical meaning: the wave profile mustbe a periodic function ofξ1 andξ2. Mathematically, using these two variables, thetimet should not appearexplicitlyfor a fully developed wave system. In other words,one can express the potential functionϕ(x,y,z, t) = φ(ξ1,ξ2,z) and the wave surfaceζ (x,y, t) = η(ξ1,ξ2), respectively, for the nonlinear interaction of a fully developedwave system composed of the two primary traveling waves.

Then,

u = ∇ϕ = i∂ϕ∂x

+ j∂ϕ∂y

+ k∂ϕ∂z

= k1∂φ∂ξ1

+ k2∂φ∂ξ2

+ k∂φ∂z

= ∇φ , (16.9)

where

∇ = k1∂

∂ξ1+ k2

∂∂ξ2

+ k∂∂z

. (16.10)

Thus,

u2 = ∇ϕ ·∇ϕ = ∇φ · ∇φ

= k21

(

∂φ∂ξ1

)2

+2k1 ·k2∂φ∂ξ1

∂φ∂ξ2

+ k22

(

∂φ∂ξ2

)2

+

(

∂φ∂z

)2

, (16.11)

wherek1 ·k = k2 ·k = 0 is used, andk1 ·k2 = k1k2cos(α1−α2). In general, it holds

∇φ · ∇ψ = k21

∂φ∂ξ1

∂ψ∂ξ1

+ k1 ·k2

(

∂φ∂ξ1

∂ψ∂ξ2

+∂ψ∂ξ1

∂φ∂ξ2

)

+ k22

∂φ∂ξ2

∂ψ∂ξ2

+∂φ∂z

∂ψ∂z

(16.12)

for arbitrary functionsφ(ξ1,ξ2,z) andψ(ξ1,ξ2,z).Similarly, we have

∇2ϕ = ∇2φ = k21

∂ 2φ∂ξ 2

1

+2k1 ·k2∂ 2φ

∂ξ1∂ξ2+ k2

2∂ 2φ∂ξ 2

2

+∂ 2φ∂z2 . (16.13)

Then, the original governing equation reads

∇2φ = k21

∂ 2φ∂ξ 2

1

+2k1 ·k2∂ 2φ

∂ξ1∂ξ2+ k2

2∂ 2φ∂ξ 2

2

+∂ 2φ∂z2 = 0, (16.14)

which has the general solution

φ = [A cos(mξ1+nξ2)+B sin(mξ1+nξ2)]e|mk1+nk2|z, (16.15)

wherem,n are integers,A,B are integral constants, and−∞ < z≤ η(ξ1,ξ2), respec-tively.

For the sake of simplicity, define


f =12

∇φ · ∇φ =u2

2

=12

[

k21

(

∂φ∂ξ1

)2

+2k1 ·k2∂φ∂ξ1

∂φ∂ξ2

+ k22

(

∂φ∂ξ2

)2

+

(

∂φ∂z

)2]

. (16.16)

Using the new variablesξ1 andξ2, the dynamic boundary condition (16.2) becomes

η =1g

(

σ1∂φ∂ξ1

+σ2∂φ∂ξ2

− f

)

, onz= η(ξ1,ξ2). (16.17)

On the free surfacez= η(ξ1,ξ2), the kinematic boundary condition (16.4) reads

σ21

∂ 2φ∂ξ 2

1

+2σ1σ2∂ 2φ

∂ξ1∂ξ2+σ2

2∂ 2φ∂ξ 2

2

+g∂φ∂z

− 2

(

σ1∂ f∂ξ1

+σ2∂ f∂ξ2

)

+ ∇φ · ∇ f = 0, (16.18)

where∂ f∂ξ1

= ∇φ · ∇(

∂φ∂ξ1

)

,∂ f∂ξ2

= ∇φ · ∇(

∂φ∂ξ2

)

(16.19)

and

∇φ · ∇ f = k21

∂φ∂ξ1

∂ f∂ξ1

+ k1 ·k2

(

∂φ∂ξ1

∂ f∂ξ2

+∂ f∂ξ1

∂φ∂ξ2

)

+ k22

∂φ∂ξ2

∂ f∂ξ2

+∂φ∂z

∂ f∂z

.

On the bottom, it holds∂φ∂z

= 0, asz→−∞. (16.20)

Given two physically reasonable angular frequenciesσ1 andσ2, our aim is tofind out the corresponding unknown potential functionφ(ξ1,ξ2,z) and the unknownfree surfaceη(ξ1,ξ2), which are governed by the linear PDE (16.14) subject to twononlinear boundary conditions (16.17) and (16.18) on theunknownfree surfacez= η(ξ1,ξ2), and one linear boundary condition (16.20) on the bottom. Here, itshould be emphasized that, by means of the two independent variablesξ1 andξ2,the timet does not appearexplicitly in the unknown potential function and wavesurface. This greatly simplifies resolving the problem, as shown below.

As mentioned before, the two variablesξ1 andξ2 have clear physical meaningsand the solutions of this problem should be periodic functions of ξ1 andξ2. Fromphysical points of view, the wave surface should be in the form

η(ξ1,ξ2) =+∞

∑m=0

+∞

∑n=−∞

am,ncos(mξ1+nξ2), (16.21)

wheream,n is the amplitude of the wave component cos(mξ1 + nξ2). Note that(16.15) is the general solution of the governing equation (16.14). So, the corre-


sponding potential function should be in the form

φ(ξ1,ξ2,z) =+∞

∑m=0

+∞

∑n=−∞

bm,n Ψm,n(ξ1,ξ2,z) (16.22)

whereΨm,n(ξ1,ξ2,z) = sin(mξ1+nξ2) e|mk1+nk2|z, (16.23)

and bm,n is unknown coefficient independent ofξ1,ξ2,z. Note that the potentialfunctionφ(ξ1,ξ2,z) defined by (16.22)automaticallysatisfies the governing equa-tion (16.14) and the bottom condition (16.20). The above expressions (16.21) and(16.22) are called thesolution-expressionsof η andφ , respectively, which play im-portant roles in the frame of the HAM, as shown below.

For simplicity, define a nonlinear operator

N [φ(ξ1,ξ2,z)] = σ21

∂ 2φ∂ξ 2

1

+2σ1σ2∂ 2φ

∂ξ1∂ξ2+σ2

2∂ 2φ∂ξ 2

2

+g∂φ∂z

−2

(

σ1∂ f∂ξ1

+σ2∂ f∂ξ2

)

+ ∇φ · ∇ f , (16.24)

corresponding to the kinematic boundary condition (16.18), where the angular fre-quenciesσ1,σ2 are given. Note that it contains a linear operator

L0 (φ) = σ21

∂ 2φ∂ξ 2

1

+2σ1σ2∂ 2φ

∂ξ1∂ξ2+σ2

2∂ 2φ∂ξ 2

2

+g∂φ∂z

. (16.25)

Let L denote an auxiliary linear differential operator with the propertyL [0] =0. As mentioned in previous chapters, the HAM provides us theextremely largefreedom to choose the auxiliary linear operator. Based on the results of the linearwave theory, i.e.

σ1 ≈√

g k1 = σ1, σ2 ≈√

g k2 = σ2, (16.26)

we choosesuch an auxiliary linear operator

L φ = σ21

∂ 2φ∂ξ 2

1

+2σ1σ2∂ 2φ

∂ξ1∂ξ2+ σ2

2∂ 2φ∂ξ 2

2

+g∂φ∂z

. (16.27)

Then, letq ∈ [0,1] denote the embedding parameter,c0 6= 0 the convergence-control parameter,φ0(ξ1,ξ2,z) an initial guess of the potential function satisfying∇2φ0 = 0 and the bottom condition (16.20), respectively. We construct the zeroth-order deformation equation

∇2 φ(ξ1,ξ2,z;q) = 0, −∞ < z≤ η(ξ1,ξ2;q), (16.28)

subject to the two boundary conditions onz= η(ξ1,ξ2;q):

(1−q)L[

φ (ξ1,ξ2,z;q)−φ0(ξ1,ξ2,z)]

= q c0 N[

φ (ξ1,ξ2,z;q)]

, (16.29)


and

(1−q)η(ξ1,ξ2;q)

= q c0

η(ξ1,ξ2;q)− 1g

[

σ1∂ φ (ξ1,ξ2,z;q)

∂ξ1+σ2

∂ φ(ξ1,ξ2,z;q)∂ξ2

− f

]

, (16.30)

where

f =12

∇φ(ξ1,ξ2,z;q) · ∇φ(ξ1,ξ2,z;q).

Besides, at the bottom, it holds

∂ φ (ξ1,ξ2,z;q)∂z

= 0, asz→−∞. (16.31)

Whenq= 0, since the initial guessφ0(ξ1,ξ2,z) satisfies the governing equation(16.14) and the bottom condition (16.20), we have due to the propertyL (0) = 0 ofthe auxiliary linear operator (16.27) that

φ(ξ1,ξ2,z;0) = φ0(ξ1,ξ2,z), (16.32)

andη(ξ1,ξ2;0) = 0, (16.33)

which provide us the initial approximations of the potential functionφ(ξ1,ξ2,z) andthe free surfaceη(ξ1,ξ2). Whenq= 1, sincec0 6= 0, (16.28) to (16.31) are equiv-alent to the original equations (16.14), (16.17), (16.18) and (16.20), respectively,provided

φ (ξ1,ξ2,z;1) = φ(ξ1,ξ2,z), η(ξ1,ξ2;1) = η(ξ1,ξ2). (16.34)

So, asq ∈ [0,1] increases from 0 to 1,φ(ξ1,ξ2,z;q) deforms (or varies)continu-ouslyfrom the initial approximationφ0(ξ1,ξ2,z) to the unknown potential functionφ(ξ1,ξ2,z), so doesη(ξ1,ξ2;q) from 0 to the unknown wave profileη(ξ1,ξ2), re-spectively. Mathematically speaking, (16.28) to (16.31) define two homotopies:

φ (ξ1,ξ2,z;q) : φ0(ξ1,ξ2,z)∼ φ(ξ1,ξ2,z),

η(ξ1,ξ2;q) : 0∼ η(ξ1,ξ2),

Note that we have freedom to choose the convergence-controlparameterc0. As-suming thatc0 is so properly chosen that the homotopy-Maclaurin series

φ(ξ1,ξ2,z;q) = φ0(ξ1,ξ2,z)++∞

∑n=1

φn(ξ1,ξ2,z) qn, (16.35)

η(ξ1,ξ2;q) =+∞

∑n=1

ηn(ξ1,ξ2) qn, (16.36)


exist and absolutely converge atq= 1, we have due to (16.34) the homotopy-seriessolution

φ(ξ1,ξ2,z) = φ0(ξ1,ξ2,z)++∞

∑n=1

φn(ξ1,ξ2,z), (16.37)

η(ξ1,ξ2) =+∞

∑n=1

ηn(ξ1,ξ2), (16.38)

where

φn(ξ1,ξ2,z) =1n!

∂ nφ (ξ1,ξ2,z;q)∂qn

∣

∣

∣

∣

q=0= Dn

[

φ(ξ1,ξ2,z;q)]

,

ηn(ξ1,ξ2) =1n!

∂ nη(ξ1,ξ2;q)∂qn

∣

∣

∣

∣

q=0= Dn [η(ξ1,ξ2;q)] ,

are thenth-order homotopy-derivatives,Dn is the nth-order homotopy-derivativeoperator. Themth-order homotopy-approximations read

φ(ξ1,ξ2,z) ≈ φ0(ξ1,ξ2,z)+m

∑n=1

φn(ξ1,ξ2,z),

η(ξ1,ξ2) ≈m

∑n=1

ηn(ξ1,ξ2).

The PDEs for the unknownφn(ξ1,ξ2,z) andηn(ξ1,ξ2) can be derived directlyfrom the zeroth-order deformation equations (16.28) to (16.31). Substituting thehomotopy-Maclaurin series (16.35) into the governing equation (16.28) and theboundary condition (16.31) at bottom, and equating the like-power of the embed-ding parameterq, we have

∇2 φm(ξ1,ξ2,z) = 0, (16.39)

subject to the boundary condition at bottom

∂φm(ξ1,ξ2,z)∂z

= 0, asz→−∞, (16.40)

wherem≥ 1. It should be emphasized that the two boundary conditions (16.29)and (16.30) are satisfied on the unknown boundaryz= η(ξ1,ξ2;q), which itselfis dependent upon the embedding parameterq, too. So, it is relatively more com-plicated to deduce the corresponding equations. Briefly speaking, substituting theseries (16.35) and (16.36) into the boundary condition (16.29) and (16.30) withz= η(ξ1,ξ2;q), then equating the like-power ofq, we have two linear boundaryconditions onz= 0:

L (φm) = c0 ∆ φm−1+ χm Sm−1− Sm, m≥ 1, (16.41)


and

ηm(ξ1,ξ2) = c0 ∆ ηm−1+ χm ηm−1, (16.42)

where

∆ ηm−1 = ηm−1−

1g

[(

σ1 φ1,0m−1+σ2 φ0,1

m−1

)

−Γm−1,0

]

(16.43)

and

L (φm) =

(

σ21

∂ 2φm

∂ξ 21

+2 σ1σ2∂ 2φm

∂ξ1∂ξ2+ σ2

2∂ 2φm

∂ξ 22

+g∂φm

∂z

)∣

∣

∣

∣

z=0

. (16.44)

The detailed derivation of the above equations and the definitions of ∆ φm−1, Sm−1,

Sm, χm, Γm−1,0, φ1,0m−1, φ0,1

m−1 are given in the Appendix 16.1. Note that, the sub-problems forφm andηm are not onlylinear but alsodecoupled: given φm−1 andηm−1, it is straightforward to getηm directly, and thenφm is obtained by solving thelinear Laplace equation (16.39) with two linear boundary conditions (16.40) and(16.41). Thus, the high-order deformation equations can beeasily solved, especiallyby means of the computer algebra system such as Mathematica.

In a similar way as shown in Chapter 2, we can prove the following theorem:

Convergence TheoremThe homotopy-series solution (16.37) and (16.38) sat-isfy the original governing equation (16.14) and the boundary conditions (16.17),(16.18) and (16.20), provided that

+∞

∑m=0

∆ φm = 0,

+∞

∑m=0

∆ ηm = 0, (16.45)

where∆ φm,∆ η

m are defined by (16.120) and (16.43), respectively.

Because the terms∆ φm and∆ η

m are by-products in solving the high-order deformationequations, the above theorem provides us a convenient way tocheck the convergenceand accuracy of the homotopy-series solution. For this reason, we define the squaredresiduals

Eφm =

1π2

∫ π

0

∫ π

0

(

m

∑n=0

∆ φn

)2

dξ1 dξ2, (16.46)

Eηm =

1π2

∫ π

0

∫ π

0

(

m

∑n=0

∆ ηn

)2

dξ1 dξ2, (16.47)

for themth-order approximations ofφ andη , respectively. According to the above-mentioned convergence theorem, ifEφ

m → 0 andEηm → 0 asm→ +∞, then the

homotopy-series solution (16.37) and (16.38) satisfy the original equation and all


boundary conditions. Besides, the values ofEφm andEη

m also indicate the accuracyof themth-order homotopy-approximation ofφ andη , respectively.

Note that the auxiliary linear operator (16.27) has the property

LΨm,n =[

g|mk1+nk2|− (mσ1+nσ2)2]Ψm,n, (16.48)

whereΨm,n is given by (16.23), which automatically satisfies the governing equation(16.39) and the bottom condition (16.40). Thus, mathematically speaking, the linearmth-order deformation equation has an infinite number of eigenfunctionsΨm,n andthe corresponding eigenvalue

λm,n = g|mk1+nk2|− (mσ1+nσ2)2 (16.49)

with the propertyL (Ψm,n) = λm,n Ψm,n.

Therefore, its inverse operatorL −1 is defined by

L−1(Ψm,n) =

Ψm,n

λm,n, λm,n 6= 0. (16.50)

Note that the inverse operatorL −1 has definitiononly for non-zero eigenvalueλm,n 6= 0. Whenλm,n = 0, we have

g|mk1+nk2|= (mσ1+nσ2)2, (16.51)

which is exactly the criterion of the so-called “wave resonance” given by Phillipsand Longuet-Higgins. LetNλ denote the number of eigenfunctions whose eigenval-ues are zero. As shown below, the value ofNλ is a key of this problem.

Whenn= 0 in (16.51), it holds

λm,0 = g k1(

|m|−m2) ,

which equals to zero only when|m|= 1 (note thatm= n= 0 corresponds toφ = 0and thus is not considered here). Similarly, whenm= 0, the eigenvalueλ0,n equals tozero only when|n|= 1. So, there exist at least two eigenfunctionsΨ1,0 = ek1zsinξ1

andΨ0,1 = ek2zsinξ2 whose eigenvalues are zero, i.e.

L

(

C1 ek1zsinξ1+C2 ek2zsinξ2

)

= 0 (16.52)

for any constantsC1 andC2. Therefore, it holdsNλ ≥ 2 in case of two primarywaves. As mentioned by Phillips [25] and Longuet-Higgins [22], the criterion(16.51) of wave resonance can be satisfied for some special wave numbers and an-gular frequencies. Thus, when the criterion (16.51) is satisfied in case ofm= m′ andn= n′, wherem′ andn′ are integers withm′2+n′2 6= 1, there exist three eigenfunc-tionsΨ1,0 = ek1zsinξ1, Ψ0,1 = ek2zsinξ2 and


Ψm′,n′ = e|m′k1+n′k2|zsin(m′ξ1+n′ξ2)

whose eigenvalues are zero, i.e.

L

[

C1 ek1zsinξ1+C2 ek2zsinξ2+C3 e|m′k1+n′k2|zsin(m′ξ1+n′ξ2)

]

= 0 (16.53)

for any constantsC1,C2 andC3. Without loss of generality, Longuet-Higgins [22]discussed a special casem′ = 2 andn′ =−1, corresponding to the eigenfunction

Ψ2,−1 = exp(|2k1− k2|z)sin(2ξ1− ξ2).

In this section, it impliesm′ = 2 andn′ =−1 when there exist three zero eigenvalues(Nλ = 3) in case of two primary waves, if not mentioned.

According to the definitions (16.27) and (16.44), it holds

L φ = (L φ)|z=0 .

Thus,L (Ψm,n) = λm,n Ψm,n|z=0 = λm,nsin(mξ1+nξ2),

which gives the definition of the linear inverse operator

L−1 [sin(mξ1+nξ2)] =

Ψm,n

λm,n, λm,n 6= 0. (16.54)

Using this inverse operator, it is easy to solve the linear Laplace equation (16.39)with two linear boundary conditions (16.40) and (16.41), asillustrated below. Here,we emphasize that the above inverse operator has definition only for non-zero eigen-valueλm,n 6= 0. For more details, please refer to Liao [20].

16.2.2 Non-resonant waves

First, let us consider the case that there exist only two eigenfunctions

Ψ1,0 = exp(k1z)sinξ1, Ψ0,1 = exp(k2z)sinξ2

whose eigenvalues are zero, i.e.λ1,0 = λ0,1 = 0. Using these two eigenfunctions andaccording to the linear wave theory, we construct the initial approximation of thepotential function

φ0(ξ1,ξ2,z) = A0

√

gk1

Ψ1,0+B0

√

gk2

Ψ0,1, (16.55)

whereA0 andB0 are unknown constants.


The corresponding first-order deformation equation about the potential functionφ1(ξ1,ξ2,z) reads

∇2φ1(ξ1,ξ2,z) = 0, (16.56)

subject to the boundary condition onz= 0:

L (φ1) = b1,01 sin(ξ1)+b0,1

1 sin(ξ2)+b1,11 sin(ξ1+ ξ2)+d1,1

1 sin(ξ1− ξ2)

+ b2,11 sin(2ξ1+ ξ2)+d2,1

1 sin(2ξ1− ξ2)

+ b1,21 sin(ξ1+2ξ2)+d1,2

1 sin(ξ1−2ξ2), (16.57)

and the boundary condition on the bottom:

∂φ1

∂z

∣

∣

∣

∣

z=0= 0, (16.58)

whereL is defined by (16.44), andbi, j1 ,bi, j

1 are constants. Especially, we have

b1,01 = c0A0

√

gk1

[

gk1−σ21 +gk1(A0k1)

2+2gk1(B0k2)2+

gB20k2

1k2

2sin2(α1−α2)

]

,

b0,11 = c0B0

√

gk2

[

gk2−σ22 +gk2(B0k2)

2+2gk2(A0k1)2+

gA20k2

2k1

2sin2(α1−α2)

]

.

Sinceλ1,0 = λ0,1 = 0, according to (16.54), it must hold

b1,01 = b0,1

1 = 0

so as to avoid the so-called “secular” termsξ1sinξ1 andξ2sinξ2. This provides usthe following algebraic equations

(A0k1)2+

[

2+k1

2k2sin2(α1−α2)

]

(B0k2)2 =

σ21

gk1−1, (16.59)

[

2+k2

2k1sin2(α1−α2)

]

(A0k1)2+(B0k2)

2 =σ2

2

gk2−1, (16.60)

whose solutions are

A0 = ±(

εk1

)

√

(

σ22

gk2−1

)[

2+k1

2k2sin2(α1−α2)

]

−(

σ21

gk1−1

)

, (16.61)

B0 = ±(

εk2

)

√

(

σ21

gk1−1

)[

2+k2

2k1sin2(α1−α2)

]

−(

σ22

gk2−1

)

, (16.62)

where


ε =

([

2+k1

2k2sin2(α1−α2)

][

2+k2

2k1sin2(α1−α2)

]

−1

)−1/2

.

Note thatA0 andB0 have multiple values: they can be either positive or negative.Then, by means of the linear inverse operatorL −1 defined by (16.54), the com-

mon solution ofφ1(ξ1,ξ2,z) reads

φ1 = A1

√

gk1

Ψ1,0+B1

√

gk2

Ψ0,1+b1,11

(

Ψ1,1

λ1,1

)

+d1,11

(

Ψ1,−1

λ1,−1

)

+ b2,11

(

Ψ2,1

λ2,1

)

+d2,11

(

Ψ2,−1

λ2,−1

)

+b1,21

(

Ψ1,2

λ1,2

)

+d1,21

(

Ψ1,−2

λ1,−2

)

, (16.63)

whereA1 andB1 are unknown coefficients, the eigenfunctionΨm,n and eigenvalueλm,n are defined by (16.23) and (16.49), respectively. In other words,φ1 is a sum( or linear combination) of eigenfunctions. Note thatφ1 automatically satisfies theLaplace equation (16.39) and the bottom condition (16.40) for anyconstantsA1 andB1. On the other hand, given the initial guessφ0, it is straightforward to calculateη1(ξ1,ξ2) directly by means of the formula (16.42).

The above approach has general meaning. In a similar way, we can obtainηm(ξ1,ξ2) andφm(ξ1,ξ2,z), successively, in the order ofm= 1,2,3, and so on. Notethat, the two unknown coefficientsAm andBm (m≥ 1) can be determined exactly inthe same way likeA0 andB0 by means of avoiding the “secular” termsξ1sinξ1 andξ2sinξ2. Note that only fundamental operations are needed in the above approachso that it is convenient to gain high-order approximations by means of computeralgebraic system such as Mathematica.

Without loss of generality, let us consider here such a special case of the twoprimary waves that

σ1√gk1

=σ2√gk2

= 1.0003, α1 = 0, α2 =π36

, k2 =π5, (16.64)

with different ratios ofk1/k2. Here, the number 1.0003 is chosen so that the per-turbation theory is valid with high accuracy and thus we can compare our resultswith those given by Phillips [25] and Longuet-Higgins [22],who suggested that thewave resonance (with amplitude growing in time) occurs if the resonance criterion(16.51) is satisfied, i.e.

k2

k1≈ 0.8925,

σ1

σ2=

√

k1

k2≈ 1.0585

in the current case. As mentioned before, there are three eigenfunctions (Nλ = 3)whose eigenvalues are zero when the above criterion is satisfied. To avoid this, weconsider here the non-resonant waves with different wave numberk1 exceptk2/k1 =0.8925.

Let us first consider the casek2/k1 = 1. Without loss of generality, we first usethe negative values ofA0 andB0 given by (16.61) and (16.62), respectively. It is


Fig. 16.1 Squared residualEφ

m versusc0 in case of (16.64)with k2/k1 =1. Solid line: 1st-order approximation; Dashedline: 3rd-order approximation;Dash-dotted line: 5th-orderapproximation.

c0

Squ

ared

resi

dual

-2 -1.5 -1 -0.5 010-19

10-17

10-15

10-13

10-11

10-9

10-7

Table 16.1 Squared residuals of the two nonlinear boundary conditionsin case of (16.64) whenk2/k1 = 1

Order of approximationm Eφm Eη

m

1 1.9×10−8 5.1×10−4

3 3.5×10−12 1.2×10−9

5 2.0×10−16 4.3×10−14

8 2.8×10−22 3.8×10−20

10 4.7×10−26 6.4×10−24

Table 16.2 Analytic approximations of some components of wave-amplitude in case of (16.64)whenk2/k1 = 1

Order of approximationm a1,0 = a0,1 a2,−1

1 -0.022502 02 -0.022846 0.000597393 -0.022814 0.000571984 -0.022816 0.000572086 -0.022816 0.000572268 -0.022816 0.0005722610 -0.022816 0.00057226

found that, as the order of approximationm increases, the squared residualEφm de-

creases in the interval−1.8 ≤ c0 < 0, and the optimal value ofc0 is close to -1,as shown in Fig. 16.1. Indeed, whenc0 = −1, the corresponding squared residualsof the two boundary conditions decrease rather quickly to the level 10−24 at the10th-order approximation, as listed in Table 16.1. According to the ConvergenceTheorem mentioned above, the corresponding homotopy-series (16.37) and (16.38)must be the solutions.

Let a1,0,a0,1 anda2,−1 denote the amplitudes of wave components cosξ1, cosξ2

and cos(2ξ1−ξ2), respectively. Obviously,a1,0 = a0,1 in case ofk1 = k2. As shown


Table 16.3 Analytical approximations of some components of wave-amplitude given by the[m,m]homotopy-Pade method in case of (16.64) whenk2/k1 = 1

m a1,0 = a0,1 a2,−1

2 -0.022816 0.000572113 -0.022816 0.000572264 -0.022816 0.000572265 -0.022816 0.00057226

Table 16.4 Components of wave-amplitude in case of (16.64) with different values ofk2/k1

k2/k1 a1,0 a0,1 a2,−1

1.00 -0.0228 -0.0228 0.000570.95 -0.0229 -0.0218 0.000840.93 -0.0230 -0.0215 0.001160.92 -0.0231 -0.0214 0.001500.91 -0.0231 -0.0215 0.002190.905 -0.0232 -0.0216 0.002900.90 -0.023 -0.022 0.004330.8925 -0.0205 -0.0232 0.008980.88 -0.0235 -0.0179 0.003070.86 -0.0221 -0.0189 0.001200.85 -0.0226 -0.0187 0.000870.83 -0.0229 -0.0182 0.000540.80 -0.0232 -0.0172 0.000330.70 -0.0239 -0.0121 0.00010

in Table 16.2, each wave component converges rather quickly, which agree well withthose obtained by the homotopy-Pade method [13, 21], givenin Table 16.3. All ofthese indicate the validity of the HAM-based analytic approach mentioned above.

Similarly, we can get convergent series solutions in case of(16.64) with differentratio of k2/k1, as shown in Table 16.4. Note that, as the ratio ofk2/k1 decreases,the wave amplitude of the component cos(2ξ1−ξ2) first increases monotonically tothe maximum value atk2/k1 = 0.8925 (the corresponding result is given by the ap-proach described below), and then decreases monotonically. Note that the variationof a1,0 anda0,1 is not monotonic.

Phillips [25] pointed out that, when the resonance criterion (16.51) is satisfied,wave resonance occurs so that the amplitude of wave component cos(2ξ1 − ξ2)grows in time, say, it contains more and more wave energy. However, as shownin Table 16.4,|a1,0| and|a0,1| are muchlarger than|a2,1| for all values ofk2/k1,even if the resonance criterion (16.51) is satisfied. Physically, it means that the res-onant wave isunnecessaryto have most of wave energy! We will discuss this veryinteresting phenomena later in details.


16.2.3 Resonant waves

Now, let us further consider the case of (16.64) with a special ratio k2/k1 = 0.8925,i.e. the resonance criterion (16.51) is satisfied. It was suggested first by Phillips[25] and then confirmed by Longuet-Higgins [22] that the so-called wave resonanceoccurs in this case so that the amplitude of wave component cos(2ξ1−ξ2) grows intime, if a2,−1 is initially zero.

In this case, the criterion (16.51) is satisfied, i.e.

g|2k1− k2|= (2σ1− σ2)2.

According to (16.64), we have

k1 = 0.703998, k2 = 0.628319, k3 = |2k1− k2|= 0.783981.

Thus, according to (16.49), we have an additional zero eigenvalueλ2,−1 = 0 withthe corresponding eigenfunction

Ψ2,−1 = e|2k1−k2|zsin(2ξ1− ξ2).

So, we have nowthreeeigenfunctionsΨ1,0,Ψ0,1 andΨ2,−1 whose eigenvalues arezero, i.e.λ1,0 = 0, λ0,1 = 0 andλ2,−1 = 0. Here, it should be emphasized that thewave resonance criterion given by Phillips [25] is mathematically equivalent to

λm,n = 0, m2+n2 > 1.

Note that the additional zero eigenvalueλ2,−1 breaks down the approach mentionedin §16.2.2, because the termΨ2,−1/λ2,−1 in (16.63) becomes infinite: this is exactlythe reason why Phillips [25] and Longuet-Higgins [22] suggested the existence ofthe so-called wave-resonance with amplitude growing in time, which, however, isphysically possible only for small time, since the wave energy can not be infiniteand besides the wave breaking occurs for large enough amplitude wave.

Note that, in case of non-resonant waves investigated in§16.2.2, the initial guess(16.55) is a linear combination of thetwoeigenfunctionsΨ1,0 andΨ0,1 whose eigen-valuesλ1,0 andλ0,1 are zero. In the current case of wave resonance, the only dif-ference is that we have an additional eigenfunctionΨ2,−1 whose eigenvalueλ2,−1 iszero, too. Note that the HAM provides us with great freedom tochoose the initialguess. With such kind of freedom, why not use these three eigenfunctions (with zeroeigenvalue) to express the initial guessφ0? In other words, we can express the initialguessφ0 by all eigenfunctions whose eigenvalues are zero, i.e.

φ0(ξ1,ξ2,z) = A0

√

gk1

Ψ1,0+ B0

√

gk2

Ψ0,1+C0

√

gk3

Ψ2,−1, (16.65)

whereA0, B0,C0 areunknownconstants. Similarly, substituting the above expres-sion into the deformation equations (16.39) to (16.41), we have the same first-order


deformation equation (16.56) with the same boundary condition (16.58) at bottom,but a more complicated equation corresponding to the boundary condition onz= 0,i.e.

L (φ1) = b1,01 sin(ξ1)+ b0,1

1 sin(ξ2)+ b2,01 sin(2ξ1)+ b3,0

1 sin(3ξ1)

+ b1,11 sin(ξ1+ ξ2)+ d1,1

1 sin(ξ1− ξ2)

+ b2,11 sin(2ξ1+ ξ2)+ d2,1

1 sin(2ξ1− ξ2)

+ b1,21 sin(ξ1+2ξ2)+ d1,2

1 sin(ξ1−2ξ2)

+ d2,21 sin(2ξ1−2ξ2)+ d2,3

1 sin(2ξ1−3ξ2)

+ d3,11 sin(3ξ1− ξ2)+ d3,2

1 sin(3ξ1−2ξ2)

+ d4,11 sin(4ξ1− ξ2)+ d4,2

1 sin(4ξ1−2ξ2)+ d4,31 sin(4ξ1−3ξ2)

+ d5,21 sin(5ξ1−2ξ2)+ d6,3

1 sin(6ξ1−3ξ2), (16.66)

wherebm,n1 , dm,n

1 are constant coefficients, and the linear operatorL is defined by(16.44). Note that there exist nowthree zero eigenvalues, i.e.λ1,0 = 0, λ0,1 = 0andλ2,−1 = 0. Therefore, according to the definition (16.54) of the inverse operatorL −1, not onlythe two coefficientsb1,0

1 andb0,11 but alsothe additional coefficient

d2,11 must be zero, so that the secular terms can be avoided. Enforcing

b1,01 = 0, b0,1

1 = 0, d2,11 = 0

gives a set of nonlinear algebraic equations

12.7576A20+20.3675B2

0+31.6768C20+25.6718B0C0 = 0.01545,

24.1456A20+9.6004B2

0+30.0398C20+14.9558A2

0C0/B0 = 0.01459,26.9621A2

0+21.6116B20+16.6956C2

0+10.7158A20B0/C0 = 0.01630,

(16.67)

for the special case mentioned above. The set of these nonlinear algebraic equationshas four complex and twelve real solutions. Because the complex solutions have nophysical meanings, we list only its twelve real roots in Table 16.5. It is found that thetwelve roots fall into three groups, and different groups give different solutions, asshown later. After solving this set of nonlinear algebraic equations, the initial guessφ0(ξ1,ξ2,z) is known and therefore it is straightforward to getη1(ξ1,ξ2) directlyby means of (16.42). More importantly, on the right-hand side of (16.66), the termssinξ1, sinξ2 and especially sin(2ξ1 − ξ2) disappear now. Then, using the inverseoperator (16.54), it is straightforward to get the common solution of the first-orderapproximation

φ1 = A1

√

gk1

Ψ1,0+ B1

√

gk2

Ψ0,1+C1

√

gk3

Ψ2,−1

+ b2,01

(

Ψ2,0

λ2,0

)

+ b3,01

(

Ψ3,0

λ3,0

)

+ b1,11

(

Ψ1,1

λ1,1

)

+ d1,11

(

Ψ1,−1

λ1,−1

)


Table 16.5 Roots of (16.67)

Series number of roots,K A0 B0 C0

(Group-I)1 -0.0156112 0.0282054 -0.00849732 -0.0156112 -0.0282054 0.00849733 0.0156112 0.0282054 -0.00849734 0.0156112 -0.0282054 0.0084973(Group-II)5 -0.0155774 -0.0141927 -0.01138006 -0.0155774 0.0141927 0.01138007 0.0155774 -0.0141927 -0.01138008 0.0155774 0.0141927 0.0113800(Group-III)9 -0.0155626 0.0106109 -0.022635310 -0.0155626 -0.0106109 0.022635311 0.0155626 0.0106109 -0.022635312 0.0155626 -0.0106109 0.0226353

+ b2,11

(

Ψ2,1

λ2,1

)

+ b1,21

(

Ψ1,2

λ1,2

)

+ d1,21

(

Ψ1,−2

λ1,−2

)

+ d2,21

(

Ψ2,−2

λ2,−2

)

+ d2,31

(

Ψ2,−3

λ2,−3

)

+ d3,11

(

Ψ3,−1

λ3,−1

)

+ d3,21

(

Ψ3,−2

λ3,−2

)

+ d4,11

(

Ψ4,−1

λ4,−1

)

+ d4,21

(

Ψ4,−2

λ4,−2

)

+ d4,31

(

Ψ4,−3

λ4,−3

)

+ d5,21

(

Ψ5,−2

λ5,−2

)

+ d6,31

(

Ψ6,−3

λ6,−3

)

. (16.68)

It should be emphasized that all eigenvaluesλm,n in the above expression arenonzero so thatφ1(ξ1,ξ2,z) is finite. Besides, all coefficients in the above expressionare independent of the time so that the corresponding wave profile doesnot growin time. Like the initial guessφ0(ξ1,ξ2,z) defined by (16.65), the common solutionφ1(ξ1,ξ2,z) given by (16.68) has three unknown coefficientsA1, B1 andC1, whichcan be determined similarly by avoiding the “secular” termsin φ2(ξ1,ξ2,z). So,the above approach has general meanings. Therefore, in a similar way, we can getηm(ξ1,ξ2) andφm(ξ1,ξ2,z) successively, in the order ofm= 1,2,3 and so on.

Note that the wave amplitude componentsa1,0 anda0,1 in Table 16.4 are negative.To calculate the corresponding wave amplitude components in case of (16.64) withk2/k1 = 0.8925, we choose the 2nd root in Group-I, corresponding toK = 2, i.e.

A0 =−0.0156112, B0 =−0.0282054, C0 = 0.00849726.

Similarly, the corresponding squared residualEφm versusc0 is given in Fig. 16.2,

which indicates that the optimal value ofc0 is close to -1. Indeed, by means ofc0 = −1, the squared residualsEφ

m and Eηm of the two boundary conditions de-

crease rapidly to the level 10−19 at the 20th-order homotopy-approximation, asshown in Table 16.6. According to the Convergence Theorem mentioned above,these homotopy-series (16.37) and (16.38) satisfy the original governing equation


Fig. 16.2 Squared residualEφ

m versusc0 in case of (16.64)with k2/k1 = 0.8925. Solidline: 1st-order approxima-tion; Dashed line: 3rd-orderapproximation; Dash-dottedline: 5th-order approximation.

c0

Squ

ared

resi

dual

-2 -1.5 -1 -0.5 010-12

10-11

10-10

10-9

10-8

10-7

10-6

Table 16.6 Squared residual of the two boundary conditions in case of (16.64) whenk2/k1 =0.8925

Order of approximationm Eφm Eη

m

1 4.0×10−7 4.8×10−4

3 1.8×10−8 1.0×10−6

5 2.4×10−10 1.2×10−8

8 1.2×10−13 1.5×10−11

10 1.3×10−14 1.2×10−12

15 1.2×10−17 3.4×10−16

20 1.2×10−20 1.9×10−19

(16.14) and all boundary conditions (16.17), (16.18) and (16.20). Besides, it is foundthat the corresponding wave amplitude componentsa1,0,a0,1 anda2,−1 converge to -0.02051, -0.023212, 0.0089752, respectively, as shown in Table 16.7. To confirm theconvergence, we further employ the homotopy-Pade technique [13,21] to acceleratethe convergence and obtain the same convergent wave amplitude components

a1,0 =−0.0205119, a0,1 =−0.0232118, a2,−1 = 0.0089752,

as shown in Table 16.8. Therefore, we indeed get convergent series solution ofresonant waves withconstantamplitudes even when the wave-resonance criterion(16.51) is exactly satisfied.

Combining the above result with those listed in Table 16.4, we obtain the wholepattern of the dimensionless wave amplitude componentk3(a2,−1) versusk2/k1 incase of (16.64), as shown in Fig. 16.3. It is true that, as the ratio k2/k1 goes to0.8925, corresponding to the criterion (16.51) of wave resonance, the dimensionlesswave amplitude componentk3 a2,−1 increases to its maximum. Note that the am-plitude |a2,−1| of the wave component cos(2ξ1− ξ2) is afinite constant, even if theresonance criterion (16.51) is satisfied exactly. Besides,it is very interesting that,in case ofk2/k1 = 0.8925, the amplitude|a2,−1| of the resonant wave component


Table 16.7 Homotopy-approximations of wave-amplitude components incase of (16.64) whenk2/k1 = 0.8925

Order of approximationm a1,0 a0,1 a2,−1

1 -0.015616 -0.028214 0.00849993 -0.020587 -0.023469 0.00941435 -0.020533 -0.023231 0.00900837 -0.020504 -0.023220 0.00897559 -0.020511 -0.023213 0.008975711 -0.020512 -0.023212 0.008975313 -0.020512 -0.023212 0.008975215 -0.020512 -0.023212 0.008975218 -0.020512 -0.023212 0.0089752

Table 16.8 Homotopy-approximations of wave-amplitude components given by the [m,m]homotopy-Pade method in case of (16.64) withk2/k1 = 0.8925

m a1,0 a0,1 a2,−1

2 -0.0206010 -0.0232058 0.00896603 -0.0204911 -0.0232251 0.00895684 -0.0205102 -0.0232172 0.00897805 -0.0205122 -0.0232118 0.00897526 -0.0205119 -0.0232118 0.00897527 -0.0205119 -0.0232118 0.00897528 -0.0205119 -0.0232118 0.00897529 -0.0205119 -0.0232118 0.0089752

Fig. 16.3 The dimensionlesswave amplitude componentk3(a2,−1) versusk2/k1 incase of (16.64). Filled circle:result whenk2/k1 = 0.8925;Open circles: results whenk2/k1 6= 0.8925.

k2/k1

a 2,-

1k 3

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

10-4

10-3

10-2

α1 = 0 degree

α2 = 5 degree

is even muchsmallerthan the wave amplitudes|a1,0| and|a0,1| of the two primarywaves!

The above results are obtained by means of the 2nd root of Group I in Table 16.5,corresponding toK = 2. Similarly, using different roots in Table 16.5, we can searchfor the corresponding convergent series solutions. It is found that the four different


Table 16.9 Multiple amplitudes of resonant waves in case of (16.64) with k2/k1 = 0.8925 givenby different roots of (16.67) listed in Table 16.5.

|a1,0| |a0,1| |a2,−1| |a1,−2|

Group I 0.02051186921 0.02321179687 0.00897520547 0.00022907754Group II 0.01475607438 0.01002488089 0.01464333477 0.00096077706Group III 0.00971236473 0.01032248128 0.02576462018 0.00059968416

roots of each group in Table 16.5 give the same absolute values |a1,0|, |a0,1| and|a2,−1| of wave components, as listed in Table 16.9. Considering thefact that thewave energy spectrum is determined by amplitude square of wave components, weregard the four different solutions in each group as the same. Thus, in case of (16.64)with k2/k1 = 0.8925, there arethreedifferent resonant-wave patterns withdifferentwave energy spectrums. Here, it should be emphasized that the amplitude|a2,−1| ofthe resonant wave is thesmallestin Group I, and is the middle in Group II, althoughit is the largest in Group III. So, the amplitude|a2,−1| of the resonant wave isnotspecial at all: it is just normal as the wave amplitude components|a1,0| and|a0,1| ofthe two primary waves! It is very interesting that, for a fully developed wave system,there existmultiplesteady-state solutions even if Phillips’ resonance criterion is ex-actly satisfied, and besides the resonant wave amplitude maybe muchsmallerthanprimary wave amplitudes. To the best of our knowledge, theseinteresting results hasneverbeen reported by other numerical and analytic methods. Thisillustrates thatthe HAM can be used as a useful tool to solve complicated nonlinear PDEs so asdeepen our physical understanding about some nonlinear phenomena.

Let Π denote the sum of amplitude square of all wave components. Write

Π0 = a21,0+a2

0,1+a22,−1.

In case of (16.64) withk2/k1 = 0.8925, corresponding to the occurrence of waveresonance, it is found thatΠ0/Π equals to 98.82%,98.27% and 99.76% for Group I,II and III, respectively. This is mainly because amplitudesof other wave componentssuch asa1,−2 are much smaller, as shown in Table 16.9. Thus, these three wavecomponents nearly contain the whole wave energy. Note that,given two primarytraveling waves with wave numbersk1 andk2, there exist an infinite number ofdifferent wave componentsam,ncos(mξ2+nξ2) with the wave numbermk1+nk2,wheremandn are arbitrary integers. Let

K = mk1+nk2|m,n are integers

denote a set of all these wave numbers. Each wave number inK corresponds to aneigenfunction defined by (16.23) with an eigenvalue defined by (16.49). Our com-putations suggest that, for a fully developed wave system composed of two travel-ing primary waves withsmallamplitudes, the main of wave energy focuses on thewave components whose eigenvalues are zero (or close to zero). This provides us analternative explanation for the so-called “wave resonance”. According to this expla-


Table 16.10 Wave energy distribution of the resonant wave ( Group I ) in case ofα1 = 0,α2 =π/36,k2/k1 = 0.8925 with different ratios ofσ1/

√gk1 = σ2/

√gk2.

σ1/√

gk1 = σ2/√

gk2 a21,0/Π a2

0,1/Π a22,−1/Π Π0/Π

1.0001 36.97% 53.95% 8.99% 99.91%1.0002 38.40% 52.74% 8.42% 99.56%1.0003 39.97% 51.19% 7.65% 98.82%1.0004 41.59% 49.28% 6.70% 97.57%1.0005 43.10% 47.09% 5.60% 95.78%1.0006 44.38% 44.75% 4.43% 93.56%1.0007 45.65% 42.73% 3.58% 91.95%1.0008 46.23% 39.07% 2.22% 87.52%

Table 16.11 Wave energy distribution of the resonant wave ( Group II ) in case ofα1 = 0,α2 =π/36,k2/k1 = 0.8925 with different ratios ofσ1/

√gk1 = σ2/

√gk2.

σ1/√

gk1 = σ2/√

gk2 a21,0/Π a2

0,1/Π a22,−1/Π Π0/Π

1.0001 42.30% 20.43% 37.07% 99.79%1.0002 41.26% 19.50% 38.44% 99.20%1.0003 40.17% 18.54% 39.56% 98.27%1.0004 39.05% 17.56% 40.44% 97.06%1.0005 37.93% 16.56% 41.12% 95.61%1.0006 36.79% 15.54% 41.64% 93.97%1.0007 35.65% 14.51% 42.02% 92.18%1.0008 34.49% 13.46% 42.29% 90.24%

Table 16.12 Wave energy distribution of the resonant wave ( Group III ) incase ofα1 = 0,α2 =π/36,k2/k1 = 0.8925 with different ratios ofσ1/

√gk1 = σ2/

√gk2.

σ1/√

gk1 = σ2/√

gk2 a21,0/Π a2

0,1/Π a22,−1/Π Π0/Π

1.0001 12.13% 12.47% 75.37% 99.97%1.0002 11.53% 12.38% 75.99% 99.90%1.0003 10.88% 12.29% 76.58% 99.76%1.0004 10.20% 12.20% 77.15% 99.55%1.0005 9.47% 12.11% 77.68% 99.26%1.0006 8.72% 12.01% 78.15% 98.88%1.0007 7.94% 11.90% 78.53% 98.38%1.0008 7.15% 11.78% 78.82% 97.75%

nation, the resonant wavea2,−1cos(2ξ1−ξ2) is just as important as the two primarywavesa1,0cosξ1 anda0,1cosξ2, and therefore isnot very special. This viewpointcan well explain why the amplitude of resonant wave is often smaller than those ofnon-resonant waves.

The wave energy distribution of the fully-developed wave system composed ofthe two traveling primary waves (corresponding to Group I) in case ofα1 = 0,α2 = π/36,k2/k1 = 0.8925 with different ratios ofσ1/

√gk1 = σ2/

√gk2 is given

in Table 16.10. It is found that the two primary waves containmost of the totalwave energy. Besides, as the ratio ofσ1/

√gk1 = σ2/

√gk2 increases, the resonant


Fig. 16.4 a22,−1/Π versus

σ1/√

gk1 in case of (16.64)when k2/k1 = 0.8925 andσ1/

√gk1 = σ2/

√gk2 for

different group of solutions.Solid line: Group I; Dashedline: Group II; Dash-dottedline: Group III.

σ1 / ( g k1 )1/2

a2 2,-

1/Π

1 1.0002 1.0004 1.0006 1.00080

0.2

0.4

0.6

0.8

Group III

Group II

Group I

wave componenta2,−1cos(2ξ1− ξ2) contains less and less percentage of the totalwave energy. Especially, whenσ1/

√gk1 = σ2/

√gk2 = 1.0008, the resonant wave

of Group I contains only 2.22% of the total wave energy. This result is very inter-esting but surprising, because, according to traditional theory, the resonance waveis supposed to have a large wave amplitude and thus to containthe main of thetotal wave energy. Similarly, the wave energy distributionof the fully-developedwave system (corresponding to Group II) is given in Table 16.11. It is found thatthe resonant wave of Group II has the comparable wave amplitudes with the com-parable percentage of the total wave energy to one of primarywaves. Among threegroups, there exists only one group (i.e. Group III) such that the resonant wave hasthe largest wave amplitude and thus contain the main part of the total wave energy,as shown in Fig. 16.4 and Table 16.12.

Note that, the primary and resonant waves as a whole contain most of the totalwave energy, especially when all wave amplitude componentsare small, as shown inFig. 16.5. However, as wave amplitudes increase, they contain less and less percent-age of the total wave energy, as shown in Fig. 16.5. This also suggests that Phillips’wave resonance criterion (16.6) is valid only for small-amplitude traveling waves.

Phillips [25] considered two small-amplitude primary waves a1,0 cosξ1 anda0,1 cosξ2 att = 0, and investigated the nonlinear interaction between them. Phillips[25] revealed that, when the wave componentam,ncos(mξ1+nξ2) is initially zero,i.e. am,n = 0, the wave-amplitude componentam,n increases linearly in time, if theresonance criterion (16.51) is satisfied. So, Phillips’ resonance criterion is valid forthe evaluation of wave system in case of small time, say,t ≪ 1. Since it is impossi-ble in physics for the wave amplitude to increase to infinity,it is unclear whether ornot the resonant wave componentam,n cos(mξ1+nξ2) contains the main of the totalwave energy ast →+∞. Different from Phillips [25], we consider here the “steady-state” nonlinear interaction of two small-amplitude primary waves, correspondingto the case oft →+∞ for the evaluation of waves, say, a fully-developed wave sys-tem. Our computations suggest that, if all wave components are fully developed sothat they are in equilibrium each other, and if Phillips’ resonance criterion (16.51)


Fig. 16.5 Π0/Π versusσ1/

√gk1 in case of (16.64)

when k2/k1 = 0.8925 andσ1/

√gk1 = σ2/

√gk2 for

different group of solutions.Solid line: Group I; Dashedline: Group II; Dash-dottedline: Group III.

σ1 / ( g k1 )1/2

Π0/Π

1 1.0002 1.0004 1.0006 1.00080.85

0.9

0.95

1

Group III

Group II

Group I

is satisfied, there existmultipleresonant wave componentam,n cos(mξ1+nξ2), andbesides the amplitudeam,n of the resonant wave component may be much smallerthan the primary onesa1,0 anda0,1, i.e. the resonant wave component isnot neces-sary to have most of the total wave energy. Therefore, Phillips’ resonance criterionis a not sufficient condition for a large resonant wave: even if Phillips’ resonancecriterion is satisfied, the amplitude of the resonant wave may be much smaller thanthe two primary ones.

The above results have general meaning, although they are obtained in a specialcase (16.64). They strongly suggest that, for a fully developed system composedof two traveling primary waves, there exist steady-state multiple “resonant waves”even if Phillips’ wave resonance criterion is exactly satisfied, besides the amplitudeof the resonant wave component may be much smaller than thoseof the primarywaves. These conclusions are still true forarbitrary number of traveling primarywaves, as shown below. For more details, please refer to Liao[20].

Currently, by means of DNS (direct numerical simulation) ofthe evolution ofnonlinear random water wave fields with a continuous spectrum, Annenkov et al. [2]investigated the role of exactly resonant, nearly resonantand non-resonant waveinteractions, and their results indicate that the amplitudes of wave packets tend toconstants. Their results, although obtained for a continuous wave spectrum, supportsome of our conclusions mentioned above.

16.3 Resonance criterion of arbitrary number of primary waves

In the following part of this chapter, the wave-resonance criterion in§ 16.2 is gener-alized in such a way that it is valid forarbitrary number of primary traveling waves.

16.3 Resonance criterion of arbitrary number of primary waves 553

16.3.1 Resonance criterion of small-amplitude waves

We verify in § 16.2 by means of the HAM that, for a fully developed wave systemcomposed of two primary traveling waves, Phillips’ resonance criterion (16.51) forsmall-amplitude waves is mathematically equivalent to a zero eigenvalue, say,

λm,n = g(mk1+nk2)− (mσ1+nσ2)2 = 0, (16.69)

wherem,n are integers andσi =√

g |k i | for i = 1,2. This conclusion has generalmeanings, and can be easily expanded in the frame of the HAM togive a resonancecriterion forarbitrary number of travel waves with small amplitude. The key is togive an explicit expression of the eigenvalue in case of arbitrary number of travelingwaves.

Let us consider the nonlinear interaction ofκ periodic traveling waves with smallamplitudes, where 2≤ κ <+∞ is an arbitrary integer. Assume that the wave systemis fully developed, i.e. the amplitude of each wave component is independent of thetime. Letkn andσn (1≤ n≤ κ) denote the wave number and angular frequency ofthenth periodic traveling waves in deep water. Define the variable

ξn = kn · r −σn t.

Then, we have the velocity potential and wave elevation

ϕ(x,y,z, t) = φ(ξ1,ξ2, · · · ,ξκ ,z), ζ (x,y, t) = η(ξ1,ξ2, · · · ,ξκ),

respectively. Similarly, we have

∂ 2ϕ∂ t2 =

κ

∑m=1

κ

∑n=1

σmσn∂ 2φ

∂ξm∂ξn,

∇ϕ =

(

κ

∑m=1

km∂φ∂ξm

)

+ k∂φ∂z

= u = ∇φ ,

∇2ϕ =

(

κ

∑m=1

κ

∑n=1

km ·kn∂ 2φ

∂ξm∂ξn

)

+∂ 2φ∂z2 = ∇2φ ,

∇φ · ∇φ =

(

κ

∑m=1

κ

∑n=1

km ·kn∂φ∂ξm

∂φ∂ξn

)

+

(

∂φ∂z

)2

= u2,

∇φ · ∇ψ =

(

κ

∑m=1

κ

∑n=1

km ·kn∂φ∂ξm

∂ψ∂ξn

)

+∂φ∂z

∂ψ∂z

.

Note that

Ψm1,m2,···,mκ ,z = exp

(∣

∣

∣

∣

∣

κ

∑n=1

mnkn

∣

∣

∣

∣

∣

z

)

sin

(

κ

∑n=1

mnξn

)

(16.70)


satisfies the Laplace equation∇φ = 0, i.e.

∇2Ψm1,m2,···,mκ = 0

and the boundary condition (16.5) at the bottom. The two nonlinear boundary con-ditions on the free surface can be written by means of the operators defined above ina similar way. Certainly, the corresponding PDE and boundary conditions are muchmore complicated than those in§ 16.2. Even so, in a rather similar way as men-tioned above, we can construct the zeroth-order deformation equations and deducethe corresponding high-order deformation equations.

Similarly, we choose such an auxiliary linear operator

L φ =

(

κ

∑m=1

κ

∑n=1

σm σn∂ 2φ

∂ξm∂ξn

)

+g∂ 2φ∂z2 , (16.71)

whereσm =√

g |km| is based on the linear theory for small-amplitude waves. Ac-cording to (16.70) and (16.71), it holds

L (Ψm1,m2,···,mκ ) = λm1,m2,···,mκ Ψm1,m2,···,mκ , (16.72)

where

λm1,m2,···,mκ = g

∣

∣

∣

∣

∣

κ

∑n=1

mnkn

∣

∣

∣

∣

∣

−(

κ

∑n=1

mnσn

)2

(16.73)

is the eigenvalue andΨm1,m2,···,mκ defined by (16.70) is the eigenfunction of the cor-responding linear high-order deformation equation. Similarly, the inverse operatorof (16.71) satisfies

L−1 (Ψm1,m2,···,mκ ) =

Ψm1,m2,···,mκ

λm1,m2,···,mκ, λm1,m2,···,mκ 6= 0. (16.74)

Note that the inverse operator (16.74) is valid only for the non-zero eigenvalueλm1,m2,···,mκ 6= 0. Besides, the eigenvalue of each primary traveling waves is zero,i.e.

λm1,m2,···,mκ = 0, whenκ

∑n=1

m2n = 1.

Therefore, there exist at leastκ zero eigenvalues forκ primary waves. Thus, theso-called wave resonance occurs when there are more thanκ zero eigenvalues, i.e.

λm1,m2,···,mκ = 0, whenκ

∑n=1

m2n > 1.

Then, according to (16.73), we have the generalized wave-resonance criterion


g

∣

∣

∣

∣

∣

κ

∑n=1

mnkn

∣

∣

∣

∣

∣

=

(

κ

∑n=1

mnσn

)2

, whenκ

∑n=1

m2n > 1, (16.75)

whereσn =√

g |kn| is based on the linear theory for small-amplitude waves.Note that (16.51) is a special case of the above generalized wave-resonance cri-

terion. Besides, it contains the resonance criterion givenby Phillips [25]. Thus, it ismore general. Substitutingσn =

√

g |kn| into (16.75) gives the generalized wave-resonance criterion

∣

∣

∣

∣

∣

κ

∑n=1

mnkn

∣

∣

∣

∣

∣

=

(

κ

∑n=1

mn

√

|kn|)2

, whenκ

∑n=1

m2n > 1, (16.76)

for waves with small-amplitude, which is independent of theacceleration of gravity.Assume that, for givenκ primary traveling waves, there areNλ ≥ κ eigenfunc-

tions whose eigenvalues are zero. WhenNλ = κ , there is no wave resonance. How-ever, whenNλ > κ , there exist the so-called resonant wave. For simplicity, letΨ∗

m(1 ≤ m≤ Nλ ) denote themth eigenfunction with zero eigenvalue. According to(16.72), it holds

L

(

Nλ

∑m=1

Am Ψ∗m

)

= 0, Nλ ≥ κ (16.77)

for any constantAm. So, we can always choose such an initial guess that

φ0 =Nλ

∑m=1

B0,m Ψ∗m, (16.78)

whereB0,m is unknown. Similarly, theNλ unknown constantsB0,m (1≤ m≤Nλ ) aredetermined by avoiding the “secular” terms inφ1. Besides, according to (16.77), thecommon solution ofφ1 containsNλ unknown constantsB1,m, which are similarlydetermined by avoiding the “secular” terms inφ2. In this way, one can solve thecorresponding high-order deformation equations successively. Similarly, an optimalconvergence-control parameterc0 can be found to guarantee the convergence of thehomotopy-series. The above approach is general, and works for arbitrary numberof periodic traveling primary waves with small amplitudes.It provides us a wayto investigate the fully-developed wave system composed ofarbitrary number ofprimary traveling waves with small amplitudes.

16.3.2 Resonance criterion of large-amplitude waves

Note that the generalized wave resonance criterion (16.75)holds whenσn =√

g |kn|only, corresponding to small-amplitude gravity waves. What is the resonance crite-rion for arbitrary number of traveling gravity waves withlarge amplitude?


Fig. 16.6 The resonance of asimple pendulum

θ

F

m

l

To answer this question, let us consider the physical meanings of (16.75). Ingeneral, the so-called resonance of a dynamic system occurswhen the frequency ofan external force (or disturbance) equals to the “natural” frequency of the dynamicsystem. For example, let us consider the resonance of a simple pendulum, as shownin Fig. 16.6, whereF = A cos(ωt +α) is the external force with the frequencyω and the phase differenceα. Let ω0 denote the natural frequency of the simplependulum. When the maximum angle of oscillationθmax is small so that sinθ ≈ θ ,ω0 ≈

√

g/l is a very good approximation. So, if the frequencyω of the externalforceF is equal to the natural frequencyω0 of the simple pendulum, i.e.ω =

√

g/l ,the total energy of the pendulum (and thereforeθmax) quickly increases in case ofthe phase differenceα = 0 (or decreases in case ofα = π): the so-called resonanceoccurs. However,ω0 ≈

√

g/l is only valid for smallθmax: the natural frequencyω0 increases asθmax becomes larger. So, asθmax becomes larger and larger so thatthe natural frequencyω0 departs more and more from the frequencyω =

√

g/l ofthe external forceF, then the simple pendulum gains less and less energy from theexternal force: the maximum angle of oscillationθmax stops increasing when thesimple pendulum can not gain energy fromF any more in a period of oscillation.

The phenomenon of gravity wave resonance is physically similar to it in essence.For a single traveling wave with the wave numberk′ and the “natural” angular fre-quencyσ ′

0, the resonance occurs when there exists an “external” periodic distur-bance with the same angular frequencyσ ′, i.e. σ ′ = σ ′

0. As mentioned above, thisresonance mechanism is physically reasonable even for large wave amplitude.

Let us considerκ primary traveling waves with wave numberkn and angularfrequencyσn, where 1≤ n≤ κ . Due to nonlinear interaction, there exist an infinitenumber of wave components

cos

(

κ

∑n=1

mn ξn

)

,

wheremn is an integer that can be negative, zero, or positive. Note that


κ

∑n=1

mn ξn =

(

κ

∑n=1

mnkn

)

· r −(

κ

∑n=1

mn σn

)

t.

So,

k′ =κ

∑n=1

mn kn (16.79)

is the wavenumber and

σ ′ =

∣

∣

∣

∣

∣

κ

∑n=1

mn σn

∣

∣

∣

∣

∣

(16.80)

is the corresponding angular frequency of the nonlinear-interaction wave. For thesake of simplicity, we callk′ the nonlinear-interaction wavenumberand σ ′ the

nonlinear-interaction angular frequency, respectively, whereκ∑

n=1m2

n > 1.

For small-amplitude waves, we haveσn ≈√

g kn = σn, which leads to∣

∣

∣

∣

∣

κ

∑n=1

mn σn

∣

∣

∣

∣

∣

≈∣

∣

∣

∣

∣

κ

∑n=1

mn σn

∣

∣

∣

∣

∣

= σ ′ (16.81)

Substituting the above expression and (16.79) into (16.75), we have the resonancecriterion (forκ small-amplitude primary waves) in the form:

g|k′|= σ ′2, (16.82)

i.e.σ ′ =

√

g|k′|. (16.83)

The above resonance criterion clearly reveals the physicalrelationship between the

nonlinear-interaction wavenumberk′ =κ∑

n=1mn kn and the nonlinear-interaction an-

gular frequencyσ ′ =

∣

∣

∣

∣

κ∑

n=1mn σn

∣

∣

∣

∣

.

Let σ ′0 denote the “natural” angular frequency of a single traveling wave with the

wavenumberk′ and the wave amplitudea′. In case of small wave amplitudes, ac-cording to the linear theory, we have the “natural” angular frequencyσ ′

0 ≈√

g |k′|.Then, the above wave-resonance criterion becomes

σ ′ = σ ′0, (16.84)

i.e.∣

∣

∣

∣

∣

κ

∑n=1

mn σn

∣

∣

∣

∣

∣

= σ ′0,

κ

∑n=1

m2n > 1. (16.85)

Physically speaking, the wave resonance occurs when the nonlinear-interaction an-

gular frequencyσ ′ =

∣

∣

∣

∣

κ∑

n=1mn σn

∣

∣

∣

∣

of the corresponding nonlinear-interaction wave


with wavenumberk′ =κ∑

n=1mn kn equals to its “natural” angular frequencyσ ′

0. Note

that, different from the nonlinear-interaction angular frequencyσ ′ that is a kind ofsum of angular frequencies of primary waves, the “natural” angular frequencyσ ′

0 ofthe corresponding wave numberk′ depends only upon the wavenumberk′ and itsamplitudea′, but has nothing to do with the angular frequencies of primary waves.Thus, in general, the nonlinear-interaction angular frequencyσ ′ is not equal to the“natural” angular frequencyσ ′

0 of the nonlinear-interaction wave with wavenumberk′. So, the wave resonance criterion is indeed rather special.This physical expla-nation agrees well with the traditional resonance theory. Thus, (16.84) and (16.85)reveal the physical essence of the gravity wave resonance.

Although (16.85) is derived from the wave-resonance criterion (16.75) forsmallwave amplitudes, this physical mechanism of gravity wave resonance has generalmeanings and holds for large wave amplitudes even ifσn ≈

√g kn is not a good ap-

proximation. So, (16.85) is the generalized wave-resonance criterion forarbitrarynumber of periodic traveling primary waves withlargeamplitudes. It should be em-phasized that (16.85) logically contains the resonance criterion (16.75) for arbitrarynumber of small-amplitude waves, and besides Phillips’ resonance criterion (16.6)for four small-amplitude waves. Thus, it is rather general.

When the wave resonance criterion (16.85) is satisfied so that the wave energytransfers from the primary waves to a resonant one, the amplitudes of primary wavesdecreases and the amplitude of the resonant wave increases.Then, the angular fre-quencyσn of each primary wave decreases but the “natural” angular frequency ofthe resonant wave increases so that the resonance criterion(16.85) does not holdany more. As a result, the “natural” frequencyσ ′

0 departs more and more from thenonlinear-interaction frequencyσ ′, so that the nonlinear-interaction wave gains lessand less energy from the primary waves, until the whole wave system is in equi-librium, i.e. fully developed. This explains why a resonantgravity wave has finitevalue of amplitude.

As mentioned above, a resonant simple pendulum acted by an external force withthe phase differenceπ , as shown in Fig. 16.6,losesits energy so that the maximumangel of oscillationθmax decreases. Similarly, when the wave resonance criterion(16.85) is satisfied, it is also possible that the wave energytransfers from the res-onant wave to primary ones so that the amplitude of resonant wave decreases andthe amplitudes of primary waves increase: this well explains why the amplitude ofa resonant wave may be much smaller than primary ones, as shown in Table 16.10.

For example, consider two primary waves denoted by the wave numberk1 andk2, and a resonant wave 2k1− k2, whose amplitudes are denoted bya1,0,a0,1 anda2,−1, respectively. Assume that Phillips’ wave resonance criterion is satisfied. Ifa2,−1 is initially zero, thena2,−1 increases linearly in time for small timet ≪ 1.This is the case investigated by Phillips [25] and Longuet-Higgins [22]. However,if a2,−1 is initially not equal to zero, i.e.a2,−1 6= 0 at t = 0, thena2,−1 may eitherincrease, i.e. the wave energy transfers from the primary waves to the resonant one,or decrease, i.e. the wave energy transfers from the resonant wave to the primaryones, depending on the initial condition att = 0. From this viewpoint, it is easy to

16.4 Concluding remark and discussions 559

understand why a resonant wave may have a very smaller amplitude than primarywaves in a fully developed wave system.

For more discussions, please refer to Liao [20].

16.4 Concluding remark and discussions

In this chapter we verify the validity of the HAM for a rather complicated non-linear PDE about the nonlinear interaction of arbitrary number of traveling waterwaves. In the frame of the HAM, the wave-resonance criterionfor arbitrary num-ber of small amplitude waves is gained, for the first time, which logically containsthe famous Phillips’ criterion for four small amplitude waves. Besides, it is foundfor the first time that, when the wave-resonance criterion issatisfied and the wavesystem is fully developed, there existmultiplesteady-state resonant waves, whoseamplitude might be muchsmallerthan primary waves so that a resonant wave maycontain muchsmallpercentage of the total wave energy. Thus, Phillips’ resonancecriterion is not sufficient to guarantee a large resonant wave amplitude. In addition,from the physical viewpoints, the above-mentioned wave-resonance criterion is fur-ther generalized forarbitrary number oflargeamplitude waves, which opens a newway to study the strongly nonlinear interactions of arbitrary number of primary trav-eling gravity waves with large amplitudes. This example illustrates that the HAMcan be used as a tool to deepen and enrich our understandings about some rathercomplicated nonlinear phenomena.

Note that the wave-resonance criterion (16.85) for arbitrary number of travelingwater waves with large amplitudes is given only from the physical view-points ofresonance. Although it explains well why the amplitude of a resonant wave is finiteand why it may be much smaller than primary ones, physical experiments and/or an-alytical/numerical results are needed to support it. Besides, it is interesting to studythe evolution of a system of arbitrary number of traveling water waves far from equi-librium, especially the wave energy transfer between different wave components.

In this chapter, the so-called homotopy multiple-variabletechnique [20] is em-ployed. Different from perturbation techniques used by Phillips [25] and Longuet-Higgins [22], it does not depend upon any small physical parameters, and besidesprovides us a convenient way to guarantee the convergence ofsolution series. Likemultiple-scale perturbation method, its solution has clear physical meanings. Forexample, using the homotopy multiple-variable technique [20], the timet does notexplicitly appear for a fully developed wave system: this not only greatly simpli-fies resolving the problem mathematically, but also contributes a lot to revealing thephysical meanings1 clearly, as shown in this chapter.

The homotopy multiple-variable method [20] is more generalthan the famousmultiple-scale perturbation technique. By means of the multiple-scale perturbation

1 Some researchers solved nonlinear wave-type PDEs by simplyexpanding the solution in Tay-lor series with respect to the timet. Unfortunately, this often leads to very complicated solutionexpressions with rather little physical meanings.


technique, one often rewrites a unknown solutionu(t) in the form

u(t) = u0(T0,T1,T2)+u1(T0,T1,T2) ε +u2(T0,T1,T2) ε2+ · · · ,

whereT0 = t, T1 = ε t, T2 = ε2 t

denote different timescales with the small physical parameter ε, and then transfersa nonlinear problem into a sequence of linear perturbed problems via the smallphysical parameterε. Using the homotopy multiple-variable method [20], we canalso rewriteu(t) by u(ξ0,ξ1,ξ2) with the same definition

ξn = εn t.

However, different from the multiple-scale perturbation techniques, we neednotany small physical parameters. Furthermore, it is easier togain high-order approx-imations, and especially, if the multiple-variables are properly defined with clearphysical meanings, it gives results with important physical meanings, as illustratedin this chapter. Thus, the homotopy multiple-variable method has general meanings.Considering the fact that the multiple-scale perturbationtechnique has been widelyemployed, the homotopy multiple-variable technique may bealso applied to solvemany types of nonlinear problems in science and engineering, such as the famousnatural phenomena about freak wave [1, 5, 6], the nonlinear interaction of arbitrarynumber of traveling water waves far from equilibrium, and soon, although furthermodifications are needed in future.

In Part III of this book, we verify the validity of the HAM by some nonlinearPDEs. In Chapter 13, the HAM is employed to solve a famous problem in finance,i.e. the American input option. Explicit analytic approximations of the optimal ex-ercise boundaryB(τ) are gained in polynomials of

√τ to o(τM), which are often

valid a couple of dozen years prior to expiry, whereas other asymptotic/perturbationformulas only a couple of days or weeks. Such kind of formula valid in so long timehas never been reported and is helpful for businessmen. In Chapter 14, the 2nd-order2D (or 3D) nonlinear Gelfand equation is solved in a rather easy way by means oftransforming it into an infinite number of 4th (or 6th) order linear PDEs. Such kindof transform has never beed used by other numerical and analytic methods. It alsosuggests that we human being might have much larger freedom to solve nonlinearproblems than we traditionally thought and believed. All ofthese show the origi-nality and flexibility of the HAM. In Chapter 15, the HAM is applied successfullyto solve a complicated nonlinear PDE about nonlinear interaction between waterwave and an exponentially shear current. It is found, for thefirst time, that the tra-ditional wave break criterion is still valid even when an exponentially shear currentexists. In the current chapter, the HAM is employed to give, for the first time, thewave-resonance criterion for arbitrary number of traveling water waves. All of theseillustrate that the HAM can be applied to solve some complicated nonlinear PDEs soas to deepen and enrich our physical understandings for someinteresting nonlinearphenomena.


It must be emphasized that the HAM isnotvalid forall nonlinear problems, sinceour aim is to develop a analytic approach valid foras manynonlinear problemsaspossibleonly. It is even an open question if there exists an analytic approximationmethod valid for all nonlinear problems. In essence, it is rather hard to understandcomplicated nonlinear phenomena, especially those related to chaos and turbulence.“The small truth has words which are clear; the great truth hasgreat silence”,as pointed out by Rabindranth Tagore (1861–1941). However,although nonlinearODEs and PDEs are still more difficult to solve than linear ones, the HAM providesus an useful, alternative tool to investigate them.


Appendix 16.1 Detailed derivation of high-order equation

Write(

+∞

∑i=1

ηi qi

)m

=+∞

∑n=m

µm,n qn, (16.86)

with the definitionµ1,n(ξ1,ξ2) = ηn(ξ1,ξ2), n≥ 1. (16.87)

Then,

(

+∞

∑i=1

ηi qi

)m+1

=

(

+∞

∑n=m

µm,n qn

)(

+∞

∑i=1

ηi qi

)

=+∞

∑n=m+1

µm+1,n qn, (16.88)

which gives

µm,n(ξ1,ξ2) =n−1

∑i=m−1

µm−1,i(ξ1,ξ2) ηn−i(ξ1,ξ2), m≥ 2, n≥ m. (16.89)

Define

ψn,mi, j (ξ1,ξ2) =

∂ i+ j

∂ξ i1∂ξ j

2

(

1m!

∂ mφn

∂zm

∣

∣

∣

∣

z=0

)

.

By Taylor series, we have for anyz that

φn(ξ1,ξ2,z) =+∞

∑m=0

(

1m!

∂ mφn

∂zm

∣

∣

∣

∣

z=0

)

zm =+∞

∑m=0

ψn,m0,0 zm (16.90)

and

∂ i+ jφn

∂ξ i1∂ξ j

2

=+∞

∑m=0

∂ i+ j

∂ξ i1∂ξ j

2

(

1m!

∂ mφn

∂zm

∣

∣

∣

∣

z=0

)

zm =+∞

∑m=0

ψn,mi, j zm. (16.91)

Then, onz= η(ξ1,ξ2;q), we have using (16.86) that

∂ i+ jφn

∂ξ i1∂ξ j

2

=+∞

∑m=0

ψn,mi, j

(

+∞

∑s=1

ηs qs

)m

= ψn,0i, j +

+∞

∑m=1

ψn,mi, j

(

+∞

∑s=m

µm,s qs

)

=+∞

∑m=0

β n,mi, j (ξ1,ξ2) qm, (16.92)

where

β n,0i, j = ψn,0

i, j , (16.93)

β n,mi, j =

m

∑s=1

ψn,si, j µs,m, m≥ 1, (16.94)


Similarly, onz= η(ξ1,ξ2;q), it holds

∂ i+ j

∂ξ i1∂ξ j

2

(

∂φn

∂z

)

=+∞

∑m=0

γn,mi, j (ξ1,ξ2) qm, (16.95)

∂ i+ j

∂ξ i1∂ξ j

2

(

∂ 2φn

∂z2

)

=+∞

∑m=0

δ n,mi, j (ξ1,ξ2) qm, (16.96)

where

γn,0i, j = ψn,1

i, j , (16.97)

γn,mi, j =

m

∑s=1

(s+1)ψn,s+1i, j µs,m, m≥ 1, (16.98)

δ n,0i, j = 2ψn,2

i, j , (16.99)

δ n,mi, j =

m

∑s=1

(s+1)(s+2)ψn,s+2i, j µs,m, m≥ 1, (16.100)

Then, onz= η(ξ1,ξ2;q), it holds using (16.92) that

φ (ξ1,ξ2, η ;q) =+∞

∑n=0

φn(ξ1,ξ2, η) qn =+∞

∑n=0

qn

[

+∞

∑m=0

β n,m0,0 (ξ1,ξ2) qm

]

=+∞

∑n=0

+∞

∑m=0

β n,m0,0 (ξ1,ξ2) qm+n =

+∞

∑s=0

qs

[

s

∑m=0

β s−m,m0,0 (ξ1,ξ2)

]

=+∞

∑n=0

φ0,0n (ξ1,ξ2) qn, (16.101)

where

φ0,0n (ξ1,ξ2) =

n

∑m=0

β n−m,m0,0 . (16.102)

Similarly, we have

∂ i+ j φ∂ξ i

1∂ξ j2

=+∞

∑n=0

φ i, jn (ξ1,ξ2) qn, (16.103)

∂ i+ j

∂ξ i1∂ξ j

2

(

∂ φ∂z

)

=+∞

∑n=0

φ i, jz,n(ξ1,ξ2) qn, (16.104)

∂ i+ j

∂ξ i1∂ξ j

2

(

∂ 2φ∂z2

)

=+∞

∑n=0

φ i, jzz,n(ξ1,ξ2) qn, (16.105)

where


φ i, jn (ξ1,ξ2) =

n

∑m=0

β n−m,mi, j . (16.106)

φ i, jz,n(ξ1,ξ2) =

n

∑m=0

γn−m,mi, j , (16.107)

φ i, jzz,n(ξ1,ξ2) =

n

∑m=0

δ n−m,mi, j . (16.108)

Then, onz= η(ξ1,ξ2;q), it holds using (16.103) and (16.104) that

f =12

∇φ · ∇φ

=k2

1

2

(

∂ φ∂ξ1

)2

+ k1 ·k2∂ φ∂ξ1

∂ φ∂ξ2

+k2

2

2

(

∂ φ∂ξ2

)2

+12

(

∂ φ∂z

)2

=+∞

∑m=0

Γm,0(ξ1,ξ2) qm, (16.109)

where

Γm,0(ξ1,ξ2) =k2

1

2

m

∑n=0

φ1,0n φ1,0

m−n+ k1 ·k2

m

∑n=0

φ1,0n φ0,1

m−n

+k2

2

2

m

∑n=0

φ0,1n φ0,1

m−n+12

m

∑n=0

φ0,0z,n φ0,0

z,m−n. (16.110)

Similarly, it holds onz= η(ξ1,ξ2;q) that

∂ f∂ξ1

= ∇φ · ∇(

∂ φ∂ξ1

)

= k21

∂ φ∂ξ1

∂ 2φ∂ξ 2

1

+ k22

∂ φ∂ξ2

∂ 2φ∂ξ1∂ξ2

+∂ φ∂z

∂∂ξ1

(

∂ φ∂z

)

+ k1 ·k2

(

∂ φ∂ξ1

∂ 2φ∂ξ1∂ξ2

+∂ φ∂ξ2

∂ 2φ∂ξ 2

1

)

=+∞

∑m=0

Γm,1(ξ1,ξ2) qm, (16.111)

∂ f∂ξ2

= ∇φ · ∇(

∂ φ∂ξ2

)

= k21

∂ φ∂ξ1

∂ 2φ∂ξ1∂ξ2

+ k22

∂ φ∂ξ2

∂ 2φ∂ξ 2

2

+∂ φ∂z

∂∂ξ2

(

∂ φ∂z

)

+ k1 ·k2

(

∂ φ∂ξ1

∂ 2φ∂ξ 2

2

+∂ φ∂ξ2

∂ 2φ∂ξ1∂ξ2

)


=+∞

∑m=0

Γm,2(ξ1,ξ2) qm, (16.112)

where

Γm,1(ξ1,ξ2) =m

∑n=0

(

k21 φ1,0

n φ2,0m−n+ k2

2 φ0,1n φ1,1

m−n+ φ0,0z,n φ1,0

z,m−n

)

+ k1 ·k2

m

∑n=0

(

φ1,0n φ1,1

m−n+ φ2,0n φ0,1

m−n

)

, (16.113)

Γm,2(ξ1,ξ2) =m

∑n=0

(

k21 φ1,0

n φ1,1m−n+ k2

2 φ0,1n φ0,2

m−n+ φ0,0z,n φ0,1

z,m−n

)

+ k1 ·k2

m

∑n=0

(

φ1,0n φ0,2

m−n+ φ0,1n φ1,1

m−n

)

. (16.114)

Besides, onz= η(ξ1,ξ2;q), we have by means of (16.103), (16.104) and (16.105)that

∂ f∂z

= ∇φ · ∇(

∂ φ∂z

)

= k21

∂ φ∂ξ1

∂∂ξ1

(

∂ φ∂z

)

+ k22

∂ φ∂ξ2

∂∂ξ2

(

∂ φ∂z

)

+∂ φ∂z

∂ 2φ∂z2

+ k1 ·k2

[

∂ φ∂ξ1

∂∂ξ2

(

∂ φ∂z

)

+∂ φ∂ξ2

∂∂ξ1

(

∂ φ∂z

)]

=+∞

∑m=0

Γm,3(ξ1,ξ2) qm, (16.115)

where

Γm,3(ξ1,ξ2) =m

∑n=0

(

k21 φ1,0

n φ1,0z,m−n+ k2

2 φ0,1n φ0,1

z,m−n+ φ0,0z,n φ0,0

zz,m−n

)

+ k1 ·k2

m

∑n=0

(

φ1,0n φ0,1

z,m−n+ φ0,1n φ1,0

z,m−n

)

. (16.116)

Furthermore, using (16.103), (16.111), (16.112) and (16.115), we have

∇φ · ∇ f = k21

∂ φ∂ξ1

∂ f∂ξ1

+ k22

∂ φ∂ξ2

∂ f∂ξ2

+∂ φ∂z

∂ f∂z

+ k1 ·k2

(

∂ φ∂ξ1

∂ f∂ξ2

+∂ φ∂ξ2

∂ f∂ξ1

)

=+∞

∑m=0

Λm(ξ1,ξ2) qm, (16.117)


where

Λm(ξ1,ξ2) =m

∑n=0

(

k21 φ1,0

n Γm−n,1+ k22 φ0,1

n Γm−n,2+ φ0,0z,n Γm−n,3

)

+ k1 ·k2

m

∑n=0

(

φ1,0n Γm−n,2+ φ0,1

n Γm−n,1)

. (16.118)

Then, using (16.103), (16.104), (16.111), (16.112) and (16.117), we have onz=η(ξ1,ξ2;q) that

N[

φ(ξ1,ξ2,z;q)]

= σ21

∂ 2φ∂ξ 2

1

+2σ1σ2∂ 2φ

∂ξ1∂ξ2+σ2

2∂ 2φ∂ξ 2

2

+g∂ φ∂z

−2

(

σ1∂ f∂ξ1

+σ2∂ f∂ξ2

)

+ ∇φ · ∇ f

=+∞

∑m=0

∆ φm(ξ1,ξ2) qm, (16.119)

where

∆ φm(ξ1,ξ2) = σ2

1 φ2,0m +2σ1σ2 φ1,1

m +σ22 φ0,2

m +gφ0,0z,m

− 2(σ1 Γm,1+σ2 Γm,2)+Λm (16.120)

for m≥ 0.Using (16.35) and (16.92), we have onz= η(ξ1,ξ2;q) that

φ −φ0 =+∞

∑n=1

φn(ξ1,ξ2, η) qn =+∞

∑n=1

qn

(

+∞

∑m=0

β n,m0,0 qm

)

=+∞

∑n=1

qn

(

n−1

∑m=0

β n−m,m0,0

)

(16.121)

and similarly

∂∂z

(

φ −φ0)

=+∞

∑n=1

∂φn

∂zqn =

+∞

∑n=1

qn

(

+∞

∑m=0

γn,m0,0 qm

)

=+∞

∑n=1

qn

(

n−1

∑m=0

γn−m,m0,0

)

, (16.122)

respectively. Then, onz= η(ξ1,ξ2;q), it holds due to the linear property of theoperator (16.27) that

L(

φ −φ0)

=+∞

∑n=1

Sn(ξ1,ξ2) qn, (16.123)


where

Sn(ξ1,ξ2)

=n−1

∑m=0

(

σ21 β n−m,m

2,0 +2σ1σ2 β n−m,m1,1 + σ2

2 β n−m,m0,2 +g γn−m,m

0,0

)

. (16.124)

Then, onz= η(ξ1,ξ2;q), it holds

(1−q)L(

φ −φ0)

= (1−q)+∞

∑n=1

Sn qn =+∞

∑n=1

(Sn− χn Sn−1)qn, (16.125)

where

χn =

0, whenn≤ 1,1, whenn> 1.

(16.126)

Substituting (16.125), (16.119) into (16.28) and equatingthe like-power ofq, wehave the boundary condition:

Sm(ξ1,ξ2)− χm Sm−1(ξ1,ξ2) = c0 ∆ φm−1(ξ1,ξ2), m≥ 1. (16.127)

Define

Sn(ξ1,ξ2)

=n−1

∑m=1

(

σ21 β n−m,m

2,0 +2σ1σ2 β n−m,m1,1 + σ2

2 β n−m,m0,2 +g γn−m,m

0,0

)

. (16.128)

Then,

Sn =(

σ21 β n,0

2,0 +2σ1σ2 β n,01,1 + σ2

2 β n,00,2 +g γn,0

0,0

)

+ Sn

=

(

σ21

∂ 2φn

∂ξ 21

+2σ1σ2∂ 2φn

∂ξ1∂ξ2+ σ2

2∂ 2φn

∂ξ 22

+g∂φn

∂z

)∣

∣

∣

∣

z=0

+ Sn. (16.129)

Substituting the above expression into (16.127) gives the boundary condition onz= 0:

L (φm) = c0 ∆ φm−1+ χm Sm−1− Sm, m≥ 1, (16.130)

whereL is defined by (16.44).Substituting the series (16.36), (16.103) and (16.109) into (16.30), equating the

like-power ofq, we have

ηm(ξ1,ξ2) = c0 ∆ ηm−1+ χm ηm−1, m≥ 1, (16.131)

where

∆ ηm = ηm− 1

g

(

σ1 φ1,0m +σ2 φ0,1

m −Γm,0)

.


References

1. Adcock, T.A.A., Taylor, P.H.: Focusing of unidirectional wave groups on deep water: an ap-proximate nonlinear Schrodinger equation-based model. Proceedings of the Royal Society:A465, 3083 – 3102 (2009)

2. Annenkov, S.Y., Shrira, V.I.: Role of non-resonant interactions in the evolution of nonlinearrandom water wave fields. J. Fluid Mech.561, 181 – 207 (2006)

3. Benney, D.T.: Non-linear gravity wave interactions. J. Fluid Mech.14, 577 – 584 (1962)4. Bretherton, F.P.: Resonant interactions between waves:the case of discrete oscillations. J.

Fluid Mech.20, 457 – 479 (1964)5. Gibbs, R.H., Taylor, P.H.: Formation of walls of water in ‘fully’ nonlinear simulations. Ap-

plied Ocean Research27, 142 – 257 (2005)6. Kharif, C., Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J.

Mech. B - Fluids.22, 603 – 634 (2003)7. Liao, S.J.: The proposed Homotopy Analysis Technique forthe Solution of Nonlinear Prob-

lems. PhD dissertation, Shanghai Jiao Tong University (1992)8. Liao, S.J.: A kind of approximate solution technique which does not depend upon small pa-

rameters (II) – an application in fluid mechanics. Int. J. Nonlin. Mech.32, 815 – 822 (1997)9. Liao, S.J.: An explicit, totally analytic approximationof Blasius viscous flow problems. Int.

J. Nonlin. Mech.34, 759 – 778 (1999)10. Liao, S.J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat

plate. J. Fluid Mech.385, 101 – 128 (1999)11. Liao, S.J., Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous

flow problems. J. Fluid Mech.453, 411 – 425 (2002)12. Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids

over a stretching sheet. J. Fluid Mech.488, 189-212 (2003)13. Liao, S.J.: Beyond Perturbation – Introduction to the Homotopy Analysis Method. Chapman

& Hall/ CRC Press, Boca Raton (2003)14. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput.

147, 499 – 513 (2004)15. Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched

plate. Int. J. Heat Mass Tran.48, 2529 – 2539 (2005)16. Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Stud.

Appl. Math.117, 2529 – 2539 (2006)17. Liao, S.J.: Notes on the homotopy analysis method - some definitions and theorems. Com-

mun. Nonlinear Sci. Numer. Simulat.14, 983 – 997 (2009)18. Liao, S.J.: On the relationship between the homotopy analysis method and Eu-

ler transform. Commun. Nonlinear Sci. Numer. Simulat.15, 1421 – 1431 (2010).doi:10.1016/j.cnsns.2009.06.008


20. Liao, S.J.: On the homotopy multiple-variable method and its applications in the interactionsof nonlinear gravity waves. Commun. Nonlinear Sci. Numer. Simulat.16, 1274 – 1303 (2011)


22. Longuet-Higgins, M.S.: Resonant interactions betweentwo trains of gravity waves. J. FluidMech.12, 321 – 332 (1962)

23. Longuet-Higgins, M.S., Smith, N.D.: An experiment on third order resonant wave interac-tions. J. Fluid Mech.25, 417 – 435 (1966)

24. McGoldrick, L.F., Phillips, O.M., Huang, N., Hodgson, T.: Measurements on resonant waveinteractions. J. Fluid Mech.25, 437 – 456 (1966)

25. Phillips, O.M.: On the dynamics of unsteady gravity waves of finite amplitude. Part 1. Theelementary interactions. J. Fluid Mech.9, 193 – 217 (1960)

26. Phillips, O.M.: Wave interactions - the evolution of an idea. J. Fluid Mech.106, 215 – 227(1981)

Index

2D Gelfand equationhomotopy-approximation, 474mathematical modeling, 468

3D Gelfand equationhomotopy-approximation, 483mathematical modeling, 480

American put optionasymptotic/perturbation formulas, 442code for businessmen, 449, 461HAM code, 454mathematical formulas based on HAM, 434mathematical modeling, 431

Auxiliary functionBVPh1.0, 254generalized, 155, 160, 161

Auxiliary linear operatorBVPh1.0, 252algebraically decaying base function, 253,

379exponentially decaying base function, 106,

253, 372, 396, 417flexibility, 69, 70, 474, 475, 483general form, 181hybrid-base function, 252, 337, 341periodic base function, 34, 35, 69, 181, 252,

292, 324, 328polynomial as base function, 181, 252, 295,

298, 304, 325, 346, 350, 474, 483systematic description, 181wave-current interaction, 507wave-wave interaction, 534

Boundary-value Problemsalgebraically decaying solution, 375American put option, 431exponentially decaying solution, 371

infinite interval, 369interaction of gravity waves, 529interaction of wave and currents, 499multiple solutions, 287non-similarity flows, 389one dimensional Gelfand equation, 337optimal HAM, 99

Convergence theoremfor general case, 175for residual of equation, 172systematic description, 172

Convergence-control parameterBVPh1.0, 254, 2552D Gelfand equation, 4753D Gelfand equation, 483complex number, 352definition, 43effective-region, 43feedback loop in control theory, 54its essence, 47optimal, 184systematic description, 184

Convergence-control vectordefinition, 43

Criterion of wave breakingon a non-uniform current, 520Stokes’ theory, 520

Criterion of wave resonancefor arbitrary number of large-amplitude

waves, 555for arbitrary number of small waves, 531,

553for four small waves by Phillips, 530

Deformation-functiondefinition, 162, 195

569

570 Index

Deformation-operatordefinition, 164

Eigenvalue problemsBVPh1.0, 3192D Gelfand equation, 4683D Gelfand equation, 480multipoint boundary condition, 345non-uniform beam, 326Orr-Sommerfeld stability equation, 349unsteady flows, 409with imaginaryc0, 349with imaginary coefficient, 349with variable coefficients, 340

Embedding parameterdefinition, 13

Euler transformdefinition, 215relation to homotopy transform, 216, 219

Generalized Taylor seriesdefinition, 199systematics description, 194Theorem 5.2, 200Theorem 5.3 for an unique singularity, 201Theorem 5.4 for an unique singularity, 204Theorem 5.1, 196

High nonlinearity2D Gelfand equation, 4683D Gelfand equation, 480non-uniform beam, 326one dimensional Gelfand equation, 337with variable coefficients, 340

High-order deformation equation1st-order deformation equation, 212nd-order deformation equation, 21definition, 20examples, 169–171systematic description, 157Theorem 4.15 for normal form, 158Theorem 4.16 for generalized form, 159Theorem 4.18 for generalized form, 164Theorem 4.17 for generalized form, 161Theorem 4.19 for generalized form, 165Theorem 4.20 for generalized form, 167

Homotopyconcept, 14definition, 15example, 14, 15homotopy of equation, 16homotopy of function, 16, 30

Homotopy multiple-variable methodcompared to multiple scale method, 559

introduction, 531Homotopy parameter

definition, 16Homotopy transform

definition, 215relation to Euler transform, 216, 219systematic description, 215

Homotopy-approximation1st-order homotopy-approximation, 21, 232nd-order homotopy-approximation, 22, 233rd-order homotopy-approximation, 23definition, 20Example 2.2, 33iteration approach, 186optimal, 184

Homotopy-derivativekth-order homotopy-derivative, 191st-order homotopy-derivative, 17property of uniqueness, 140systematic description, 136, 137

Homotopy-derivative operatorbasic properties, 138definition, 19for sin(q φ )/q, 151for N (φ ,ψ,q), 150for f (u), 146, 147for f (u,u′,u′′), 150for f (u,w), 149for exponential function, 143for polynomial, 141for trigonometric function, 144property of commutativity, 139property of linear superposition, 139property of uniqueness, 140systematic description, 136, 137

Homotopy-iteration techniqueBVPh1.0, 256Example 2.2, 62systematic description, 186

Homotopy-Maclaurin seriesdefinition, 20Example 2.1, 19Example 2.2, 32systematic description, 136

Homotopy-Pade techniqueExample 2.2, 55Example 2.1, 24systematic description, 186, 187

Homotopy-seriesdefinition, 20systematic description, 136

Homotopy-series solutionconvergence theorem, 37, 38definition, 20

Index 571

Example 2.1, 19Example 2.2, 32systematic description, 136

Infinite intervalalgebraically decaying solution, 375American put option, 431exponentially decaying solution, 99, 371interaction of gravity waves, 529interaction of wave and current, 499non-similarity flows, 389optimal HAM, 99unsteady flows, 409

Initial approximationgeneral form, 180optimal, 180, 185systematics description, 179

Laplace transformAmerican put option, 438

Multiple solutiona three-point boundary-value problem, 298channel flows, 303infinite number of solutions, 375interaction of gravity waves, 529multipoint boundary condition, 345non-uniform beam, 326nonlinear diffusion-action model, 291resonance of gravity waves, 549

Multiple-solution-control parameterBVPh1.0, 250, 256channel flows, 305–307

diffusion-rection, 295, 297, 298multipoint boundary condition, 299, 300,

302uniform beam, 328with variable coefficient, 345

Multipoint boundary conditionsa three-point boundary-value problem, 298multiple solutions, 345

Residual of equationdefinition, 180definition in general, 184

Singularityone dimensional Gelfand equation, 337with variable coefficients, 340

Solution-expressiondefinition, 29, 178Example 2.2, 28systematic description, 177

Zeroth-order deformation equationdefinition, 16example, 16Example 2.1, 18Example 2.2, 32generalized form in 1999, 155generalized form in 2003, 155, 156generalized form in 2008 by Marinca et al.,

156initial form in 1992, 154modified form in 1997, 154systematic description, 135

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