Partial Differential Equations

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

R. W. BROCKETT, Harvard University, Cambridge, Mass., U.S.A. J. CORONES, Iowa State University, U.S.A. and Ames Laboratory, U.S. Department of

Energy, Iowa, U.S.A. Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands G.-c. ROTA, M.l T., Cambridge, Mass., U.S.A.

Richard Bellman Dept. of Electrical Engineering, University of Southern California, Los Angeles, U.S.A.; Center for Applied Mathematics, The University of Georgia, Athens, Georgia, U.S.A.

and

George Adomian Center for Applied Mathematics, The University of Georgia, Athens, Georgia, U.S.A.

Partial Differential Equations New Methods/or Their Treatment and Solution

D. Reidel Publishing Company ~.

A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP " Dordrecht / Boston / Lancaster

Librvy of Congress Cataloging in Publication Dall

BeUman, Richard Ernest, 1920-Partial differential equations..

(Mathematics and its applications) Bibliography: p. Includes index. 1. Differential eqUlltions, Partial. I. Adomian, G. II . Title.

III. Series: MathematiC$ and its applications (D. Reidel Publishing Company) QA374.B33 1984 S IS.3'S3 84-18241

ISBN-13: 978-94-010-8804-6 e-ISBN- 13: 978-94-009-5209-6 DOl: 10.1007/978-94-009-5209-6

Published by D. Reidel Publishing Company P.O. Box 17, 3300 AA Dordrecht, HoUand

Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers 190 Old Derby SUeet, Hingttam, MA 02043, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht , Holland

All Righu Reserved CI 1985 by D. Reidel Publishing Company . Softcover reprint of the hardcover I It edition 1985 No part of the material protected by this copyright notice may be reproduced or utili:ted in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without wrilten permission from the oopyriglu owner

To Peter D. Lax

TABLE OF CONTENTS

PREFACE Xlll

PREFACE BY SECOND AUTHOR XVll

CHAPTER I / MONOTONE CONVERGENCE AND POSITIVE OPERATORS 1. Introduction 2. Monotone Operators 3. Monotonicity 2 4. Convergence 2 5. Differential Equations with Initial Conditions 2 6. Two-Point Boundary Conditions 3 7. Nonlinear Heat Equation 3 8. The Nonlinear Potential Equation 3 Bibliography and Comments 4

CHAPTER II/CONSERVATION 5 1. Introduction 5 2. Analytic and Physical Preliminaries 5 3. The Defining Equations 9 4. Limiting Differential Equations 11 5. Conservation for the Discrete Approximation 11 6. Existence of Solutions for Discrete Approximation 12 7. Conservation for Nonlinear Equations 16 8. The Matrix Riccati Equation 16 9. Steady-State Neutron Transport with Discrete

Energy Levels 17 10. Analytic Preliminaries 18 11. Reflections, Transmission, and Loss Matrices 19 12. Existence and Uniqueness of Solutions 22 13. Proof of Conservation Relation 23 14. Proof of Nonnegativ~ty 24 15. Statement of Result 26 Bibliography and Comments 26

CHAPTER III/DYNAMIC PROGRAMMING AND PARTIAL DIFFERENTIAL EQUATIONS 28

1. Introduction 28 2. Calculus of Variations as a Multistage Decision

Process 28

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3. A New Formalism 4. Layered Functionals 5. Dynamic Programming Approach 6. Quadratic Case 7. Bounds Bibliography and Comments

TABLE OF CONTENTS

29 31 32 34 34 34

CHAPTER IV / THE EULER-LAGRANGE EQUATIONS AND CHARACTERISTICS 36

1. Introduction 36 2. Preliminaries 37 3. The Fundamental Relations of the Calculus of

Variations 38 4. The Variational Equations 39 5. The Eulerian Description 43 6. The Lagrangian Description 47 7. The Hamiltonian Description 52 8. Characteristics 53 Bibliography and Comments 58

CHAPTER V / QUASILINEARIZATION AND A NEW METHOD OF SUCCESSIVE APPROXIMATIONS 59

1. Introduction 59 2. The Fundamental Variational Relation 59 3. Successive Approximations 60 4. Convergence 60 Bibliography and Comments 61

CHAPTER VI/THE VARIATION OF CHARACTERISTIC VALUES AND FUNCTIONS 62

1. Introduction 62 2. Variational Problem 63 3. Dynamic Programming Approach 65 4. Variation of the Green's Function 66 5. Justification of Equating Coefficients 69 6. Change of Variable 69 7. Analytic Continuation 71 8. Analytic Character of Green's Function 72 9. Alternate Derivation of Expression for ¢(x) 73

10. Variation of Characteristic Values and Characteristic Functions 74

11. Matrix Case 76 12. Integral Equations 80 Bibliography and Comments 81

TABLE OF CONTENTS

CHAPTER VII/THE HADAMARD VARIATIONAL FORMULA 1. Introduction 2. Preliminaries 3. A Minimum Problem 4. A Functional Equation 5. The Hadamard Variation 6. Laplace-Beltrami Operator 7. Inhomogeneous Operator Bibliography and Comments

CHAPTER VIII/THE TWO-DIMENSIONAL POTENTIAL EQUATION 1. Introduct ion 2. The Euler-Lagrange Equation 3. Inhomogeneous and Nonlinear Cases 4. Green's Function 5. Two-Dimensional Case 6. Discretization 7. Rectangular Region 8. Associated Minimization Problem 9. Approximation from Above

10. Discussion 11. Semidiscretization 12. Solution of the Difference Equations 13. The Potential Equation 14. Discretization 15. Matrix-Vector Formulation 16. Dynamic Programming 17. Recurrence Equations 18. The Calculations 19. Irregular Regions Bibliography and Comments

CHAPTER IX / THE T~EE-DIMENSIONAL POTENTIAL EQUATION 1. Introduction 2. Discrete Variational Problems 3. Dynamic Programming 4. Boundary Conditions 5. Recurrence Relations 6. General Regions 7. Discussion Bibliography and Comments

CHAPTER X / THE HEAT EQUATION 1. Introduction 2. The One-Dimensional Heat Equation 3. The Transform Equation

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82 82 82 83 85 86 86 86 87

88 88 88 89 90 90 91 91 93 93 93 94 94 95 96 97 98

100 102 102 102

103 103 103 105 107 107 108 109 109

110 110 110 112

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4. Some Numerical Results 5. Multidimensional Case Bibliography and Comments

T ABLE OF CONTENTS

113 118 119

CHAPTER XI/NONLINEAR PARABOLIC EQUATIONS 120 120 120 121 121 123 125 126 126 127

1. Introduction 2. Linear Equation 3. The Non-negativity of the Kernel 4. Monotonicity of Mean Values 5. Positivity of the Parabolic Operator 6. Nonlinear Equations 7. Asymptotic Behavior 8. Extensions Bibliography and Comments

CHAPTER XII/DIFFERENTIAL QUADRATURE 129 1. Introduct ion 129 2. Differential Quadrature 129 3. Determination of Weighting Coefficients 130 4. Numerical Results for First Order Problems 132 5. Systems of Nonlinear Partial Differential Equation~ 135 6. Higher Order Problems 138 7. Error Representation 140 8. Hodgkin-Huxley Equation 141 9. Equations of the Mathematical Model 143

10. Numerical Method 145 11. Conclusion 146 Bibliography and Comments 147

CHAPTER XIII/ADAPTIVE GRIDS AND NONLINEAR EQUATIONS 148 1. Introduction 148 2. The Equation u = -uu 148

t x 4 3. An Example 1 9 4. Discussion 150 5. Extension 150 6. Higher Order Approximations 151 Bibliography and Comments 152

CHAPTER XIV / INFINITE SYSTEMS OF DIFFERENTIAL EQUATIONS 153 1. Introduction 153 2. Burgers' Equation 154 3. Some Numerical Examples '156 4. Two-Dimensional Case 160 5. Closure Techniques 162 6. A Direct Method 163

TABLE OF CONTENTS

7. Extrapolation 8. Difference Approximations 9. An Approximating Algorithm

10. Numerical Results 11. Higher Order Approximation 12. Truncation 13. Associated Equation 14. Discussion of Convergence of u(N) 15. The Fejer Sum 16. The Modified Truncation Bibliography and Comments

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163 166 166 167 170 170 172 173 173 174 175

CHAPTER XV j GREEN'S FUNCTIONS 176 1. Introduction 176 2. The Concept of the Green's Function 177 3. Sturm-Liouville Operator 180 4. Properties of the Green's Function for the

Sturm-Liouville Equation 181 5. Properties of the 0 Function 187 6. Distributions 188 7. Symbolic Functions 189 8. Derivative of Symbolic Functions 189 9. What Space Are We Considering? 194

10. Boundary Conditions 197 11. Properties of Operator L 203 12. Adjoint Operators 203 13. n-th Order Operators 207 14. Boundary Conditions for the Sturm-Liouville

Equation 208 15. Green's Function for Sturm-Liouville Operator 209 16. Solution of the Inhomogeneous Equation 210 17. Solving Non-Homogeneous Boundary Conditions 213 18. Boundary Conditions Specified on Finite

Interval [a, b] 215 19. Scalar Products 220 20. Use of Green's Function to Solve a Second-order

Stochastic Differential Equation 222 21. Use of Green's Function in Quantum Physics 226 22. Use of Green's Functions in Transmission Lines 226 23. Two-Point Green's Functions - Generalization to

n-point Green's Functions 24. Evaluation of Arbitrary Functior,;:; for

Nonhomogeneous Boundary Conditions by Matrix Equations

25. Mixed Boundary Conditions

227

230 231

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26. Some General Properties 232 1. Nonnegativity of Green's Functions and Solutions 232 2. Variation-Diminishing Properties of Green's

Functions 233 Notes 235 Bibliography 235

CHAPTER XVI/APPROXIMATE CALCULATION OF GREEN'S FUNCTIONS 237

CHAPTER XVII/GREEN'S FUNCTIONS FOR PARTIAL DIFFERENTIAL EQUATIONS 243

1. Introduction 243 2. Green's Functions for Multidimensional Problems in

Cartesian Coordinates 243 3. Green's Functions in Curvilinear Coordinates 244 4. Properties of 0 Functions for Multi-dimensional

Case 246

CHAPTER XVIII/THE ITO EQUATION AND A GENERAL STOCHASTIC MODEL FOR DYNAMICAL SYSTEMS 248

Bibliography 252

CHAPTE~ XIX / NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND THE DECOMPOSITION METHOD 254

1. Parametrization and the A Polynomials 262 n

2. Inverses for Non-simple Differential Operators 274 3. Multidimensional Green's Functions by Decomposition

Method 275 4. Relationships Between Green's Functions and the

Decomposition Method for Partial Differential Equations 277

5. Separable Systems 281 6. The partitioning Method of Butkovsky 281 7. Computation of the ~ 282

8. The Question of Convergence 284 Bibliography 287

INDEX 289

PREFACE

The purpose of this book is to present some new methods in the treatment of partial differential equations. Some of these methods lead to effective numerical algorithms when combined with the digital computer. Also presented is a useful chapter on Green's functions which generalizes, after an introduction, to new methods of obtaining Green's functions for partial differential operators. Finally some very new material is presented on solving partial differential equations by Adomian's decomposition methodology. This method can yield realistic computable solutions for linear or non­linear cases even for strong nonlinearities, and also for deterministic or stochastic cases - again even if strong stochasticity is involved. Some interesting examples are discussed here and are to be followed by a book dealing with frontier applications in physics and engineering.

In Chapter I, it is shown that a use of positive operators can lead to monotone convergence for various classes of nonlinear partial differential equations.

In Chapter II, the utility of conservation technique is shown. These techniques are suggested by physical principles. In Chapter III, it is shown that dyn~mic programming applied to variational problems leads to interesting classes of nonlinear partial differential equations. In Chapter IV, this is investigated in greater detail. In Chapter V, we show. that the use of a transformation suggested by dynamic programming leads to a new method of successive approximations.

In Chapter VI, we consider the variation of characteristic values and characteristic functions with the domain, restrict­ing our attention to the one-dimensional case. We point out how the method may be applied to the multidimensional case. In Chapter VII, we apply this method to obtain the classic Hadamard variational formula. It is pointed out that the same method can treat multidimensional variational problems.

In Chapter VIII, we consider the discretized form of the two-dimensional potential equation. This equation arises from the minimization of a quadratic form. The minimization is carried out by means of dynamic programming and the dimen-

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xiv PREFACE

sionality difficulties are circumvented observing that the minimum itself is a quadratic form. In Chapter IX, we consider the discretized form of the three-dimensional potential equation. Dimensionality difficulties are now avoided by using a more general domain. In Chapter X, we consider the linear heat equation. Our procedure has two parts. First, we use the Laplace transform, obtaining an equation of potential type. Then, we employ a numerical inver3ion method. In Chapter XI, we consider nonlinear parabolic partial differential equations, obtaining an analogue of the classical Poincare-Lyapunov theorem for ordinary differential equations. The results required concerning the linear eq~ation are of interest in themselves.

In Chapter XII, we give some examples of the use of differential quadrature. In Chapter XIII, we consider an adaptive grid to treat nonlinear partial differential equations.

In Chapter XIV, we consider the expansion of the solu­tion of a partial differential equation in orthogonal func­tions. This yields an infinite system of ordinary differential equations. To obtain numerical results from this system, some method of truncation must be employed. Many interesting stability questions arise in this way.

The concluding chapters deal with Green's functions, methods of determining Green's functions for partial differential equations, and new and valuable methods of dealing with nonlinearity and/or randomness in multidimensional problems using the basic decomposition method first discussed by Adomian in Stochastic Systems (Academic Press, 1983); Stochastic Systems II and also Applications of Stochastic Systems Theory to Physics and Engineering will appear shortly. Chapter XV deals with Green's functions; Chapter XVI shows a method for approximate calculation of Green's functions which is very useful in computation; Chapter XVII deals with Green's functions for partial differential equations; Chapter XVIII deals with the Ito equation and a general stochastic model for dynamical systems. The last chapter is completely new and deals with solution of nonlinear partial differential equations by the decomposition method and provides examples as well. Randolph Rach of Raytheon Company has been very helpful here.

A great deal of work has been done in the field of partial differential equations. We have made no attempt to cover it all. Rather, we have restricted our attention to areas where we have worked ourselves. Even here, as will be apparent from the text, there is still a great deal to be done.

RICHARD E. BELLMAN

PREFACE BY SECOND AUTHOR

Before the final galley corrections to this book, Richard E. Bellman passed into history on March 19, 1984 at the age of 63.

To the very end, his mind was filled with exciting ideas, and he planned further contributions. Having had the privilege of knowing him for over 20 years, and of close collaboration in recent years, it is clear that the loss to science, engineering, and mathematics is immeasurable and profound. Richard Bellman will be remembered around the world as one of the foremost mathematical scientists of this century.

Richard Bellman's influence on present and future researchers and generations of students to come will be pervasive and deep. His incredibly productive pioneering work and creative thought in mathematics and its applica­tions to science, engineering, medicine, and economics have made him a leader in his field. His helpfulness and compassion and courage endeared him to o~r hearts.

G. ADOMIAN

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