HANDBOOK OF DIFFERENTIAL EQUATIONS

Third Edition

Daniel Zwillinger Department of Mathematical Sciences

Rensselaer Polytechnic Institute Troy,NY

ACADEMIC PRESS San Diego Boston London

New York Sydney Tokyo Toronto

Contents

Preface xiv Introduction xvi Introduction to the Electronic Version xviii How to Use This Book xix

LA Dennitions and Concepts 1 Definition of Terms 1 2 Alternative Theorems 13 3 Bifurcation Theory 16 4 A Caveat for Partial Differential Equations 24 5 Chaos in Dynamical Systems 26 6 Classification of Partial Differential Equations 33 7 Compatible Systems 39 8 Conservation Laws 43 9 Differential Resultants 46 10 Existence and Uniqueness Theorems 49 11 Fixed Point Existence Theorems 54 12 Hamilton-Jacobi Theory 56 13 Integrability of Systems 60 14 Internet Resources 66 15 Inverse Problems 69 16 Limit Cycles 72 17 Natural Boundary Conditions for a PDE 76 18 Normal Forms: Near-Identity Transformations 78 19 Random Differential Equations 83 20 Self-Adjoint Eigenfunction Problems 86 21 Stability Theorems 92 22 Sturm-Liouville Theory 94

22.1 Classification of Sturm-Liouville Problems 97 23 Variational Equations 99 24 Well Posed Differential Equations 104 25 Wronskians and Fundamental Solutions 108 26 Zeros of Solutions 111

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vi Contents

I.B Transformat ions 27 Canonical Forms 115 28 Canonical Transformations 118 29 Darboux Transformation 121 30 An Involutory Transformation 125 31 Liouville Transformation - 1 127 32 Liouville Transformation - 2 130 33 Reduction of Linear ODEs to a First Order System 131 34 Prüfer Transformation 133 35 Modified Prüfer Transformation 135 36 Transformations of Second Order Linear ODEs - 1 137 37 Transformations of Second Order Linear ODEs - 2 141 38 Transformation of an ODE to an Integral Equation 143 39 Miscellaneous ODE Transformations 146 40 Reduction of PDEs to a First Order System 149 41 Transforming Partial Differential Equations 151 42 Transformations of Partial Differential Equations 156

II Exact Analytical Methods 43 Introduction to Exact Analytical Methods 161 44 Look-Up Technique 162

44.1 Ordinary Differential Equations 163 44.1.1 First Order Equations 163 44.1.2 Second Order Equations 164 44.1.3 Higher Order Equations 171

44.2 Partial Differential Equations 172 44.2.1 Linear Equations 172 44.2.2 Second Order Nonlinearity 174 44.2.3 Higher Order/Variable Order Nonlinearities 176

44.3 Systems of Differential Equations 182 44.3.1 Systems of ODEs 182 44.3.2 Systems of PDEs 183

44.4 The Laplacian in Different Coordinate Systems 187 44.5 Parametrized Equations at Specific Values 188

45 Look-Up ODE Forms 201 45.1 Equation Form: y" + c(x)y = 0 202 45.2 Equation Form: y" + b(x)y' + c(x)y = 0 203 45.3 Equation Form: xy" + b(x)y' + c(x)y = 0 203 45.4 Equation Form: (1 - x2)y" + b(x)y' + c(x)y = 0 203 45.5 Equation Form: x2y" + b(x)y' + c(x)y = 0 203 45.6 Equation Form: x(l - x)y" + b(x)y' + c(x)y = 0 203

ILA Exact Methods for ODEs 46 An Nth Order Equation 205 47 Use of the Adjoint Equation* 207 48 Autonomous Equations - Independent Variable Missing 210

Contents vii

49 Bernoulli Equation 214 50 Clairaut's Equation 216 51 Computer-Aided Solution 218 52 Constant Coefficient Linear Equations 225 53 Contact Transformation 227 54 Delay Equations 230 55 Dependent Variable Missing 237 56 Differentiation Method 239 57 Differential Equations with Discontinuities* 240 58 Eigenfunction Expansions* 243 59 Equidimensional-in-x Equations 250 60 Equidimensional-in-y Equations 253 61 Euler Equations 256 62 Exact First Order Equations 258 63 Exact Second Order Equations 260 64 Exact Nth Order Equations 263 65 Factoring Equations* 265 66 Factoring Operators* 266 67 Factorization Method 272 68 Fokker-Planck Equation 275 69 Fractional Differential Equations* 279 70 Free Boundary Problems* 282 71 Generating Functions* 285 72 Green's Functions* 288 73 Homogeneous Equations 297 74 Method of Images* 299 75 Integrable Combinations 303 76 Integral Representation: Laplace's Method* 304 77 Integral Transforms: Finite Intervals* 309 78 Integral Transforms: Infinite Intervals* 314 79 Integrating Factors* 322 80 Interchanging Dependent and Independent Variables 326 81 Lagrange's Equation 328 82 Lie Groups: ODEs 331 83 Operational Calculus* 344 84 PfafHan Differential Equations 348 85 Reduction of Order 352 86 Riccati Equations 354 87 Matrix Riccati Equations 357 88 Scale Invariant Equations 360 89 Separable Equations 363 90 Series Solution* 364 91 Equations Solvable for x 370 92 Equations Solvable for y 371 93 Superposition* 373 94 Method of Undetermined Coefficients* 375 95 Variation of Parameters 378 96 Vector Ordinary Differential Equations 381

viii Contents

II.B Exact Methods for PDEs 97 Bäcklund Transformations 387 98 Method of Characteristics 390 99 Characteristic Strip Equations 396 100 Conformal Mappings 399 101 Method of Descent 403 102 Diagonalization of a Linear System of PDEs 406 103 DuhamePs Principle 408 104 Exact Equations 411 105 Hodograph Transformation 412 106 Inverse Scattering 416 107 Jacobi's Method 419 108 Legendre Transformation 422 109 Lie Groups: PDEs 426 110 Poisson Formula 433 111 Riemann's Method 436 112 Separation of Variables 441 113 Separable Equations: Stäckel Matrix 447 114 Similarity Methods 450 115 Exact Solutions to the Wave Equation 454 116 Wiener-Hopf Technique 457

III Approximate Analytical Methods 117 Introduction to Approximate Analysis 463 118 Chaplygin's Method 464 119 Collocation 467 120 Dominant Balance 469 121 Equation Splitting 471 122 Floquet Theory 474 123 Graphical Analysis: The Phase Plane 477 124 Graphical Analysis: The Tangent Field 483 125 Harmonie Balance 486 126 Homogenization 489 127 Integral Methods 493 128 Interval Analysis 496 129 Least Squares Method 499 130 Lyapunov Functions 501 131 Equivalent Linearization and Nonlinearization 504 132 Maximum Principles 509 133 McGarvey Iteration Technique 514 134 Moment Equations: Closure 516 135 Moment Equations: Itö Calculus 520 136 Monge's Method 522 137 Newton's Method 524 138 Pade Approximants 528 139 Perturbation Method: Method of Averaging 531 140 Perturbation Method: Boundary Layer Method 535

Contents ix

141 Perturbation Method: Functional Iteration 542 142 Perturbation Method: Multiple Scales 549 143 Perturbation Method: Regulär Perturbation 553 144 Perturbation Method: Strained Coordinates 556 145 Picard Iteration 560 146 Reversion Method 562 147 Singular Solutions 564 148 Soliton-Type Solutions 567 149 Stochastic Limit Theorems 569 150 Taylor Series Solutions 572 151 Variational Method: Eigenvalue Approximation 575 152 Variational Method: Rayleigh-Ritz 578 153 WKB Method 581

IV.A Numerical Methods: Concepts 154 Introduction to Numerical Methods 587 155 Definition of Terms for Numerical Methods 589 156 Available Software 592 157 Finite Difference Forrnulas 599

157.1 One Dimension: Rectilinear Grid 600 157.2 Two Dimensions: Rectilinear Grid 601 157.3 Two Dimensions: Irregulär Grid 601 157.4 Two Dimensions: Triangulär Grid 602 157.5 Numerical Schemes for the ODE: y' = f{x,y) 603 157.6 Explicit Numerical Schemes for the PDE: aux + ut = 0 . . . . 604 157.7 Implicit Numerical Schemes for the PDE: aux + ut = S(x,t). . 604 157.8 Numerical Schemes for the PDE: F(u)x + ut = 0 604 157.9 Numerical Schemes for the PDE: ux — Utt 605

158 Finite Difference Methodology 607 159 Grid Generation 611 160 Richardson Extrapolation 615 161 Stability: ODE Approximations 618 162 Stability: Courant Criterion 623 163 Stability: Von Neumann Test 626 164 Testing Differential Equation Routines 627

IV.B Numerical Methods for ODEs 165 Analytic Continuation* 631 166 Boundary Value Problems: Box Method 634 167 Boundary Value Problems: Shooting Method* 638 168 Continuation Method* 641 169 Continued Fractions 644 170 Cosine Method* 647 171 Differential Algebraic Equations 650 172 Eigenvalue/Eigenfunction Problems 655 173 Euler's Forward Method .' 659

x Contents

174 Finite Element Method* 663 175 Hybrid Computer Methods* 673 176 Invariant Imbedding* 675 177 Multigrid Methods 680 178 Parallel Computer Methods 682 179 Predictor-Corrector Methods 686 180 Runge-Kutta Methods 690 181 Stiff Equations* 697 182 Integrating Stochastic Equations 702 183 Symplectic Integration 706 184 Use of Wavelets 710 185 Weighted Residual Methods* 712

IV.C Numerical Methods for PDEs 186 Boundary Element Method 717 187 Differential Quadrature 721 188 Domain Decomposition 724 189 Elliptic Equations: Finite Differences 729 190 Elliptic Equations: Monte-Carlo Method 733 191 Elliptic Equations: Relaxation 739 192 Hyperbolic Equations: Method of Characteristics 742 193 Hyperbolic Equations: Finite Differences 746 194 Lattice Gas Dynamics 750 195 Method of Lines 752 196 Parabolic Equations: Explicit Method 756 197 Parabolic Equations: Implicit Method 760 198 Parabolic Equations: Monte-Carlo Method 765 199 Pseudospectral Method 771

Mathematical Nomenclature 777 Differential Equation Index 779 Index 785

List of Tables XI

List of Tables 13.1 First integrals for the Lorenz equations 64 48.1 Changing the independent variable: u(y) = yx(x) 212 59.1 Changing the dependent variable: x = ef 252 60.1 Changing the independent variable: y(x) = eu(^ 255 61.1 Changing the dependent variable: x = e 257 72.1 Green's functions for common partial differential equations. . . 290 77.1 Different finite integral transform pairs 312 78.1 Different infinite integral transform pairs 317 80.1 Switching dependent and independent variables 328 82.1 Some common Lie group generators 337 82.2 All possible cases for a two-dimensional Lie algebra 340 82.3 Generators for some classes of first order ODEs 340 82.4 Generators for some classes of second order ODEs 340 82.5 Lie groups for some second order ODEs 341 140.1 Possible behaviors for ey" — p(x)y' — q(x)y = g(x) as a BVP . 540 156.1 The GAMS taxonomy of differential equations Software 593 157.1 Numerical schemes for the ODE: y' = f{x,y) 603 157.2 Explicit numerical schemes for the PDE: aux + ut = 0 604 157.3 Implicit numerical schemes for the PDE: aux + ut = S(x, t). . . 605 157.4 Numerical schemes for the PDE: F{u)x + ut = 0 605 157.5 Numerical schemes for the PDE: ux = utt 606 160.1 Numerical approximations using Richardson extrapolation. . . 617 182.1 Comparison of different stochastic equation approximations. . . 704 182.2 Test problems for stochatic equation methods 705 183.1 Symplectic Integration schemes for separable Hamiltonians. . . 708 196.1 Explicit parabolic technique, varying N and At 759 197.1 Implicit parabolic technique, varying N and At 764

List of P r o g r a m s 14.1 Overture program for a reaction diffusion problem 68 48.1 Macsyma program to change variables 213 48.2 Mathematica program to change variables: u(y) = yx{x). . . . 214 59.1 Macsyma program to change variables 253 60.1 Mathematica program to change variables: y(x) = eu(x^ 255 90.1 Macsyma program to produce series Solution 369 166.1 Portran program for box method 637 167.1 Portran program for shooting method 641 170.1 Fortran program for cosine method 650 173.1 C program for Euler method 660 173.2 Fortran program for Euler method 661 179.1 Fortran program for predictor-corrector method 689 180.1 C program for Runge-Kutta method 692 180.2 Fortran program for Runge-Kutta method 693 181.1 Fortran program for stiff ODEs 699 182.1 Fortran program for stochastic equation Integration 704 187.1 Fortran program for differential quadrature 723 189.1 ELLPACK program for an elliptic problem 733 190.1 Fortran: Monte-Carlo method applied to elliptic equations. . . 738

Xll List of Figures

191.1 Fortran program for relaxation method 741 193.1 Fortran: finite differences applied to hyperbolic equations. . . . 749 196.1 C: explicit method applied to parabolic equations 758 196.2 Fortran: explicit method applied to parabolic equations 759 197.1 Fortran: implicit method applied to parabolic equations 763 198.1 Fortran: Monte-Carlo method applied to parabolic equations. . 770 199.1 Fortran program for spectral method 774

List of Figures 3.1 A bead on a spinning semi-circular wire 18 3.2 Bifurcation diagram for a bead on a spinning wire 20 3.3 Graphical Output generated automatically by the Bifurcation

Interpreter 22 3.4 Diagrams of some types of bifurcations 23 4.1 Characteristics and the ränge of validity of a Solution 26 5.1 DufHng equation with T = 0.20. (Period 1 Solution.) 29 5.2 Duffing equation with F = 0.28. (Period 2 Solution.) 29 5.3 Duffing equation with F = 0.50. (Chaotic Solution.) 29 5.4 The three Lyapunov exponents for Duffing's equation 30 5.5 The canonical piecewise-linear circuit and the voltage-current

characteristic of the nonlinear resistor GN 31 50.1 Solution curves for a differential equation 218 70.1 Diagram illustrating the location of the freezing boundary . . . 283 74.1 Image of a grounded sphere with a point source 300 74.2 The original source and the image source 302 76.1 A contour integral Solution to a differential equation 307 98.1 Depiction of the characteristics for a quasilinear equation. . . . 392 100.1 Domain before and after conformal mapping 400 100.2 Domain before and after Schwartz-Christoffel transformation . 401 111.1 Generic domain of the problem 437 111.2 Domain for the example problem 439 123.1 The different types of behavior in the phase plane: (a) and (c)

are nodes, (b) is a saddle point, (d) is a center, and (e) is a spiral. 479 123.2 The different types of behavior in the phase plane, as a function

of the trace and determinant of the 2 x 2 matrix 480 123.3 Phase plane for a specific equation 481 124.1 Tangent field for a specific equation 484 124.2 Tangent field for a specific equation 485 140.1 Comparison of exact and approximation Solutions 538 140.2 Different Solutions to ey" + yy' — y = 0 for varying BCs . . . . 541 141.1 Diagrammatic representation of differential equation Solution . 544 141.2 Rules for creating and interpreting diagrams 547 141.3 Two steps in the diagrammatic expansion of an equation . . . 547 142.1 Comparison of exact Solution and approximation obtained by

multiple scales technique 551 143.1 Comparison of the exact Solution and a two term approximation 555 152.1 Comparison of exact Solution and approximation obtained by

Rayleigh-Ritz method 580 155.1 Computational molecules for two different approximations. . . 590

List of Figures xm

157.1 Spacing on an irregulär domain 602 157.2 Definition of the coordinate System for a triangulär domain. . . 602 159.1 Two common computational grids, for rectilinear coordinates

and for polar coordinates 612 159.2 A grid and a refined grid on a domain 612 159.3 Six different grids for a domain 613 161.1 Stability diagrams for Euler's forward and backward method. . 621 162.1 Characteristics that are included in the numerical domain of

dependence 624 162.2 Characteristics that are not included in the numerical domain

of dependence 625 173.1 Different numerical techniques applied to y' = —6y + 5e _ f . . . 662 174.1 The finite element "hat functions" 665 174.2 Exact Solution and finite element approximation 669 174.3 Finite elements used in example 3 669 174.4 The functions for the cubic Hermite approximation 672 175.1 Block diagram for the analog Solution of x" + ax' + bx2 — f(t). 674 181.1 The exact Solution to the example equation 698 188.1 Generic domain for domain decomposition 725 188.2 Example domain and its decomposition 727 189.1 Numerical grid on which a problem is solved 731 189.2 The grid on which the example equation is solved 732 190.1 The domain in which Laplace's equation is solved 737 192.1 Depiction of the characteristics for a typical calculation 744 194.1 Blocking of the rectilinear array at different time steps 751 194.2 All possible motions and interactions on the rectilinear grid in

one time step (up to rotations) 751 195.1 Subdivision for the method of lines 753 195.2 Subdivision of the domain 755