Handbook of Differential Equations (Errata added)

Daniel Zwillinger0127843965, 9780127843964, 0127843973, 9780127843971

This book and CD-ROM compile the most widely applicable methods for solving and approximating differential equations. The CD-ROM provides convenient access to these methods through electronic search capabilities, andtogether the book and CD-ROM contain numerous examples showing the methods use. Topics include ordinary differential equations, symplectic integration of differential equations, and the use of wavelets when numerically solving differential equations.

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Table of contents :
Titlepage……Page 1
Contents……Page 2
Preface……Page 8
Introduction……Page 10
Introduction to the Electronic Version……Page 13
How to Use This Book……Page 14
1 Definition of Terms……Page 20
2 Alternative Theorems……Page 33
3 Bifurcation Theory……Page 37
4 A Caveat for Partial Differential Equations……Page 45
5 Chaos in Dynamical Systems……Page 47
6 Classification of Partial Differential Equations……Page 54
7 Compatible Systems……Page 61
8 Conservation Laws……Page 65
9 Differential Resultants……Page 68
10 Existence and Uniqueness Theorems……Page 71
11 Fixed Point Existence Theorems……Page 76
12 Hamilton-Jacobi Theory……Page 79
13 Integrability of Systems……Page 83
14 Internet Resources……Page 89
15 Inverse Problems……Page 93
16 Limit Cycles……Page 96
17 Natural Boundary Conditions for a PDE……Page 101
18 Normal Forms: Near-Identity Transformations……Page 104
19 Random Differential Equations……Page 109
20 Self-Adjoint Eigenfunction Problems……Page 113
21 Stability Theorems……Page 119
22 Sturm-Liouville Theory……Page 121
23 Variational Equations……Page 127
24 Well Posed Differential Equations……Page 133
25 Wronskians and Fundamental Solutions……Page 137
26 Zeros of Solutions……Page 141
27 Canonical Forms……Page 145
28 Canonical Transformations……Page 149
29 Darboux Transformation……Page 152
30 An Involutory Transformation……Page 156
31 Liouville Transformation – 1……Page 158
32 Liouville Transformation – 2……Page 161
33 Reduction of Linear ODEs to a First Order System……Page 163
34 Prufer Transformation……Page 165
35 Modified Prufer Transformation……Page 167
36 Transformations of Second Order Linear ODEs – 1……Page 169
37 Transformations of Second Order Linear ODEs – 2……Page 174
38 Transformation of an ODE to an Integral Equation……Page 176
39 Miscellaneous ODE Transformations……Page 179
40 Reduction of PDEs to a First Order System……Page 183
41 Transforming Partial Differential Equations……Page 185
42 Transformations of Partial Differential Equations……Page 190
43 Introduction to Exact Analytical Methods……Page 194
44 Look-Up Technique……Page 195
45 Look-Up ODE Forms……Page 235
46 An Nth Order Equation……Page 239
47 Use of the Adjoint Equation……Page 241
48 Autonomous Equations – Independent Variable Missing……Page 245
49 Bernoulli Equation……Page 250
50 Clairaut’s Equation……Page 252
51 Computer-Aided Solution……Page 255
52 Constant Coefficient Linear Equations……Page 262
53 Contact Transformation……Page 264
54 Delay Equations……Page 268
55 Dependent Variable Missing……Page 275
56 Differentiation Method……Page 277
57 Differential Equations with Discontinuities……Page 279
58 Eigenfunction Expansions……Page 283
59 Equidimensional-in-x Equations……Page 290
60 Equidimensional-in-y Equations……Page 293
61 Euler Equations……Page 296
62 Exact First Order Equations……Page 299
63 Exact Second Order Equations……Page 302
64 Exact Nth Order Equations……Page 305
65 Factoring Equations……Page 307
66 Factoring Operators……Page 309
67 Factorization Method……Page 315
68 Fokker-Planck Equation……Page 318
69 Fractional Differential Equations……Page 323
70 Free Boundary Problems……Page 326
71 Generating Functions……Page 330
72 Green’s Functions……Page 333
73 Homogeneous Equations……Page 342
74 Method of Images……Page 345
75 Integrable Combinations……Page 349
76 Integral Representation: Laplace’s Method……Page 351
77 Integral Transforms: Finite Intervals……Page 357
78 Integral Transforms: Infinite Intervals……Page 362
79 Integrating Factors……Page 371
80 Interchanging Dependent and Independent Variables……Page 375
81 Lagrange’s Equation……Page 378
82 Lie Groups: ODEs……Page 381
83 Operational Calculus……Page 394
84 Pfaffian Differential Equations……Page 399
85 Reduction of Order……Page 404
86 Riccati Equations……Page 407
87 Matrix Riccati Equations……Page 410
88 Scale Invariant Equations……Page 413
89 Separable Equations……Page 416
90 Series Solution……Page 418
91 Equations Solvable for x……Page 424
92 Equations Solvable for y……Page 426
93 Superposition……Page 428
94 Method of Undetermined Coefficients……Page 430
95 Variation of Parameters……Page 433
96 Vector Ordinary Differential Equations……Page 436
97 Backlund Transformations……Page 442
98 Method of Characteristics……Page 446
99 Characteristic Strip Equations……Page 452
100 Conformal Mappings……Page 455
101 Method of Descent……Page 460
102 Diagonalization of a Linear System of PDEs……Page 463
103 Duhamel’s Principle……Page 465
104 Exact Equations……Page 468
105 Hodograph Transformation……Page 470
106 Inverse Scattering……Page 474
107 Jacobi’s Method……Page 478
108 Legendre Transformation……Page 481
109 Lie Groups: PDEs……Page 485
110 Poisson Formula……Page 492
111 Riemann’s Method……Page 495
112 Separation of Variables……Page 501
113 Separable Equations: Stackel Matrix……Page 508
114 Similarity Methods……Page 511
115 Exact Solutions to the Wave Equation……Page 515
116 Wiener-Hopf Technique……Page 519
117 Introduction to Approximate Analysis……Page 523
118 Chaplygin’s Method……Page 524
119 Collocation……Page 527
120 Dominant Balance……Page 530
121 Equation Splitting……Page 533
122 Floquet Theory……Page 536
123 Graphical Analysis: The Phase Plane……Page 539
124 Graphical Analysis: The Tangent Field……Page 545
125 Harmonic Balance……Page 548
126 Homogenization……Page 551
127 Integral Methods……Page 555
128 Interval Analysis……Page 558
129 Least Squares Method……Page 562
130 Lyapunov Functions……Page 564
131 Equivalent Linearization and Nonlinearization……Page 568
132 Maximum Principles……Page 573
133 McGarvey Iteration Technique……Page 579
134 Moment Equations: Closure……Page 581
135 Moment Equations: Ito Calculus……Page 585
136 Monge’s Method……Page 588
137 Newton’s Method……Page 591
138 Pade Approximants……Page 595
139 Perturbation Method: Method of Averaging……Page 599
140 Perturbation Method: Boundary Layer Method……Page 603
141 Perturbation Method: Functional Iteration……Page 611
142 Perturbation Method: Multiple Scales……Page 618
143 Perturbation Method: Regular Perturbation……Page 623
144 Perturbation Method: Strained Coordinates……Page 627
145 Picard Iteration……Page 631
146 Reversion Method……Page 634
147 Singular Solutions……Page 636
148 Soliton-Type Solutions……Page 639
149 Stochastic Limit Theorems……Page 642
150 Taylor Series Solutions……Page 645
151 Variational Method: Eigenvalue Approximation……Page 648
152 Variational Method: Rayleigh-Ritz……Page 651
153 WKB Method……Page 655
154 Introduction to Numerical Methods……Page 660
155 Definition of Terms for Numerical Methods……Page 663
156 Available Software……Page 666
157 Finite Difference Formulas……Page 673
158 Finite Difference Methodology……Page 682
159 Grid Generation……Page 687
160 Richardson Extrapolation……Page 691
161 Stability: ODE Approximations……Page 695
162 Stability: Courant Criterion……Page 700
163 Stability: Von Neumann Test……Page 704
164 Testing Differential Equation Routines……Page 706
165 Analytic Continuation……Page 709
166 Boundary Value Problems: Box Method……Page 712
167 Boundary Value Problems: Shooting Method……Page 717
168 Continuation Method……Page 721
169 Continued Fractions……Page 724
170 Cosine Method……Page 727
171 Differential Algebraic Equations……Page 731
172 Eigenvalue-Eigenfunction Problems……Page 737
173 Euler’s Forward Method……Page 741
174 Finite Element Method……Page 745
175 Hybrid Computer Methods……Page 755
176 Invariant Imbedding……Page 758
177 Multigrid Methods……Page 763
178 Parallel Computer Methods……Page 766
179 Predictor-Corrector Methods……Page 770
180 Runge-Kutta Methods……Page 774
181 Stiff Equations……Page 781
182 Integrating Stochastic Equations……Page 786
183 Symplectic Integration……Page 791
184 Use of Wavelets……Page 795
185 Weighted Residual Methods……Page 797
186 Boundary Element Method……Page 802
187 Differential Quadrature……Page 806
188 Domain Decomposition……Page 810
189 Elliptic Equations: Finite Differences……Page 815
190 Elliptic Equations: Monte-Carlo Method……Page 820
191 Elliptic Equations: Relaxation……Page 826
192 Hyperbolic Equations: Method of Characteristics……Page 830
193 Hyperbolic Equations: Finite Differences……Page 834
194 Lattice Gas Dynamics……Page 838
195 Method of Lines……Page 841
196 Parabolic Equations: Explicit Method……Page 845
197 Parabolic Equations: Implicit Method……Page 849
198 Parabolic Equations: Monte-Carlo Method……Page 854
199 Pseudospectral Method……Page 861
Mathematical Nomenclature……Page 867
Errata……Page 868