Handbook of Differential Equations (Errata added)

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