Handbook of Differential Equations

Content: 
Front Matter, Page iii
Copyright, Page iv
Preface, Pages xi-xii
Introduction, Pages xiii-xv
How to Use This Book, Pages xvii-xx
1 - Definition of Terms, Pages 1-13
2 - Alternative Theorems, Pages 14-16
3 - Bifurcation Theory, Pages 16-24
4 - A Caveat for Partial Differential Equations, Pages 25-26
5 - Chaos in Dynamical Systems, Pages 26-32
6 - Classification of Partial Differential Equations, Pages 33-39
7 - Compatible Systems, Pages 39-43
8 - Conservation Laws, Pages 43-46
9 - Differential Resultants, Pages 46-49
10 - Existence and Uniqueness Theorems, Pages 50-54
11 - Fixed Point Existence Theorems, Pages 54-56
12 - Hamilton-Jacobi Theory, Pages 57-60
13 - Inverse Problems, Pages 61-63
14 - Limit Cycles, Pages 63-67
15 - Natural Boundary Conditions for a PDE, Pages 67-69
16 - Normal Forms: Near-Identity Transformations, Pages 70-74
17 - Self-Adjoint Eigenfunction Problems, Pages 74-80
18 - Stability Theorems, Pages 80-82
19 - Sturm—Liouville Theory, Pages 82-88
20 - Variational Equations, Pages 88-93
21 - Well-Posedness of Differential Equations, Pages 94-97
22 - Wronskians and Fundamental Solutions, Pages 97-100
23 - Canonical Forms, Pages 101-104
24 - Canonical Transformations, Pages 105-108
25 - Darboux Transformation, Pages 108-111
26 - An Involutory Transformation, Pages 111-114
27 - Liouville Transformation – 1, Pages 114-117
28 - Liouville Transformation – 2, Pages 117-118
29 - Reduction of Linear ODEs to a First Order System, Pages 118-120
30 - Prüfer Transformation, Pages 120-122
31 - Modified Prüfer Transformation, Pages 122-124
32 - Transformations of Second Order Linear ODEs – 1, Pages 124-128
33 - Transformations of Second Order Linear ODEs – 2, Pages 128-130
34 - Transformation of an ODE to an Integral Equation, Pages 130-133
35 - Miscellaneous ODE Transformations, Pages 133-136
36 - Reduction of PDEs to a First Order System, Pages 136-139
37 - Transforming Partial Differential Equations, Pages 139-143
38 - Transformations of Partial Differential Equations, Pages 144-146
39 - Introduction to Exact Analytical Methods, Pages 147-148
40 - Look Up Technique, Pages 148-181
41 - Look Up ODE Forms, Pages 181-184
42 - An N-th Order Equation, Pages 185-187
43 - Use of the Adjoint Equation*, Pages 187-190
44 - Autonomous Equations, Pages 190-194
45 - Bernoulli Equation, Pages 194-195
46 - Clairaut's Equation, Pages 196-197
47 - Computer-Aided Solution, Pages 197-204
48 - Constant Coefficient Linear Equations, Pages 204-205
49 - Contact Transformation, Pages 206-209
50 - Delay Equations, Pages 209-216
51 - Dependent Variable Missing, Pages 216-217
52 - Differentiation Method, Pages 218-219
53 - Differential Equations with Discontinuities*, Pages 219-222
54 - Eigenfunction Expansions*, Pages 223-230
55 - Equidimensional-In-x Equations, Pages 230-233
56 - Equidimensional-In-y Equations, Pages 233-235
57 - Euler Equations, Pages 235-237
58 - Exact First Order Equations, Pages 238-240
59 - Exact Second Order Equations, Pages 240-243
60 - Exact N-th Order Equations, Pages 243-245
61 - Factoring Equations, Pages 245-246
62 - Factoring Operators, Pages 246-251
63 - Factorization Method, Pages 251-254
64 - Fokker—Planck Equation, Pages 254-258
65 - Fractional Differential Equations*, Pages 258-262
66 - Free Boundary Problems, Pages 262-265
67 - Generating Functions, Pages 265-267
68 - Green's Functions, Pages 268-276
69 - Homogeneous Equations, Pages 276-279
70 - Method of Images*, Pages 279-282
71 - Integrable Combinations, Pages 283-284
72 - Integral Representations: Laplace's Method*, Pages 284-290
73 - Integral Transforms: Finite Intervals*, Pages 290-295
74 - Integral Transforms: Infinite Intervals*, Pages 295-304
75 - Integrating Factors*, Pages 305-308
76 - Interchanging Dependent and Independent Variables, Pages 308-310
77 - Lagrange's Equation, Pages 311-314
78 - Lie Groups: ODEs, Pages 314-321
79 - Operational Calculus*, Pages 322-325
80 - Pfaffian Differential Equations, Pages 326-330
81 - Reduction of Order, Pages 330-332
82 - Riccati Equation, Pages 332-335
83 - Matrix Riccati Equations, Pages 335-338
84 - Scale Invariant Equations, Pages 338-341
85 - Separable Equations, Pages 341-342
86 - Series Solution*, Pages 342-348
87 - Equations Solvable for x, Pages 349-350
88 - Equations Solvable for y, Pages 350-352
89 - Superposition*, Pages 352-354
90 - Method of Undetermined Coefficients*, Pages 354-356
91 - Variation of Parameters, Pages 356-359
92 - Vector Ordinary Differential Equations, Pages 360-363
93 - Bäcklund Transformations, Pages 365-368
94 - Method of Characteristics, Pages 368-374
95 - Characteristic Strip Equations, Pages 374-376
96 - Conformal Mappings, Pages 376-381
97 - Method of Descent, Pages 382-384
98 - Diagonalization of a Linear System of PDEs, Pages 384-386
99 - Duhamel's Principle, Pages 386-389
100 - Exact Equations, Pages 389-390
101 - Hodograph Transformation, Pages 390-393
102 - Inverse Scattering, Pages 393-397
103 - Jacobi's Method, Pages 397-400
104 - Legendre Transformation, Pages 400-404
105 - Lie Groups: PDEs, Pages 404-411
106 - Poisson Formula, Pages 411-413
107 - Riemann's Method, Pages 414-418
108 - Separation of Variables, Pages 419-424
109 - Similarity Methods, Pages 424-428
110 - Exact Solutions to the Wave Equation, Pages 429-432
111 - Wiener—Hopf Technique, Pages 432-436
112 - Introduction to Approximate Analysis, Pages 437-438
113 - Chaplygin's Method, Pages 438-441
114 - Collocation, Pages 441-443
115 - Dominant Balance, Pages 443-445
116 - Equation Splitting, Pages 446-447
117 - Floquet Theory, Pages 448-451
118 - Graphical Analysis: The Phase Plane, Pages 451-456
119 - Graphical Analysis: The Tangent Field, Pages 457-459
120 - Harmonic Balance, Pages 460-462
121 - Homogenization, Pages 463-467
122 - Integral Methods, Pages 467-470
123 - Interval Analysis, Pages 470-473
124 - Least Squares Method, Pages 473-475
125 - Lyapunov Functions, Pages 476-479
126 - Equivalent Linearization and Nonlinearization, Pages 479-484
127 - Maximum Principles, Pages 484-488
128 - McGarvey Iteration Technique, Pages 488-490
129 - Moment Equations: Closure, Pages 491-494
130 - Moment Equations: Itô Calculus, Pages 494-497
131 - Monge's Method, Pages 497-500
137 - Newton's Method, Pages 500-503
133 - Padé Approximants, Pages 503-507
134 - Perturbation Method: Method of Averaging, Pages 507-510
135 - Perturbation Method: Boundary Layer Method, Pages 510-517
136 - Perturbation Method: Functional Iteration, Pages 518-524
137 - Perturbation Method: Multiple Scales, Pages 524-528
138 - Perturbation Method: Regular Perturbation, Pages 528-531
139 - Perturbation Method: Strained Coordinates, Pages 532-535
140 - Picard Iteration, Pages 535-538
141 - Reversion Method, Pages 538-540
142 - Singular Solutions, Pages 540-543
143 - Soliton Type Solutions, Pages 543-545
144 - Stochastic Limit Theorems, Pages 545-548
145 - Taylor Series Solutions, Pages 548-551
146 - Variational Method: Eigenvalue Approximation, Pages 551-554
147 - Variational Method: Rayleigh—Ritz, Pages 554-558
148 - WKB Method, Pages 558-563
149 - Introduction to Numerical Methods, Pages 565-567
150 - Definition of Terms for Numerical Methods, Pages 567-569
151 - Available Software, Pages 570-573
152 - Finite Difference Methodology, Pages 573-577
153 - Finite Difference Formulas, Pages 578-585
154 - Excerpts from GAMS, Pages 586-605
155 - Grid Generation, Pages 606-609
156 - Richardson Extrapolation, Pages 609-613
157 - Stability: ODE Approximations, Pages 613-617
158 - Stability: Courant Criterion, Pages 618-620
159 - Stability: Von Neumann Test, Pages 621-622
160 - Analytic Continuation, Pages 623-626
161 - Boundary Value Problems: Box Method, Pages 626-630
162 - Boundary Value Problems: Shooting Method*, Pages 631-634
163 - Continuation Method, Pages 635-637
164 - Continued Fractions, Pages 637-640
165 - Cosine Method*, Pages 640-644
166 - Differential Algebraic Equations, Pages 644-649
167 - Eigenvalue/Eigenfunction Problems, Pages 650-652
168 - Euler's Forward Method, Pages 653-656
169 - Finite Element Method*, Pages 656-666
170 - Hybrid Computer Methods*, Pages 666-668
171 - Invariant Imbedding*, Pages 669-673
172 - Multigrid Methods*, Pages 673-676
173 - Parallel Computer Methods, Pages 676-679
174 - Predictor—Corrector Methods, Pages 679-683
175 - Runge—Kutta Methods, Pages 684-690
176 - Stiff Equations*, Pages 690-694
177 - Integrating Stochastic Equations, Pages 695-698
178 - Weighted Residual Methods*, Pages 699-702
179 - Boundary Element Method, Pages 703-707
180 - Differential Quadrature, Pages 708-711
181 - Domain Decomposition, Pages 711-716
182 - Elliptic Equations: Finite Differences, Pages 716-720
183 - Elliptic Equations: Monte Carlo Method, Pages 721-726
184 - Elliptic Equations: Relaxation, Pages 726-729
185 - Hyperbolic Equations: Method of Characteristics, Pages 730-733
186 - Hyperbolic Equations: Finite Differences, Pages 733-737
187 - Lattice Gas Dynamics, Pages 737-740
188 - Method of Lines, Pages 740-744
189 - Parabolic Equations: Explicit Method, Pages 744-747
190 - Parabolic Equations: Implicit Method, Pages 747-752
191 - Parabolic Equations: Monte Carlo Method, Pages 752-759
192 - Pseudo-Spectral Method, Pages 759-763
Mathematical Nomenclature, Pages 765-766
Differential Equation Index, Pages 767-772
Index, Pages 773-787