Zwillinger Handbook of Differential Equations

1. Auflage 2014
ISBN: 978-1-4832-2096-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 694 Seiten, Web PDF

Handbook of Differential Equations is a handy reference to many popular techniques for solving and approximating differential equations, including exact analytical methods, approximate analytical methods, and numerical methods. Topics covered range from transformations and constant coefficient linear equations to finite and infinite intervals, along with conformal mappings and the perturbation method. Comprised of 180 chapters, this book begins with an introduction to transformations as well as general ideas about differential equations and how they are solved, together with the techniques needed to determine if a partial differential equation is well-posed or what the 'natural' boundary conditions are. Subsequent sections focus on exact and approximate analytical solution techniques for differential equations, along with numerical methods for ordinary and partial differential equations. This monograph is intended for students taking courses in differential equations at either the undergraduate or graduate level, and should also be useful for practicing engineers or scientists who solve differential equations on an occasional basis.

Dr. Daniel Zwillinger is a Senior Principal Systems Engineer for the Raytheon Company. He was a systems requirements 'book boss” for the Cobra Judy Replacement (CJR) ship and was a requirements and test lead for tracking on the Ungraded Early Warning Radars (UEWR). He has improved the Zumwalt destroyer's software accreditation process and he was test lead on an Active Electronically Scanned Array (AESA) radar. Dan is a subject matter expert (SME) in Design for Six Sigma (DFSS) and is a DFSS SME in Test Optimization, Critical Chain Program Management, and Voice of the Customer. He is currently leading a project creating Trust in Autonomous Systems. At Raytheon, he twice won the President's award for best Six Sigma project of the year: on converting planning packages to work packages for the Patriot missile, and for revising Raytheon's timecard system. He has managed the Six Sigma white belt training program. Prior to Raytheon, Dan worked at Sandia Labs, JPL, Exxon, MITRE, IDA, BBN, and The Mathworks (where he developed an early version of their Statistics Toolbox). For ten years, Zwillinger was owner and president of Aztec Corporation. As a small business, Aztec won several Small Business Innovation Research (SBIR) contracts. The company also created several software packages for publishing companies. Prior to Aztec, Zwillinger was a college professor at Rensselaer Polytechnic Institute in the department of mathematics. Dan has written several books on mathematics on the topics of differential equations, integration, statistics, and general mathematics. He is editor-in-chief of the Chemical Rubber Company's (CRC's) 'Standard Mathematical Tables and Formulae”, and is on the editorial board for CRC's 'Handbook of Chemistry and Physics”. Zwillinger holds a bachelor's degree in mathematics from the Massachusetts Institute of Technology (MIT). He earned his doctorate in applied mathematics from the California Institute of Technology (Caltech). Zwillinger is a certified Raytheon Six Sigma Expert and an ASQ certified Six Sigma Black Belt. He also holds a pilot's license.

Autoren/Hrsg.

Zwillinger, Daniel

Inhaltsverzeichnis

1;Front Cover;1
2;Handbook of Differential Equations;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;12
6;Introduction;14
7;How to Use This Book;18
8;Part I.A: Definitions and Concepts;22
8.1;Chapter 1. Definition of Terms;22
8.1.1;Analytic;22
8.1.2;Asymptotic equivalence;22
8.1.3;Asymptotic expansions;22
8.1.4;Asymptotic series;23
8.1.5;Autonomous;23
8.1.6;Bifurcation;23
8.1.7;Boundary data;23
8.1.8;Boundary layer;23
8.1.9;Boundary value problem;23
8.1.10;Characteristics;23
8.1.11;Cauchy-Kowalewski theorem;24
8.1.12;Cauchy problem;24
8.1.13;Commutator;24
8.1.14;Complete;24
8.1.15;Complete system;24
8.1.16;Conservation form;24
8.1.17;Consistency;25
8.1.18;Coupled systems of equations;25
8.1.19;Degree;25
8.1.20;Dirichlet problem;25
8.1.21;Eigenvalues, Eigenfunctions;25
8.1.22;Elliptic operator;25
8.1.23;Euler-Lagrange equation;25
8.1.24;First integral;26
8.1.25;Fréchet derivative,
Gâteaux derivative;26
8.1.26;Fuchsian equation;26
8.1.27;Fundamental matrix;26
8.1.28;General solution;27
8.1.29;Green's function;27
8.1.30;Harmonic functions;27
8.1.31;Hodograph;27
8.1.32;Homogeneous equations - 1;27
8.1.33;Homogeneous equations - 2;27
8.1.34;Ill-posed problems;27
8.1.35;Initial value problem;27
8.1.36;Involutory transformation;27
8.1.37;L2 functions;27
8.1.38;Lagrange bracket;28
8.1.39;Lagrangian derivative;28
8.1.40;Laplacian;28
8.1.41;Leibniz's rule;28
8.1.42;Lie Algebra;28
8.1.43;Limit cycle;28
8.1.44;Linear differential equations;28
8.1.45;Linearize;28
8.1.46;Lipschitz condition;29
8.1.47;Lipschitz condition;29
8.1.48;Maximum principle;29
8.1.49;Mean value theorem;29
8.1.50;Neumann problem;29
8.1.51;Metaparabolic equations;29
8.1.52;Normal form;29
8.1.53;Nonlinear;30
8.1.54;Order of a differential equation;30
8.1.55;Orthogonal;30
8.1.56;Padé approximant;30
8.1.57;Particular solution;30
8.1.58;Poisson bracket;30
8.1.59;Quasi-linear equations;30
8.1.60;Radiation condition;30
8.1.61;Maximum principle;29
8.1.62;Mean value theorem;29
8.1.63;Neumann problem;29
8.1.64;Metaparabolic equations;29
8.1.65;Normal form;29
8.1.66;Nonlinear;30
8.1.67;Order of a differential equation;30
8.1.68;Orthogonal;30
8.1.69;Padé approximant;30
8.1.70;Particular solution;30
8.1.71;Poisson bracket;30
8.1.72;Quasi-linear equations;30
8.1.73;Radiation condition;30
8.1.74;Riemann's P function;31
8.1.75;Robbins Problem;31
8.1.76;Schwarzian derivative;31
8.1.77;Singular points;31
8.1.78;Singular solutions;32
8.1.79;Stability;32
8.1.80;Stefan problem;32
8.1.81;Superposition principle;32
8.1.82;Total differential equations;32
8.1.83;Trivial solution;32
8.1.84;Turning points;32
8.1.85;Weak solution;32
8.1.86;Well-posed problems;33
8.1.87;Wronskian;33
8.1.88;References;33
8.2;Chapter 2. Alternative Theorems;34
8.2.1;Applicable to;34
8.2.2;Idea;34
8.2.3;Procedure;34
8.2.4;Example 2;35
8.2.5;Notes;36
8.2.6;References;36
8.3;Chapter 3. Bifurcation Theory;36
8.3.1;Applicable to;36
8.3.2;Idea;37
8.3.3;Procedure;37
8.3.4;Example 1;37
8.3.5;Example 2;38
8.3.6;References;41
8.4;Chapter 4. A Caveat for Partial Differential Equations;42
8.4.1;Example;42
8.4.2;Note;43
8.4.3;Reference;43
8.5;Chapter 5. Classification of Partial Differential Equations;43
8.5.1;Second Order Equations;46
8.5.2;Hyperbolic Equations;46
8.5.3;Parabolic Equations;48
8.5.4;Elliptic Equations;49
8.5.5;References;50
8.6;Chapter 6. Compatible Systems;50
8.6.1;Applicable to;50
8.6.2;Yield;50
8.6.3;Procedure 1;51
8.6.4;Example;51
8.6.5;Procedure 2;52
8.6.6;Special Case 1;52
8.6.7;Special Case 2;53
8.6.8;Special Case 3;53
8.6.9;Notes;53
8.6.10;References;54
8.7;Chapter 7. Conservation Laws;55
8.7.1;Applicable to;55
8.7.2;Yield;55
8.7.3;Procedure;55
8.7.4;Example 1;56
8.7.5;Example 2;56
8.7.6;Notes;57
8.7.7;References;57
8.8;Chapter 8. Differential Resultants;58
8.8.1;Applicable to;58
8.8.2;Yield;58
8.8.3;Idea;58
8.8.4;Procedure;58
8.8.5;Example;59
8.8.6;Notes;60
8.8.7;References;60
8.9;Chapter 9· Fixed Point Existence Theorems;61
8.9.1;Applicable to;61
8.9.2;Yield;61
8.9.3;Idea;61
8.9.4;Procedure;61
8.9.5;Example;61
8.9.6;References;63
8.10;Chapter 10. Hamilton-Jacobi Theory;64
8.10.1;Applicable to;64
8.10.2;Yield;64
8.10.3;Procedure;64
8.10.4;Example;65
8.10.5;Notes;66
8.10.6;References;66
8.11;Chapter 11. Limit Cycles;66
8.11.1;Applicable to;66
8.11.2;Yield;66
8.11.3;Idea;67
8.11.4;Procedure;67
8.11.5;Example 1;67
8.11.6;Example 2;68
8.11.7;Notes;68
8.11.8;References;70
8.12;Chapter 12. Natural Boundary Conditions for a PDE;70
8.12.1;Applicable to;70
8.12.2;Yield;70
8.12.3;Idea;71
8.12.4;Procedure;71
8.12.5;Example;72
8.12.6;Notes;72
8.12.7;References;72
8.13;Chapter 13. Self-Adjoint Eigenfunction Problems;73
8.13.1;Applicable to;73
8.13.2;Yield;73
8.13.3;Procedure;73
8.13.4;Example 1;75
8.13.5;Case 1;75
8.13.6;Case 2;76
8.13.7;Example 2;76
8.13.8;Example 3;77
8.13.9;Example 4;77
8.13.10;Notes;77
8.13.11;References;78
8.14;Chapter 14. Sturm—Liouville Theory;78
8.14.1;Applicable t o;78
8.14.2;Yield;78
8.14.3;Procedure;79
8.14.4;Example 1;81
8.14.5;Example 2;82
8.14.6;Notes;82
8.14.7;References;82
8.15;Chapter 15. Variational Equations;83
8.15.1;Applicable to;83
8.15.2;Yield;83
8.15.3;Procedure;83
8.15.4;Example 1;84
8.15.5;Example 2;84
8.15.6;Example 3;85
8.15.7;Example 4;85
8.15.8;Example 5;85
8.15.9;Example 6;86
8.15.10;Example 7;86
8.15.11;Notes;87
8.15.12;References;88
8.16;Chapter 16. Well-Posedness of Differential Equations;88
8.16.1;Applicable to;88
8.16.2;Yield;88
8.16.3;Idea;89
8.16.4;Procedure;89
8.16.5;Example;89
8.16.6;Notes;90
8.16.7;References;91
8.17;Chapter 17. Wronskians and Fundamental Solutions;91
8.17.1;Example 1;92
8.17.2;Example 2;93
8.17.3;Notes;93
8.17.4;References;94
9;Part I.B: Transformations;96
9.1;Chapter 18. Canonical Forms;96
9.1.1;Applicable to;96
9.1.2;Idea;97
9.1.3;Procedure 1;97
9.1.4;Procedure 2;97
9.1.5;Procedure 3;98
9.1.6;Procedure 4;98
9.1.7;Notes;99
9.1.8;References;99
9.2;Chapter 19. Canonical Transformations;100
9.2.1;Applicable to;100
9.2.2;Yield;100
9.2.3;Procedure;100
9.2.4;Example;101
9.2.5;Notes;102
9.2.6;References;102
9.3;Chapter 20. Darboux Transformation;103
9.3.1;Applicable to;103
9.3.2;Yield;103
9.3.3;Procedure;103
9.3.4;Example 1;104
9.3.5;Example 2;105
9.3.6;Note;106
9.3.7;References;106
9.4;Chapter 21. An Involutory Transformation;106
9.4.1;Applicable to;106
9.4.2;Yield;106
9.4.3;Idea;106
9.4.4;Procedure;107
9.4.5;Example;108
9.4.6;Note;108
9.4.7;Reference;108
9.5;Chapter 22. Liouville Transformation — 1;109
9.5.1;Applicable to;109
9.5.2;Procedure;109
9.5.3;Example;110
9.5.4;References;111
9.6;Chapter 23· Liouville Transformation — 2;111
9.6.1;Applicable to;111
9.6.2;Procedure;111
9.6.3;Example;112
9.6.4;Notes;112
9.6.5;Reference;112
9.7;Chapter 24. Reduction of Linear ODEs to a First Order System;113
9.7.1;Applicable to;113
9.7.2;Yield;113
9.7.3;Idea;113
9.7.4;Procedure;113
9.7.5;Example;114
9.7.6;Notes;114
9.7.7;References;114
9.8;Chapter 25. Transformations of Second Order Linear ODEs — 1;115
9.8.1;Applicable to;115
9.8.2;Transformation 1;115
9.8.3;Example;115
9.8.4;Transformation 2;115
9.8.5;Example;116
9.8.6;Transformation 3;116
9.8.7;Example;117
9.8.8;Transformation 4;118
9.8.9;Notes;118
9.8.10;References;118
9.9;Chapter 26. Transformations of Second Order Linear ODEs — 2;119
9.9.1;Applicable to;119
9.9.2;Transformation 1;119
9.9.3;Transformation 2;119
9.9.4;Transformation 3;120
9.9.5;Transformation 4;120
9.9.6;Notes;121
9.9.7;References;121
9.10;Chapter 27. Transformation of an ODE to an Integral Equation;121
9.10.1;Applicable
to;121
9.10.2;Yield;121
9.10.3;Idea;121
9.10.4;Procedure;122
9.10.5;Example;123
9.10.6;Notes;123
9.10.7;References;123
9.11;Chapter 28.
Miscellaneous ODE Transformations;124
9.11.1;Transformation 1;124
9.11.2;Example;125
9.11.3;Transformation 2;125
9.11.4;Transformation 3;126
9.11.5;Transformation 4;126
9.11.6;Notes;127
9.11.7;References;127
9.12;Chapter 29. Reduction of PDEs to a First Order System;127
9.12.1;Applicable to;127
9.12.2;Yield;127
9.12.3;Idea;127
9.12.4;Procedure;128
9.12.5;Notes;129
9.12.6;Reference;129
9.13;Chapter 30· Transforming Partial Differential
Equations;130
9.13.1;Applicable to;130
9.13.2;Idea;130
9.13.3;Procedure;130
9.13.4;Example;130
9.13.5;Notes;132
9.13.6;References;133
9.14;Chapter 31. Transformations of Partial Differential Equations;134
9.14.1;Kirchoff Transformation;134
9.14.2;Transformations of Parabolic Differential Equations;134
9.14.3;Removing First Derivative Terms;135
9.14.4;Von Mises Transformation;135
9.14.5;Notes;136
9.14.6;References;136
10;Part Il: Exact Analytical Methods;138
10.1;Chapter 32. Introduction to Exact Analytical Methods;138
10.1.1;Most Useful Methods for ODEs;139
10.1.2;Most Useful Methods for PDEs;139
10.2;Chapter 33. Look Up Technique;139
10.2.1;Ordinary Differential Equations;141
10.2.2;Partial Differential Equations;150
10.2.3;Systems of Differential Equations;157
10.2.4;References;161
11;Part Il.A: Exact Methods for
ODEs;168
11.1;Chapter 34. An N-th Order Equation;168
11.1.1;Applicable to;168
11.1.2;Yield;168
11.1.3;Idea;168
11.1.4;Procedure;169
11.1.5;Example;169
11.1.6;Example;169
11.1.7;Note;170
11.1.8;Reference;170
11.2;Chapter 35. Use of the Adjoint
Equation;170
11.2.1;Applicable to;170
11.2.2;Yield;170
11.2.3;Idea;170
11.2.4;Procedure;170
11.2.5;Example;172
11.2.6;Notes;173
11.2.7;References;174
11.3;Chapter 36. Autonomous Equations;174
11.3.1;Applicable to;174
11.3.2;Yield;174
11.3.3;Idea;174
11.3.4;Procedure;174
11.3.5;Example;175
11.3.6;Table 36;176
11.3.7;Notes;176
11.3.8;References;177
11.4;Chapter 37. Bernoulli Equation;178
11.4.1;Applicable to;178
11.4.2;Yield;178
11.4.3;Idea;178
11.4.4;Procedure;178
11.4.5;Example;179
11.4.6;References;179
11.5;Chapter 38. Clairaut's Equation;179
11.5.1;Applicable to;179
11.5.2;Yield;179
11.5.3;Idea;180
11.5.4;Procedure;180
11.5.5;Example;180
11.5.6;Notes;181
11.5.7;References;181
11.6;Chapter 39. Computer-Aided Solution;181
11.6.1;Applicable to;181
11.6.2;Yield;181
11.6.3;Idea;181
11.6.4;Procedure;181
11.6.5;Example 1;182
11.6.6;Example 2;183
11.6.7;Example 3;183
11.6.8;Example 4;184
11.6.9;Example 5;184
11.6.10;Notes;185
11.6.11;References;185
11.7;Chapter 40. Constant Coefficient Linear Equations;186
11.7.1;Applicable to;186
11.7.2;Yield;186
11.7.3;Idea;186
11.7.4;Procedure;186
11.7.5;Example;187
11.7.6;References;187
11.8;Chapter 41. Contact Transformation;187
11.8.1;Applicable to;187
11.8.2;Yield;187
11.8.3;Idea;188
11.8.4;Procedure;188
11.8.5;Example;189
11.8.6;Notes;189
11.8.7;References;190
11.9;Chapter 42. Delay Equations;191
11.9.1;Applicable to;191
11.9.2;Yield;191
11.9.3;Idea;191
11.9.4;Procedure;191
11.9.5;Example 1;192
11.9.6;Notes;194
11.9.7;References;195
11.10;Chapter 43. Dependent Variable Missing;195
11.10.1;Applicable
to;195
11.10.2;Yield;195
11.10.3;Idea;196
11.10.4;Procedure;196
11.10.5;Example;196
11.10.6;Note;196
11.10.7;References;197
11.11;Chapter 44. Differentiation Method;197
11.11.1;Applicable to;197
11.11.2;Yield;197
11.11.3;Idea;197
11.11.4;Procedure;197
11.11.5;Example;197
11.11.6;Notes;198
11.11.7;Reference;198
11.12;Chapter 45.
Differential Equations with Discontinuities;198
11.12.1;Applicable to;198
11.12.2;Yield;198
11.12.3;Idea;199
11.12.4;Procedure;199
11.12.5;Example;200
11.12.6;Notes;201
11.12.7;References;201
11.13;Chapter 46. Eigenfunction
Expansions;202
11.13.1;Applicable to;202
11.13.2;Yield;202
11.13.3;Idea;202
11.13.4;Procedure;202
11.13.5;Example 1;204
11.13.6;Example 2;206
11.13.7;Example 3;207
11.13.8;Notes;208
11.13.9;References;209
11.14;Chapter 47. Equidimensional-In-x Equations;209
11.14.1;Applicable to;209
11.14.2;Yield;209
11.14.3;Idea;209
11.14.4;Procedure;210
11.14.5;Example;210
11.14.6;Table 47;211
11.14.7;Reference;212
11.15;Chapter 48. Equidimensional-In-y Equations;212
11.15.1;Applicable to;212
11.15.2;Yield;212
11.15.3;Idea;212
11.15.4;Procedure;212
11.15.5;Example;213
11.15.6;Table 48;214
11.15.7;Notes;214
11.15.8;Reference;214
11.16;Chapter 49. Euler Equations;214
11.16.1;Applicable to;214
11.16.2;Yield;214
11.16.3;Idea;214
11.16.4;Procedure;215
11.16.5;Example;215
11.16.6;Table 49;215
11.16.7;Notes;216
11.16.8;References;216
11.17;Chapter 50. Exact First Order Equations;216
11.17.1;Applicable
to;216
11.17.2;Yield;216
11.17.3;Idea;216
11.17.4;Procedure;217
11.17.5;Example;218
11.17.6;References;218
11.18;Chapter 51. Exact Second Order Equations;219
11.18.1;Applicable to;219
11.18.2;Yield;219
11.18.3;Idea;219
11.18.4;Procedure;219
11.18.5;Example;220
11.18.6;Notes;221
11.18.7;References;221
11.19;Chapter 52. Exact N-th Order Equations;222
11.19.1;Applicable to;222
11.19.2;Yield;222
11.19.3;Idea;222
11.19.4;Procedure;222
11.19.5;Special Case;222
11.19.6;Example;223
11.19.7;References;223
11.20;Chapter 53. Factoring
Equations;223
11.20.1;Applicable to;223
11.20.2;Yield;223
11.20.3;Idea;224
11.20.4;Procedure;224
11.20.5;Example;224
11.20.6;Note;224
11.20.7;References;224
11.21;Chapter 54. Factoring
Operators;225
11.21.1;Applicable to;225
11.21.2;Yield;225
11.21.3;Idea;225
11.21.4;Procedure;225
11.21.5;Example 1;225
11.21.6;Example 2;226
11.21.7;Example 3;227
11.21.8;Notes;228
11.21.9;References;229
11.22;Chapter 55· Factorization Method;230
11.22.1;Applicable
to;230
11.22.2;Yield;230
11.22.3;Idea;230
11.22.4;Procedure;231
11.22.5;Example;231
11.22.6;Notes;232
11.22.7;References;232
11.23;Chapter 56. Fokker—Planck
Equation;233
11.23.1;Applicable to;233
11.23.2;Yield;233
11.23.3;Idea;233
11.23.4;Procedure;233
11.23.5;Special Case;234
11.23.6;Example;235
11.23.7;Notes;236
11.23.8;References;237
11.24;Chapter
57. Fractional Differential Equations;237
11.24.1;Applicable to;237
11.24.2;Yield;237
11.24.3;Idea;237
11.24.4;Procedure;237
11.24.5;Example 1;238
11.24.6;Example 2;239
11.24.7;Notes;239
11.24.8;References;240
11.25;Chapter 58. Free Boundary Problems;240
11.25.1;Applicable to;240
11.25.2;Idea;240
11.25.3;Procedure;240
11.25.4;Example;240
11.25.5;Notes;242
11.25.6;References;243
11.26;Chapter 59. Generating
Functions;243
11.26.1;Applicable to;243
11.26.2;Yield;243
11.26.3;Idea;243
11.26.4;Procedure;244
11.26.5;Example;244
11.26.6;Notes;246
11.26.7;References;246
11.27;Chapter 60. Green's
Functions;246
11.27.1;Applicable to;246
11.27.2;Yield;246
11.27.3;Idea;246
11.27.4;Procedure;246
11.27.5;Method 1;248
11.27.6;Method 2;249
11.27.7;Example 1;249
11.27.8;Example 2;250
11.27.9;Table 60;251
11.27.10;Notes;252
11.27.11;References;254
11.28;Chapter 61. Homogeneous Equations;254
11.28.1;Applicable to;254
11.28.2;Yield;254
11.28.3;Idea;254
11.28.4;Procedure;254
11.28.5;Example;255
11.28.6;Notes;256
11.28.7;References;256
11.29;Chapter 62. Method of
Images;257
11.29.1;Applicable to;257
11.29.2;Yield;257
11.29.3;Idea;257
11.29.4;Procedure;257
11.29.5;Example;257
11.29.6;Notes;259
11.29.7;References;259
11.30;Chapter 63. Integrable Combinations;259
11.30.1;Applicable to;259
11.30.2;Yield;259
11.30.3;Idea;259
11.30.4;Procedure;259
11.30.5;Example 1;260
11.30.6;Example 2;260
11.30.7;Note;261
11.30.8;Reference;261
11.31;Chapter 64.
Integral Representations: Laplace's Method;261
11.31.1;Applicable to;261
11.31.2;Yield;261
11.31.3;Idea;261
11.31.4;Procedure;261
11.31.5;Special Case;262
11.31.6;Example;263
11.31.7;Notes;265
11.31.8;References;266
11.32;Chapter
65. Integral Transforms: Finite Intervals;266
11.32.1;Applicable to;266
11.32.2;Idea;267
11.32.3;Procedure;267
11.32.4;Example;267
11.32.5;Table 65;269
11.32.6;Notes;270
11.32.7;References;271
11.33;Chapter 66. Integral Transforms: Infinite Intervals;271
11.33.1;Applicable to;271
11.33.2;Idea;271
11.33.3;Procedure;271
11.33.4;Warning;272
11.33.5;Example 1;272
11.33.6;Example 2;274
11.33.7;Table 66;275
11.33.8;Notes;277
11.33.9;References;279
11.34;Chapter 67. Integrating
Factors;280
11.34.1;Applicable to;280
11.34.2;Yield;280
11.34.3;Idea;280
11.34.4;Procedure;280
11.34.5;Example;281
11.34.6;Special Case;282
11.34.7;Notes;282
11.34.8;References;283
11.35;Chapter 68. Interchanging Dependent and Independent Variables;284
11.35.1;Applicable to;284
11.35.2;Yield;284
11.35.3;Idea;284
11.35.4;Procedure;284
11.35.5;Example 1;285
11.35.6;Example 2;285
11.35.7;Table 68;286
11.35.8;Note;286
11.35.9;References;286
11.36;Chapter 69. Lagrange's Equation;286
11.36.1;Applicable to;286
11.36.2;Yield;286
11.36.3;Idea;286
11.36.4;Procedure;287
11.36.5;Example 1;288
11.36.6;Example 2;289
11.36.7;Notes;289
11.36.8;References;289
11.37;Chapter 70. Lie Groups: ODEs;290
11.37.1;Applicable to;290
11.37.2;Yield;290
11.37.3;Idea;290
11.37.4;Procedure;290
11.37.5;Example;292
11.37.6;Notes;293
11.37.7;References;295
11.38;Chapter 71. Operational
Calculus;296
11.38.1;Applicable to;296
11.38.2;Yield;296
11.38.3;Idea;296
11.38.4;Procedure;297
11.38.5;Example 1;298
11.38.6;Example 2;299
11.38.7;Notes;299
11.38.8;References;300
11.39;Chapter 72. Pfaffian Differential Equations;300
11.39.1;Applicable
to;300
11.39.2;Yield;300
11.39.3;Idea;300
11.39.4;Example;301
11.39.5;Procedure 1;301
11.39.6;Example 1;302
11.39.7;Procedure 2;302
11.39.8;Example 2;303
11.39.9;Procedure 3;303
11.39.10;Example 3;303
11.39.11;Notes;304
11.39.12;References;304
11.40;Chapter 73. Prüfer Substitution;305
11.40.1;Applicable to;305
11.40.2;Yield;305
11.40.3;Idea;305
11.40.4;Procedure;305
11.40.5;Example;306
11.40.6;Note;306
11.40.7;References;307
11.41;Chapter 74. Reduction of Order;307
11.41.1;Applicable to;307
11.41.2;Yield;307
11.41.3;Idea;307
11.41.4;Procedure;307
11.41.5;Example;308
11.41.6;Note;308
11.41.7;References;309
11.42;Chapter 75. Riccati Equation — 1;309
11.42.1;Applicable to;309
11.42.2;Yield;309
11.42.3;Idea;309
11.42.4;Procedure;309
11.42.5;Example;310
11.42.6;Notes;310
11.42.7;References;310
11.43;Chapter 76. Riccati Equation - 2;311
11.43.1;Applicable to;311
11.43.2;Yield;311
11.43.3;Idea;311
11.43.4;Procedure;311
11.43.5;Example;311
11.43.6;References;312
11.44;Chapter 77. Matrix Riccati Equations;312
11.44.1;Applicable to;312
11.44.2;Yield;312
11.44.3;Idea;312
11.44.4;Procedure;312
11.44.5;References;314
11.45;Chapter 78. Scale Invariant Equations;315
11.45.1;Applicable to;315
11.45.2;Yield;315
11.45.3;Idea;315
11.45.4;Procedure;315
11.45.5;Example;316
11.45.6;Notes;317
11.45.7;References;317
11.46;Chapter 79. Separable Equations;317
11.46.1;Applicable to;317
11.46.2;Yield;317
11.46.3;Idea;318
11.46.4;Procedure;318
11.46.5;Example;318
11.46.6;Notes;318
11.46.7;References;319
11.47;Chapter 80. Series
Solution;319
11.47.1;Applicable to;319
11.47.2;Yield;319
11.47.3;Idea;319
11.47.4;Procedure;319
11.47.5;Example 1;321
11.47.6;Example 2;322
11.47.7;Notes;322
11.47.8;References;324
11.48;Chapter 81· Equations Solvable for x;325
11.48.1;Applicable to;325
11.48.2;Yield;325
11.48.3;Idea;325
11.48.4;Procedure;325
11.48.5;Example;326
11.48.6;Reference;326
11.49;Chapter 82. Equations Solvable for y;326
11.49.1;Applicable to;326
11.49.2;Yield;326
11.49.3;Idea;326
11.49.4;Procedure;327
11.49.5;Note;327
11.49.6;Reference;327
11.50;Chapter 83.
Superposition;328
11.50.1;Applicable to;328
11.50.2;Yield;328
11.50.3;Idea;328
11.50.4;Procedure;328
11.50.5;Example 1;328
11.50.6;Example 2;329
11.50.7;Notes;329
11.50.8;References;330
11.51;Chapter 84.
Method of Undetermined Coefficients;330
11.51.1;Applicable to;330
11.51.2;Yield;330
11.51.3;Idea;330
11.51.4;Procedure;331
11.51.5;Example 1;331
11.51.6;Example 2;332
11.51.7;Example 3;332
11.51.8;Notes;332
11.51.9;References;333
11.52;Chapter 85. Variation of Parameters;333
11.52.1;Applicable to;333
11.52.2;Yield;333
11.52.3;Idea;333
11.52.4;Procedure;333
11.52.5;Example;334
11.52.6;Notes;335
11.52.7;References;336
11.53;Chapter 86. Vector Ordinary Differential Equations;336
11.53.1;Applicable to;336
11.53.2;Yield;336
11.53.3;Idea;336
11.53.4;Procedure;336
11.53.5;Example;337
11.53.6;Notes;339
12;Part II.B: Exact Methods for PDEs;342
12.1;Chapter 87. Bäcklund Transformations;342
12.1.1;Applicable to;342
12.1.2;Yield;342
12.1.3;Idea;342
12.1.4;Procedure;343
12.1.5;Example 1;343
12.1.6;Example 2;344
12.1.7;Notes;345
12.1.8;References;345
12.2;Chapter 88. Method of Characteristics;346
12.2.1;Applicable to;346
12.2.2;Yield;346
12.2.3;Idea;346
12.2.4;Procedure;346
12.2.5;Example 1;348
12.2.6;Example 2;349
12.2.7;Notes;350
12.2.8;References;351
12.2.9;89. Characteristic Strip Equations;351
12.2.10;Applicable
to;351
12.3;Chapter
89. Characteristic Strip Equations;351
12.3.1;Applicable
to;351
12.3.2;Yield;351
12.3.3;Idea;352
12.3.4;Procedure;352
12.3.5;Example;352
12.3.6;Notes;354
12.3.7;References;354
12.4;Chapter 90. Conformai Mappings;354
12.4.1;Applicable to;354
12.4.2;Yield;354
12.4.3;Idea;354
12.4.4;Procedure;354
12.4.5;Example;355
12.4.6;References;357
12.5;Chapter 91. Method of Descent;358
12.5.1;Applicable to;358
12.5.2;Yield;358
12.5.3;Idea;358
12.5.4;Procedure;358
12.5.5;Example;358
12.5.6;Notes;359
12.5.7;References;360
12.6;Chapter 92. Diagonalization of a Linear System of PDEs;360
12.6.1;Applicable to;360
12.6.2;Yield;360
12.6.3;Idea;360
12.6.4;Procedure;360
12.6.5;Example;361
12.6.6;Reference;362
12.7;Chapter 93. Duhamel's
Principle;362
12.7.1;Applicable to;362
12.7.2;Yield;362
12.7.3;Idea;363
12.7.4;Procedure;363
12.7.5;Example;364
12.7.6;Note;365
12.7.7;References;365
12.8;Chapter 94. Hodograph Transformation;365
12.8.1;Applicable to;365
12.8.2;Yield;365
12.8.3;Idea;366
12.8.4;Procedure;366
12.8.5;Example;366
12.8.6;Notes;367
12.8.7;References;367
12.9;Chapter 95. Inverse Scattering;367
12.9.1;Applicable to;367
12.9.2;Yield;367
12.9.3;Idea;368
12.9.4;Procedure;368
12.9.5;Example 1;369
12.9.6;Example 2;370
12.9.7;Notes;370
12.9.8;References;370
12.10;Chapter 96. Jacobi's Method;371
12.10.1;Applicable to;371
12.10.2;Yield;371
12.10.3;Idea;371
12.10.4;Procedure;371
12.10.5;Example;372
12.10.6;Notes;373
12.10.7;References;373
12.11;Chapter 97. Legendre Transformation;374
12.11.1;Applicable to;374
12.11.2;Yield;374
12.11.3;Idea;374
12.11.4;Procedure;374
12.11.5;Summary;375
12.11.6;Example;375
12.11.7;References;377
12.12;Chapter 98. Lie Groups: PDEs;377
12.12.1;Applicable to;377
12.12.2;Yield;377
12.12.3;Idea;378
12.12.4;Procedure;378
12.12.5;Example;379
12.12.6;Special Case 2;381
12.12.7;Notes;381
12.12.8;References;382
12.13;Chapter 99· Poisson Formula;383
12.13.1;Yield;383
12.13.2;Idea;383
12.13.3;Procedure;383
12.13.4;Example;384
12.13.5;Notes;384
12.13.6;References;386
12.14;Chapter 100. Riemann's Method;386
12.14.1;Applicable to;386
12.14.2;Yield;386
12.14.3;Idea;386
12.14.4;Procedure;386
12.14.5;Example 1;388
12.14.6;Example 2;390
12.14.7;Notes;390
12.14.8;References;390
12.15;Chapter 101. Separation of Variables;391
12.15.1;Applicable to;391
12.15.2;Yield;391
12.15.3;Idea;391
12.15.4;Procedure;391
12.15.5;Example 1;392
12.15.6;Example 2;394
12.15.7;Notes;394
12.15.8;References;395
12.16;Chapter 102· Similarity Methods;396
12.16.1;Applicable to;396
12.16.2;Yield;396
12.16.3;Idea;396
12.16.4;Procedure;396
12.16.5;Example;397
12.16.6;Notes;399
12.16.7;References;399
12.17;Chapter 103· Exact Solutions to the Wave Equation;400
12.17.1;Applicable to;400
12.17.2;Yield;400
12.17.3;Idea;400
12.17.4;Procedure;400
12.17.5;Special Case 1;401
12.17.6;Special Case 2;401
12.17.7;Special Case 3;402
12.17.8;Example;402
12.17.9;Notes;402
12.17.10;References;403
12.18;Chapter 104. Wiener–Hopf
Technique;404
12.18.1;Applicable to;404
12.18.2;Yield;404
12.18.3;Idea;404
12.18.4;Procedure;404
12.18.5;Example;405
12.18.6;References;407
13;Part III: Approximate Analytical Methods;408
13.1;Chapter 105. Introduction to Approximate Analysis;408
13.1.1;Most Useful Methods for Differential Equations;409
13.2;Chapter 106· Chaplygin's Method;409
13.2.1;Applicable to;409
13.2.2;Yield;409
13.2.3;Idea;409
13.2.4;Procedure;410
13.2.5;Technique 1;410
13.2.6;Technique 2;411
13.2.7;Example;411
13.2.8;Notes;412
13.2.9;References;412
13.3;Chapter 107. Collocation;412
13.3.1;Applicable to;412
13.3.2;Yield;412
13.3.3;Idea;412
13.3.4;Procedure;413
13.3.5;Example;413
13.3.6;Notes;414
13.3.7;References;414
13.4;Chapter 108. Dominant Balance;414
13.4.1;Applicable to;414
13.4.2;Yield;414
13.4.3;Idea;414
13.4.4;Procedure;414
13.4.5;Example;415
13.4.6;Note;416
13.4.7;References;416
13.5;Chapter 109. Equation Splitting;416
13.5.1;Applicable to;416
13.5.2;Yield;416
13.5.3;Idea;417
13.5.4;Procedure;417
13.5.5;Example;417
13.5.6;Notes;418
13.5.7;References;418
13.6;Chapter 110. Equivalent Linearization;419
13.6.1;Applicable to;419
13.6.2;Yield;419
13.6.3;Idea;419
13.6.4;Procedure;419
13.6.5;Example;420
13.6.6;Notes;421
13.6.7;References;422
13.7;Chapter 111. Equivalent Nonlinearization;422
13.7.1;Applicable to;422
13.7.2;Yield;422
13.7.3;Idea;422
13.7.4;Procedure;423
13.7.5;Example;424
13.7.6;Notes;425
13.7.7;References;425
13.8;Chapter 112. Floquet Theory;425
13.8.1;Applicable to;425
13.8.2;Yield;425
13.8.3;Idea;426
13.8.4;Procedure;426
13.8.5;Example;427
13.8.6;Notes;428
13.8.7;References;428
13.9;Chapter 113. Graphical Analysis: The Phase Plane;429
13.9.1;Applicable to;429
13.9.2;Yield;429
13.9.3;Idea;429
13.9.4;Procedure;429
13.9.5;Example;430
13.9.6;Notes;432
13.9.7;References;433
13.10;Chapter 114. Graphical Analysis: The Tangent Field;433
13.10.1;Applicable to;433
13.10.2;Yield;433
13.10.3;Idea;434
13.10.4;Procedure;434
13.10.5;Example 1;434
13.10.6;Example 2;435
13.10.7;Note;435
13.10.8;References;436
13.11;Chapter 115. Harmonic Balance;436
13.11.1;Applicable to;436
13.11.2;Yield;436
13.11.3;Idea;437
13.11.4;Procedure;437
13.11.5;Example 1;437
13.11.6;Example 2;438
13.11.7;Notes;439
13.11.8;References;439
13.12;Chapter 116. Homogenization;439
13.12.1;Applicable to;439
13.12.2;Yield;439
13.12.3;Idea;439
13.12.4;Procedure;440
13.12.5;Example;440
13.12.6;Notes;441
13.12.7;References;441
13.13;Chapter 117. Integral Methods;442
13.13.1;Applicable to;442
13.13.2;Yield;442
13.13.3;Idea;442
13.13.4;Procedure;442
13.13.5;Example;442
13.13.6;References;444
13.14;Chapter 118. Interval Analysis;445
13.14.1;Applicable to;445
13.14.2;Yield;445
13.14.3;Idea;445
13.14.4;Procedure;445
13.14.5;Example;446
13.14.6;Notes;447
13.14.7;References;447
13.15;Chapter 119. Least Squares Method;448
13.15.1;Applicable to;448
13.15.2;Yield;448
13.15.3;Idea;448
13.15.4;Procedure;448
13.15.5;Example;449
13.15.6;Notes;450
13.15.7;Reference;450
13.16;Chapter 120. Liapunov Functions;450
13.16.1;Applicable to;450
13.16.2;Yield;450
13.16.3;Idea;450
13.16.4;Procedure;450
13.16.5;Example 1;450
13.16.6;Example 2;451
13.16.7;Notes;452
13.16.8;References;453
13.17;Chapter 121. Maximum Principles;453
13.17.1;Applicable to;453
13.17.2;Yield;453
13.17.3;Idea;453
13.17.4;Procedure;453
13.17.5;Example;454
13.17.6;Notes;455
13.17.7;References;456
13.18;Chapter 122· McGarvey Iteration Technique;456
13.18.1;Applicable to;456
13.18.2;Yield;456
13.18.3;Idea;457
13.18.4;Procedure;457
13.18.5;Example;457
13.18.6;Notes;458
13.18.7;Reference;458
13.19;Chapter 123. Moment Equations: Closure;459
13.19.1;Applicable to;459
13.19.2;Yield;459
13.19.3;Idea;459
13.19.4;Procedure;459
13.19.5;Example;460
13.19.6;Notes;461
13.19.7;References;462
13.20;Chapter 124. Moment Equations:
Ito Calculus;463
13.20.1;Applicable to;463
13.20.2;Yield;463
13.20.3;Idea;463
13.20.4;Procedure;463
13.20.5;Example;464
13.20.6;Notes;465
13.20.7;References;465
13.21;Chapter 125. Mongers Method;465
13.21.1;Applicable to;465
13.21.2;Yield;465
13.21.3;Idea;465
13.21.4;Procedure;465
13.21.5;Example;466
13.21.6;Note;468
13.21.7;References;468
13.22;Chapter 126. Newton's Method;468
13.22.1;Applicable to;468
13.22.2;Yield;468
13.22.3;Idea;468
13.22.4;Procedure;468
13.22.5;Special Case;469
13.22.6;Example;469
13.22.7;Notes;470
13.22.8;References;471
13.23;Chapter 127. Padé
Approximants;471
13.23.1;Applicable to;471
13.23.2;Yield;471
13.23.3;Idea;471
13.23.4;Procedure;471
13.23.5;Example 1;472
13.23.6;Example 2;473
13.23.7;Notes;473
13.23.8;References;474
13.24;Chapter 128. Perturbation Method: Method of Averaging;475
13.24.1;Applicable to;475
13.24.2;Yield;475
13.24.3;Idea;475
13.24.4;Procedure;475
13.24.5;Example 1;477
13.24.6;Note;477
13.24.7;References;477
13.25;Chapter 129. Perturbation Method: Boundary Layer Method;478
13.25.1;Applicable to;478
13.25.2;Yield;478
13.25.3;Idea;478
13.25.4;Procedure;478
13.25.5;Example;478
13.25.6;Notes;481
13.25.7;References;482
13.26;Chapter 130. Perturbation Method: Functional Iteration;483
13.26.1;Applicable to;483
13.26.2;Yield;483
13.26.3;Idea;483
13.26.4;Procedure;483
13.26.5;Example 1;485
13.26.6;Example 2;487
13.26.7;Notes;489
13.26.8;References;489
13.27;Chapter 131. Perturbation Method: Multiple Scales;490
13.27.1;Applicable to;490
13.27.2;Yield;490
13.27.3;Idea;490
13.27.4;Procedure;490
13.27.5;Example;490
13.27.6;Notes;493
13.27.7;References;493
13.28;Chapter 132· Perturbation Method: Regular Perturbation;494
13.28.1;Applicable to;494
13.28.2;Yield;494
13.28.3;Idea;494
13.28.4;Procedure;494
13.28.5;Example;494
13.28.6;Notes;496
13.28.7;References;497
13.29;Chapter 133. Perturbation Method: Strained Coordinates;497
13.29.1;Applicable to;497
13.29.2;Yield;497
13.29.3;Idea;497
13.29.4;Procedure;497
13.29.5;Example;498
13.29.6;Notes;500
13.29.7;References;500
13.30;Chapter 134· Picard Iteration;501
13.30.1;Applicable to;501
13.30.2;Yield;501
13.30.3;Idea;501
13.30.4;Procedure;501
13.30.5;Example;502
13.30.6;Notes;502
13.30.7;References;503
13.31;Chapter 135. Modified Prüfer Substitution;503
13.31.1;Applicable to;503
13.31.2;Yield;503
13.31.3;Idea;504
13.31.4;Procedure;504
13.31.5;Example;504
13.31.6;Note;505
13.31.7;References;505
13.32;Chapter 136. Reversion Method;505
13.32.1;Applicable to;505
13.32.2;Yield;505
13.32.3;Idea;505
13.32.4;Procedure;506
13.32.5;Example;506
13.32.6;Notes;507
13.32.7;References;507
13.33;Chapter 137. Singular Solutions;507
13.33.1;Applicable to;507
13.33.2;Yield;507
13.33.3;Idea;507
13.33.4;Procedure;508
13.33.5;Example;509
13.33.6;References;510
13.34;Chapter 138. Soliton Type Solutions;510
13.34.1;Applicable to;510
13.34.2;Yield;510
13.34.3;Idea;510
13.34.4;Procedure;510
13.34.5;Example;511
13.34.6;Notes;512
13.34.7;References;512
13.35;Chapter 139. Stochastic Limit Theorems;512
13.35.1;Applicable to;512
13.35.2;Yield;512
13.35.3;Idea;513
13.35.4;Procedure;513
13.35.5;Example;513
13.35.6;References;514
13.36;Chapter 140. Taylor Series Solutions;515
13.36.1;Applicable to;515
13.36.2;Yield;515
13.36.3;Idea;515
13.36.4;Procedure;515
13.36.5;Example;516
13.36.6;Notes;517
13.36.7;References;517
13.37;Chapter 141. Variational Method: Eigenvalue Approximation;518
13.37.1;Applicable to;518
13.37.2;Yield;518
13.37.3;Idea;518
13.37.4;Procedure;518
13.37.5;Example;519
13.37.6;Notes;520
13.37.7;References;520
13.38;Chapter 142· Variational Method: Rayleigh—Ritz;521
13.38.1;Applicable to;521
13.38.2;Yield;521
13.38.3;Idea;521
13.38.4;Procedure;521
13.38.5;Example 1;522
13.38.6;Notes;524
13.38.7;References;524
13.39;Chapter 143. WKB Method;525
13.39.1;Applicable to;525
13.39.2;Yield;525
13.39.3;Idea;525
13.39.4;Procedure;525
13.39.5;Special Case;526
13.39.6;Example;527
13.39.7;Notes;527
13.39.8;References;528
14;Part IV.A: Numerical Methods: Concepts;530
14.1;Chapter 144. Introduction to Numerical Methods;530
14.1.1;Most Useful Methods for ODEs;532
14.1.2;Most Useful Methods for PDEs;532
14.1.3;References;532
14.2;Chapter 145· Definition of Terms for Numerical Methods;533
14.2.1;A-stable;533
14.2.2;Computational molecule;533
14.2.3;Consistency of a finite difference scheme;534
14.2.4;Conservative scheme;534
14.2.5;Difference scheme;534
14.2.6;Explicit method;535
14.2.7;Grid;535
14.2.8;Implicit method;535
14.2.9;Mesh;535
14.2.10;Order of a numerical method;535
14.2.11;Step size;535
14.2.12;Stiff equations;535
14.2.13;Stiffly stable;535
14.2.14;Truncation error;536
14.3;Chapter 146. Courant-Friedrichs-Lewy Consistency Criterion;536
14.3.1;Applicable to;536
14.3.2;Yield;536
14.3.3;Idea;536
14.3.4;Procedure;536
14.3.5;Example;537
14.3.6;Notes;538
14.3.7;References;539
14.4;Chapter 147· Finite Difference Schemes for ODEs;539
14.4.1;Applicable to;539
14.4.2;Yield;539
14.4.3;Procedure;539
14.4.4;Notes;542
14.4.5;References;543
14.5;Chapter 148. Richardson Extrapolation;544
14.5.1;Applicable to;544
14.5.2;Yield;544
14.5.3;Procedure;544
14.5.4;Example;544
14.5.5;Notes;545
14.5.6;References;545
14.6;Chapter 149. Software Libraries;546
14.6.1;Notes;547
14.6.2;References;547
14.6.3;150. Von Neumann Test;548
14.6.4;Applicable to;548
14.6.5;Yield;548
14.6.6;Procedure;548
14.6.7;Example;549
14.6.8;Notes;549
14.6.9;References;549
15;Part IV.B:
Numerical Methods for ODEs;550
15.1;Chapter 151. Analytic
Continuation;550
15.1.1;Applicable to;550
15.1.2;Yield;550
15.1.3;Idea;550
15.1.4;Procedure;551
15.1.5;Example;551
15.1.6;References;553
15.2;Chapter 152· Boundary Value Problems: Box Method;553
15.2.1;Applicable to;553
15.2.2;Yield;553
15.2.3;Idea;553
15.2.4;Procedure;554
15.2.5;Notes;557
15.2.6;References;557
15.3;Chapter 153.
Boundary Value Problems: Shooting Method;558
15.3.1;Applicable to;558
15.3.2;Yield;558
15.3.3;Idea;558
15.3.4;Procedure;558
15.3.5;Example;559
15.3.6;Program 153;560
15.3.7;Notes;561
15.3.8;References;561
15.4;Chapter 154. Continuation
Method;562
15.4.1;Applicable to;562
15.4.2;Yield;562
15.4.3;Idea;562
15.4.4;Procedure;562
15.4.5;Example;562
15.4.6;Notes;564
15.4.7;References;564
15.5;Chapter 155. Continued Fractions;564
15.5.1;Applicable
to;564
15.5.2;Yield;564
15.5.3;Idea;565
15.5.4;Procedure;565
15.5.5;Example;566
15.5.6;Notes;566
15.5.7;References;567
15.6;Chapter 156.
Cosine Method;567
15.6.1;Applicable to;567
15.6.2;Yield;567
15.6.3;Idea;567
15.6.4;Procedure;568
15.6.5;Example;569
15.6.6;Program 156;570
15.6.7;Notes;571
15.6.8;References;571
15.7;Chapter 157. Differential Algebraic Equations;571
15.7.1;Applicable to;571
15.7.2;Yield;571
15.7.3;Idea;572
15.7.4;Example 1;572
15.7.5;Example 2;572
15.7.6;Example 3;573
15.7.7;Notes;573
15.7.8;References;574
15.8;Chapter 158. Finite Element
Method;574
15.8.1;Applicable to;574
15.8.2;Yield;574
15.8.3;Procedure;574
15.8.4;Example 2;578
15.8.5;Notes;580
15.8.6;References;581
15.9;Chapter 159. Forward Euler's Method;582
15.9.1;Applicable to;582
15.9.2;Yield;582
15.9.3;Idea;582
15.9.4;Procedure;582
15.9.5;Example;583
15.9.6;Program 159;583
15.9.7;Notes;584
15.9.8;References;584
15.10;Chapter 160. Hybrid Computer
Methods;585
15.10.1;Applicable to;585
15.10.2;Yield;585
15.10.3;Idea;585
15.10.4;Procedure;585
15.10.5;Example;585
15.10.6;Notes;586
15.10.7;References;586
15.11;Chapter 161. Invariant
Imbedding;587
15.11.1;Applicable to;587
15.11.2;Yield;587
15.11.3;Idea;587
15.11.4;Procedure;587
15.11.5;Example;589
15.11.6;Notes;590
15.11.7;References;591
15.12;Chapter 162. Predictor—Corrector Methods;591
15.12.1;Applicable to;591
15.12.2;Yield;591
15.12.3;Idea;592
15.12.4;Procedure;592
15.12.5;Example;593
15.12.6;Program 162;593
15.12.7;Notes;594
15.12.8;References;595
15.13;Chapter 163. Runge-Kutta Methods;596
15.13.1;Applicable to;596
15.13.2;Yield;596
15.13.3;Idea;596
15.13.4;Procedure;596
15.13.5;Example;596
15.13.6;Program 163;597
15.13.7;Notes;597
15.13.8;References;599
15.14;Chapter 164. Stiff
Equations;600
15.14.1;Applicable to;600
15.14.2;Yield;600
15.14.3;Idea;600
15.14.4;Procedure;600
15.14.5;Program 164;602
15.14.6;Notes;603
15.14.7;References;604
15.15;Chapter 165. Integrating Stochastic Equations;605
15.15.1;Applicable to;605
15.15.2;Yield;605
15.15.3;Idea;605
15.15.4;Procedure;605
15.15.5;Example;606
15.15.6;Program 165;607
15.15.7;Notes;607
15.15.8;References;608
15.16;Chapter 166. Numerical Method for Sturm-Liouville Problems;609
15.16.1;Applicable to;609
15.16.2;Yield;609
15.16.3;Idea;609
15.16.4;Procedure;609
15.16.5;Example;610
15.16.6;Notes;611
15.16.7;References;611
15.17;Chapter
167. Weighted Residual Methods;612
15.17.1;Applicable to;612
15.17.2;Yield;612
15.17.3;Idea;612
15.17.4;Procedure;612
15.17.5;Example;614
15.17.6;References;615
16;Part IV.C: Numerical Methods for PDEs;616
16.1;Chapter 168. Boundary Element Method;616
16.1.1;Applicable to;616
16.1.2;Yield;616
16.1.3;Idea;616
16.1.4;Procedure;617
16.1.5;Notes;618
16.1.6;References;619
16.2;Chapter 169. Differential Quadrature;620
16.2.1;Applicable to;620
16.2.2;Yield;620
16.2.3;Idea;620
16.2.4;Procedure;620
16.2.5;Example;621
16.2.6;Program 169;622
16.2.7;Notes;623
16.2.8;References;623
16.3;Chapter 170. Elliptic Equations: Finite Differences;624
16.3.1;Applicable to;624
16.3.2;Yield;624
16.3.3;Idea;624
16.3.4;Procedure;624
16.3.5;Example;626
16.3.6;Notes;627
16.3.7;References;627
16.4;Chapter 171. Elliptic Equations: Monte Carlo Method;628
16.4.1;Applicable to;628
16.4.2;Yield;628
16.4.3;Idea;628
16.4.4;Procedure;628
16.4.5;Example;631
16.4.6;Program 171;633
16.4.7;Notes;633
16.4.8;References;633
16.5;Chapter 172. Elliptic Equations: Relaxation;634
16.5.1;Applicable to;634
16.5.2;Yield;634
16.5.3;Idea;634
16.5.4;Procedure;634
16.5.5;Example;634
16.5.6;Program 172;636
16.5.7;Notes;636
16.5.8;References;637
16.6;Chapter 173· Hyperbolic Equations: Method of Characteristics;637
16.6.1;Applicable to;637
16.6.2;Yield;637
16.6.3;Idea;637
16.6.4;Procedure;638
16.6.5;Notes;640
16.6.6;References;640
16.7;Chapter 174. Hyperbolic Equations: Finite Differences;641
16.7.1;Applicable to;641
16.7.2;Yield;641
16.7.3;Idea;641
16.7.4;Procedure;641
16.7.5;Example;641
16.7.6;Special Case;643
16.7.7;Program 174;643
16.7.8;Notes;644
16.7.9;References;644
16.8;Chapter 175. Method of Lines;644
16.8.1;Applicable to;644
16.8.2;Yield;644
16.8.3;Idea;644
16.8.4;Procedure;645
16.8.5;Example;646
16.8.6;Notes;647
16.8.7;References;647
16.9;Chapter 176. Parabolic Equations: Explicit Method;648
16.9.1;Applicable to;648
16.9.2;Yield;648
16.9.3;Idea;648
16.9.4;Procedure;648
16.9.5;Table 176;650
16.9.6;Program 176;650
16.9.7;References;651
16.10;Chapter 177. Parabolic Equations: Implicit Method;651
16.10.1;Applicable to;651
16.10.2;Yield;651
16.10.3;Idea;651
16.10.4;Procedure;652
16.10.5;Example;653
16.10.6;Program 177;655
16.10.7;Notes;655
16.10.8;References;656
16.11;Chapter 178. Parabolic Equations: Monte Carlo Method;656
16.11.1;Applicable to;656
16.11.2;Yield;656
16.11.3;Idea;656
16.11.4;Procedure;657
16.11.5;Example;660
16.11.6;Program 178;661
16.11.7;Notes;662
16.11.8;References;662
16.12;Chapter 179. Pseudo-Spectral Method;663
16.12.1;Applicable to;663
16.12.2;Yield;663
16.12.3;Idea;663
16.12.4;Procedure;663
16.12.5;Example;664
16.12.6;Program 179;665
16.12.7;Notes;666
16.12.8;References;666
16.13;Chapter 180. Schwarz's Method;667
16.13.1;Applicable to;667
16.13.2;Yield;667
16.13.3;Idea;667
16.13.4;Procedure;667
16.13.5;Example;669
16.13.6;Notes;671
16.13.7;References;671
17;Mathematical Nomenclature;672
18;Differential Equation Index;674
19;Index;680