Zwillinger Handbook of Differential Equations

2. Auflage 2014
ISBN: 978-1-4832-6396-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 808 Seiten, Web PDF

Handbook of Differential Equations, Second Edition is a handy reference to many popular techniques for solving and approximating differential equations, including numerical methods and exact and approximate analytical methods. Topics covered range from transformations and constant coefficient linear equations to Picard iteration, along with conformal mappings and inverse scattering. Comprised of 192 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;CHAPTER I.A Definitions and Concepts;22
8.1;1. Definition of Terms;22
8.2;2. Alternative Theorems;35
8.3;3. Bifurcation Theory;37
8.4;4. A Caveat for Partial Differential Equations;46
8.5;5. Chaos in Dynamical Systems;47
8.6;6. Classification of Partial Differential Equations;54
8.7;7. Compatible Systems;60
8.8;8. Conservation Laws;64
8.9;9. Differential Resultants;67
8.10;10. Existence and Uniqueness Theorems;71
8.11;11. Fixed Point Existence Theorems;75
8.12;12. Hamilton—Jacobi Theory;78
8.13;13. Inverse Problems;82
8.14;14. Limit Cycles;84
8.15;15. Natural Boundary Conditions for a PDE;88
8.16;16. Normal Forms: Near-Identity Transformations;91
8.17;17. Self-Adjoint Eigenfunction Problems;95
8.18;18. Stability Theorems;101
8.19;19. Sturm—Liouville Theory;103
8.20;20. Variational Equations;109
8.21;21. Well-Posedness of Differential Equations;115
8.22;22. Wronskians and Fundamental Solutions;118
9;CHAPTER
I.B Transformations;122
9.1;23. Canonical Forms;122
9.2;24. Canonical Transformations;126
9.3;25. Darboux Transformation;129
9.4;26. An Involutory Transformation;132
9.5;27. Liouville Transformation — 1;135
9.6;28· Liouville Transformation — 2;138
9.7;29. Reduction of Linear ODEs to a First Order System;139
9.8;30. Prüfer Transformation;141
9.9;31. Modified Prüfer Transformation;143
9.10;32. Transformations of Second Order Linear ODEs - 1;145
9.11;33. Transformations of Second Order Linear ODEs - 2;149
9.12;34. Transformation of an ODE to an Integral Equation;151
9.13;35. Miscellaneous ODE Transformations;154
9.14;36. Reduction of PDEs to a First Order System;157
9.15;37. Transforming Partial Differential Equations;160
9.16;38. Transformations of Partial Differential Equations;165
10;CHAPTER
II. Exact Analytical Methods;168
10.1;39. Introduction to Exact Analytical Methods;168
10.2;40. Look Up Technique;169
10.3;41. Look Up ODE Forms;202
11;CHAPTER
II.A Exact Methods for ODEs;206
11.1;42. An N-th Order Equation;206
11.2;43. Use of the Adjoint Equation;208
11.3;44. Autonomous Equations;211
11.4;45. Bernoulli Equation;215
11.5;46. Clairaut's Equation;217
11.6;47. Computer-Aided Solution;218
11.7;48. Constant Coefficient Linear Equations;225
11.8;49. Contact Transformation;227
11.9;50. Delay Equations;230
11.10;51. Dependent Variable Missing;237
11.11;52. Differentiation Method;239
11.12;53. Differential Equations with Discontinuities;240
11.13;54. Eigenfunction Expansions;244
11.14;55. Equidimensional-In-x Equations;251
11.15;56. Equidimensional-In-y Equations;254
11.16;57. Euler Equations;256
11.17;58. Exact First Order Equations;259
11.18;59. Exact Second Order Equations;261
11.19;60. Exact N-th Order Equations;264
11.20;61. Factoring Equations;266
11.21;62. Factoring Operators;267
11.22;63. Factorization Method;272
11.23;64. Fokker–Planck Equation;275
11.24;65. Fractional Differential Equations;279
11.25;66. Free Boundary Problems;283
11.26;67. Generating Functions;286
11.27;68. Green's Functions;289
11.28;69. Homogeneous Equations;297
11.29;70. Method of Images;300
11.30;71. Integrable Combinations;304
11.31;72. Integral Representations: Laplace's Method;305
11.32;73. Integral Transforms: Finite Intervals;311
11.33;74. Integral Transforms: Infinite Intervals;316
11.34;75. Integrating Factors;326
11.35;76. Interchanging Dependent and Independent Variables;329
11.36;77. Lagrange's Equation;332
11.37;78. Lie Groups: ODEs;335
11.38;79. Operational Calculus;343
11.39;80. Pfaffian Differential Equations;347
11.40;81. Reduction of Order;351
11.41;82. Riccati Equation;353
11.42;83. Matrix Riccati Equations;356
11.43;84. Scale Invariant Equations;359
11.44;85. Separable Equations;362
11.45;86. Series Solution;363
11.46;87. Equations Solvable for x;370
11.47;88. Equations Solvable for y;371
11.48;89. Superposition;373
11.49;90. Method of Undetermined Coefficients;375
11.50;91. Variation of Parameters;377
11.51;92. Vector Ordinary Differential Equations;381
12;CHAPTER
II.B Exact Methods for PDEs;386
12.1;93. Bäcklund Transformations;386
12.2;94. Method of Characteristics;389
12.3;95. Characteristic Strip Equations;395
12.4;96. Conformal Mappings;397
12.5;97. Method of Descent;403
12.6;98. Diagonalization of a Linear System of PDEs;405
12.7;99. Duhamel's Principle;407
12.8;100. Exact Equations;410
12.9;101. Hodograph Transformation;411
12.10;102. Inverse Scattering;414
12.11;103. Jacobi's Method;418
12.12;104. Legendre Transformation;421
12.13;105. Lie Groups: PDEs;425
12.14;106. Poisson Formula;432
12.15;107. Riemann's Method;435
12.16;108. Separation of Variables;440
12.17;109. Similarity Methods;445
12.18;110. Exact Solutions to the Wave Equation;450
12.19;111. Wiener–Hopf Technique;453
13;CHAPTER III. Approximate Analytical Methods;458
13.1;112. Introduction to Approximate Analysis;458
13.2;113. Chaplygin's Method;459
13.3;114. Collocation;462
13.4;115. Dominant Balance;464
13.5;116. Equation Splitting;467
13.6;117. Floquet Theory;469
13.7;118. Graphical Analysis: The Phase Plane;472
13.8;119. Graphical Analysis: The Tangent Field;478
13.9;120. Harmonic Balance;481
13.10;121. Homogenization;484
13.11;122. Integral Methods;488
13.12;123. Interval Analysis;491
13.13;124. Least Squares Method;494
13.14;125. Lyapunov Functions;497
13.15;126. Equivalent Linearization and Nonlinearization;500
13.16;127. Maximum Principles;505
13.17;128. McGarvey Iteration Technique;509
13.18;129. Moment Equations: Closure;512
13.19;130. Moment Equations: Itô Calculus;515
13.20;131. Monge's Method;518
13.21;132. Newton's Method;521
13.22;133. Padé Approximants;524
13.23;134. Perturbation Method: Method of Averaging;528
13.24;135. Perturbation Method: Boundary Layer Method;531
13.25;136. Perturbation Method: Functional Iteration;539
13.26;137. Perturbation Method: Multiple Scales;545
13.27;138. Perturbation Method: Regular Perturbation;549
13.28;139. Perturbation Method: Strained Coordinates;553
13.29;140. Picard Iteration;556
13.30;141. Reversion Method;559
13.31;142. Singular Solutions;561
13.32;143. Soliton Type Solutions;564
13.33;144. Stochastic Limit Theorems;566
13.34;145. Taylor Series Solutions;569
13.35;146. Variational Method: Eigenvalue Approximation;572
13.36;147. Variational Method: Rayleigh–Ritz;575
13.37;148. WKB Method;579
14;CHAPTER
IV.A Numerical Methods: Concepts;586
14.1;149. Introduction to Numerical Methods;586
14.2;150. Definition of Terms for Numerical Methods;588
14.3;151. Available Software;591
14.4;152. Finite Difference Methodology;594
14.5;153. Finite Difference Formulas;599
14.6;154. Excerpts from GAMS;607
14.7;155. Grid Generation;627
14.8;156. Richardson Extrapolation;630
14.9;157. Stability: ODE Approximations;634
14.10;158. Stability: Courant Criterion;639
14.11;159. Stability: Von Neumann Test;642
15;CHAPTER
IV.B Numerical Methods for ODEs;644
15.1;160. Analytic Continuation;644
15.2;161. Boundary Value Problems: Box Method;647
15.3;162. Boundary Value Problems: Shooting Method;652
15.4;163. Continuation Method;656
15.5;164. Continued Fractions;658
15.6;165. Cosine Method;661
15.7;166. Differential Algebraic Equations;665
15.8;167. Eigenvalue/Eigenfunction Problems;671
15.9;168. Euler's Forward Method;674
15.10;169. Finite Element Method;677
15.11;170. Hybrid Computer Methods;687
15.12;171. Invariant Imbedding;690
15.13;172. Multigrid Methods;694
15.14;173. Parallel Computer Methods;697
15.15;174. Predictor–Corrector Methods;700
15.16;175. Runge—Kutta Methods;705
15.17;176. Stiff Equations;711
15.18;177. Integrating Stochastic Equations;716
15.19;178. Weighted Residual Methods;720
16;CHAPTER
IV.C Numerical Methods for PDEs;724
16.1;179. Boundary Element Method;724
16.2;180. Differential Quadrature;729
16.3;181. Domain Decomposition;732
16.4;182. Elliptic Equations: Finite Differences;737
16.5;183. Elliptic Equations: Monte Carlo Method;742
16.6;184. Elliptic Equations: Relaxation;747
16.7;185. Hyperbolic Equations: Method of Characteristics;751
16.8;186. Hyperbolic Equations: Finite Differences;754
16.9;187. Lattice Gas Dynamics;758
16.10;188. Method of Lines;761
16.11;189. Parabolic Equations: Explicit Method;765
16.12;190. Parabolic Equations: Implicit Method;768
16.13;191. Parabolic Equations: Monte Carlo Method;773
16.14;192. Pseudo-Spectral Method;780
17;Mathematical Nomenclature;786
18;Differential Equation Index;788
19;Index;794