Handbook of Differential Equations (eBook)
808 Seiten
Elsevier Science (Verlag)
978-1-4832-6396-0 (ISBN)
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.
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 "e;natural"e; 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.
Front Cover 1
Handbook of Differential Equations 4
Copyright Page 5
Table of Contents 6
Preface 12
Introduction 14
How to Use This Book 18
CHAPTER I.A Definitions and Concepts 22
1. Definition of Terms 22
2. Alternative Theorems 35
3. Bifurcation Theory 37
4. A Caveat for Partial Differential Equations 46
5. Chaos in Dynamical Systems 47
6. Classification of Partial Differential Equations 54
7. Compatible Systems 60
8. Conservation Laws 64
9. Differential Resultants 67
10. Existence and Uniqueness Theorems 71
11. Fixed Point Existence Theorems 75
12. Hamilton—Jacobi Theory 78
13. Inverse Problems 82
14. Limit Cycles 84
15. Natural Boundary Conditions for a PDE 88
16. Normal Forms: Near-Identity Transformations 91
17. Self-Adjoint Eigenfunction Problems 95
18. Stability Theorems 101
19. Sturm—Liouville Theory 103
20. Variational Equations 109
21. Well-Posedness of Differential Equations 115
22. Wronskians and Fundamental Solutions 118
CHAPTER
122
23. Canonical Forms 122
24. Canonical Transformations 126
25. Darboux Transformation 129
26. An Involutory Transformation 132
27. Liouville Transformation — 1 135
28· Liouville Transformation — 2 138
29. Reduction of Linear ODEs to a First Order System 139
30. Prüfer Transformation 141
31. Modified Prüfer Transformation 143
32. Transformations of Second Order Linear ODEs - 1 145
33. Transformations of Second Order Linear ODEs - 2 149
34. Transformation of an ODE to an Integral Equation 151
35. Miscellaneous ODE Transformations 154
36. Reduction of PDEs to a First Order System 157
37. Transforming Partial Differential Equations 160
38. Transformations of Partial Differential Equations 165
CHAPTER
168
39. Introduction to Exact Analytical Methods 168
40. Look Up Technique 169
41. Look Up ODE Forms 202
CHAPTER
206
42. An N-th Order Equation 206
43. Use of the Adjoint Equation 208
44. Autonomous Equations 211
45. Bernoulli Equation 215
46. Clairaut's Equation 217
47. Computer-Aided Solution 218
48. Constant Coefficient Linear Equations 225
49. Contact Transformation 227
50. Delay Equations 230
51. Dependent Variable Missing 237
52. Differentiation Method 239
53. Differential Equations with Discontinuities 240
54. Eigenfunction Expansions 244
55. Equidimensional-In-x Equations 251
56. Equidimensional-In-y Equations 254
57. Euler Equations 256
58. Exact First Order Equations 259
59. Exact Second Order Equations 261
60. Exact N-th Order Equations 264
61. Factoring Equations 266
62. Factoring Operators 267
63. Factorization Method 272
64. Fokker–Planck Equation 275
65. Fractional Differential Equations 279
66. Free Boundary Problems 283
67. Generating Functions 286
68. Green's Functions 289
69. Homogeneous Equations 297
70. Method of Images 300
71. Integrable Combinations 304
72. Integral Representations: Laplace's Method 305
73. Integral Transforms: Finite Intervals 311
74. Integral Transforms: Infinite Intervals 316
75. Integrating Factors 326
76. Interchanging Dependent and Independent Variables 329
77. Lagrange's Equation 332
78. Lie Groups: ODEs 335
79. Operational Calculus 343
80. Pfaffian Differential Equations 347
81. Reduction of Order 351
82. Riccati Equation 353
83. Matrix Riccati Equations 356
84. Scale Invariant Equations 359
85. Separable Equations 362
86. Series Solution 363
87. Equations Solvable for x 370
88. Equations Solvable for y 371
89. Superposition 373
90. Method of Undetermined Coefficients 375
91. Variation of Parameters 377
92. Vector Ordinary Differential Equations 381
CHAPTER
386
93. Bäcklund Transformations 386
94. Method of Characteristics 389
95. Characteristic Strip Equations 395
96. Conformal Mappings 397
97. Method of Descent 403
98. Diagonalization of a Linear System of PDEs 405
99. Duhamel's Principle 407
100. Exact Equations 410
101. Hodograph Transformation 411
102. Inverse Scattering 414
103. Jacobi's Method 418
104. Legendre Transformation 421
105. Lie Groups: PDEs 425
106. Poisson Formula 432
107. Riemann's Method 435
108. Separation of Variables 440
109. Similarity Methods 445
110. Exact Solutions to the Wave Equation 450
111. Wiener–Hopf Technique 453
CHAPTER III. Approximate Analytical Methods 458
112. Introduction to Approximate Analysis 458
113. Chaplygin's Method 459
114. Collocation 462
115. Dominant Balance 464
116. Equation Splitting 467
117. Floquet Theory 469
118. Graphical Analysis: The Phase Plane 472
119. Graphical Analysis: The Tangent Field 478
120. Harmonic Balance 481
121. Homogenization 484
122. Integral Methods 488
123. Interval Analysis 491
124. Least Squares Method 494
125. Lyapunov Functions 497
126. Equivalent Linearization and Nonlinearization 500
127. Maximum Principles 505
128. McGarvey Iteration Technique 509
129. Moment Equations: Closure 512
130. Moment Equations: Itô Calculus 515
131. Monge's Method 518
132. Newton's Method 521
133. Padé Approximants 524
134. Perturbation Method: Method of Averaging 528
135. Perturbation Method: Boundary Layer Method 531
136. Perturbation Method: Functional Iteration 539
137. Perturbation Method: Multiple Scales 545
138. Perturbation Method: Regular Perturbation 549
139. Perturbation Method: Strained Coordinates 553
140. Picard Iteration 556
141. Reversion Method 559
142. Singular Solutions 561
143. Soliton Type Solutions 564
144. Stochastic Limit Theorems 566
145. Taylor Series Solutions 569
146. Variational Method: Eigenvalue Approximation 572
147. Variational Method: Rayleigh–Ritz 575
148. WKB Method 579
CHAPTER
586
149. Introduction to Numerical Methods 586
150. Definition of Terms for Numerical Methods 588
151. Available Software 591
152. Finite Difference Methodology 594
153. Finite Difference Formulas 599
154. Excerpts from GAMS 607
155. Grid Generation 627
156. Richardson Extrapolation 630
157. Stability: ODE Approximations 634
158. Stability: Courant Criterion 639
159. Stability: Von Neumann Test 642
CHAPTER
644
160. Analytic Continuation 644
161. Boundary Value Problems: Box Method 647
162. Boundary Value Problems: Shooting Method 652
163. Continuation Method 656
164. Continued Fractions 658
165. Cosine Method 661
166. Differential Algebraic Equations 665
167. Eigenvalue/Eigenfunction Problems 671
168. Euler's Forward Method 674
169. Finite Element Method 677
170. Hybrid Computer Methods 687
171. Invariant Imbedding 690
172. Multigrid Methods 694
173. Parallel Computer Methods 697
174. Predictor–Corrector Methods 700
175. Runge—Kutta Methods 705
176. Stiff Equations 711
177. Integrating Stochastic Equations 716
178. Weighted Residual Methods 720
CHAPTER
724
179. Boundary Element Method 724
180. Differential Quadrature 729
181. Domain Decomposition 732
182. Elliptic Equations: Finite Differences 737
183. Elliptic Equations: Monte Carlo Method 742
184. Elliptic Equations: Relaxation 747
185. Hyperbolic Equations: Method of Characteristics 751
186. Hyperbolic Equations: Finite Differences 754
187. Lattice Gas Dynamics 758
188. Method of Lines 761
189. Parabolic Equations: Explicit Method 765
190. Parabolic Equations: Implicit Method 768
191. Parabolic Equations: Monte Carlo Method 773
192. Pseudo-Spectral Method 780
Mathematical Nomenclature 786
Differential Equation Index 788
Index 794
| Erscheint lt. Verlag | 12.5.2014 |
|---|---|
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Technik | |
| ISBN-10 | 1-4832-6396-7 / 1483263967 |
| ISBN-13 | 978-1-4832-6396-0 / 9781483263960 |
| Haben Sie eine Frage zum Produkt? |
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