Daniel Zwillinger

Handbook of Differential Equations

• Broschiertes Buch

Produktbeschreibung

This book compiles the most widely applicable methods for solving and approximating differential equations. as well as numerous examples showing the methods use. Topics include ordinary differential equations, symplectic integration of differential equations, and the use of wavelets when numerically solving differential equations.

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Produktdetails

• Verlag: Elsevier Science
• 3rd edition
• Seitenzahl: 801
• Erscheinungstermin: 3. November 1997
• Englisch
• ISBN-13: 9781493302208
• ISBN-10: 1493302205
• Artikelnr.: 41641905

Autorenporträt

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.

Inhaltsangabe

1. Definitions and Concepts
2. Transformations
3. Exact Analytical Methods
4. Exact Methods for ODEs
5. Exact Methods for PDEs
6. Approximate Analytical Methods
7. Numerical Methods: Concepts
8. Numerical Methods for ODEs
9. Numerical Methods for PDEs

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I.A Definitions and Concepts. 1. Definition of Terms. 2. Alternative Theorems. 3. Bifurcation Theory. 4. Chaos in Dynamical Systems. 5. Classification of Partial Differential Equations. 6. Compatible Systems. 7. Conservation Laws. 8. Differential Equations - Diagrams. 9. Differential Equations - Symbols. 10. Differential Resultants. 11. Existence and Uniqueness Theorems. 12. Fixed Point Existence Theorems. 13. Hamilton - Jacobi Theory. 14. Infinite Order Differential Equations. 15. Integrability of Systems. 16. Inverse Problems. 17. Limit Cycles. 18. PDEs & Natural Boundary Conditions. 19. Normal Forms: Near-Identity Transformations. 20. q-Differential Equations. 21. Quaternionic Differential Equations. 22. Self-Adjoint Eigenfunction Problems. 23. Stability Theorems. 24. Stochastic Differential Equations. 25. Sturm-Liouville Theory. 26. Variational Equations. 27. Web Resources. 28. Well-Posed Differential Equations. 29. Wronskians & Fundamental Solutions. 30. Zeros of Solutions.

I.B. Transformations. 31. Canonical Forms. 32. Canonical Transformations. 33. Darboux Transformation. 34. An Involutory Transformation. 35. Liouville Transformation - 1. 36. Liouville Transformation - 2. 37. Changing Linear ODEs to a First Order System. 38. Transformations of Second Order Linear ODEs - 1. 39. Transformations of Second Order Linear ODEs - 2. 40. Transforming an ODE to an Integral Equation. 41. Miscellaneous ODE Transformations. 42. Transforming PDEs Generically. 43. Transformations of PDEs. 44. Transforming a PDE to a First Order System. 45. Prüfer Transformation. 46. Modified Prüfer Transformation. II. Exact Analytical Methods. 47. Introduction to Exact Analytical Methods. 48. Look-Up Technique. 49. Look-Up ODE Forms.

II.A Exact Methods for ODEs. 50. Use of the Adjoint Equation. 51. An Nth Order Equation. 52. Autonomous Equations - Independent Variable Missing. 53. Bernoulli Equation. 54. Clairaut's Equation. 55. Constant Coefficient Linear ODEs. 56 Contact Transformation. 57. Delay Equations. 58. Dependent Variable Missing. 59. Differentiation Method. 60. Differential Equations with Discontinuities. 61. Eigenfunction Expansions. 62. Equidimensional-in-x Equations. 63. Equidimensional-in-y Equations. 64. Euler Equations. 65. Exact First Order Equations. 66. Exact Second Order Equations. 67. Exact Nth Order Equations. 68. Factoring Equations. 69. Factoring/Composing Operators. 70. Factorization Method. 71. Fokker-Planck Equation. 72. Fractional Differential Equations. 73. Free Boundary Problems. 74. Generating Functions. 75. Green's Functions. 76. ODEs with Homogeneous Functions. 77. Hypergeometric Equation. 78. Method of Images. 79. Integrable Combinations. 80. Integrating Factors*. 81. Interchanging Dependent and Independent Variables. 82. Integral Representation: Laplace's Method. 83. Integral Transforms: Finite Intervals. 84. Integral Transforms: Infinite Intervals. 85. Lagrange's Equation. 86. Lie Algebra Technique. 87. Lie Groups: ODEs. 88. Non-normal Operators. 89. Operational Calculus. 90. Pfaffian Differential Equations. 91. Quasilinear Second Order ODEs. 92. Quasipolynomial ODEs. 93. Reduction of Order. 94. Resolvent Method for Matrix ODEs. 95. Riccati Equation - Matrices. 96. Riccati Equation - Scalars. 97. Scale Invariant Equations. 98. Separable Equations. 99. Series Solution. 100. Equations Solvable for x. 101. Equations Solvable for y. 102. Superposition. 103. Undetermined Coefficients. 104. Variation of Parameters. 105. Vector ODEs. II.B Exact Methods for PDEs. 106. Bäcklund Transformations. 107. Cagniard-de Hoop Method. 108. Method of Characteristics. 109. Characteristic Strip Equations. 110. Conformal Mappings. 111. Method of Descent. 112. Diagonalizable Linear Systems of PDEs. 113. Duhamel's Principle. 114. Exact Partial Differential Equations. 115. Fokas Method / Unified Transform. 116. Hodograph Transformation. 117. Inverse Scattering. 118. Jacobi's Method. 119. Legendre Transformation. 120. Lie Groups: PDEs. 121. Many Consistent PDEs. 122. Poisson Formula. 123. Resolvent Method for PDEs. 124. Riemann's Method 125 Separation of Variables. 126. Separable Equations: Stäckel Matrix. 127. Similarity Methods. 128. Exact Solutions to the Wave Equation. 129. Wiener-Hopf Technique.

III. Approximate Analytical Methods. 130. Introduction to Approximate Analysis. 131. Adomian Decomposition Method. 132. Chaplygin's Method. 133. Collocation. 134. Constrained Functions. 135. Differential Constraints. 136. Dominant Balance. 137. Equation Splitting. 138. Floquet Theory. 139. Graphical Analysis: The Phase Plane. 140 Graphical Analysis: Poincaré Map. 141. Graphical Analysis: Tangent Field. 142. Harmonic Balance. 143. Homogenization. 144. Integral Methods. 145. Interval Analysis. 146. Least Squares Method. 147. Equivalent Linearization and Nonlinearization. 148. Lyapunov Functional. 149. Maximum Principles. 150. McGarvey Iteration Technique. 151. Moment Equations: Closure. 152. Moment Equations: Itô Calculus. 153. Monge's Method 154. Newton's Method. 155. Padé Approximants. 156. Parametrix Method. 157. Perturbation Method: Averaging. 158. Perturbation Method: Boundary Layers. 159. Perturbation Method: Functional Iteration. 160. Perturbation Method: Multiple Scales. 161. Perturbation Method: Regular Perturbation. 162. Perturbation Method: Renormalization Group. 163. Perturbation Method: Strained Coordinates. 164. Picard Iteration. 165. Reversion Method. 166. Singular Solutions. 167. Soliton-Type Solutions. 168. Stochastic Limit Theorems. 169. Structured Guessing. 170. Taylor Series Solutions. 171. Variational Method: Eigenvalue Approximation. 172. Variational Method: Rayleigh-Ritz. 173. WKB Method.

IV.A Numerical Methods: Concepts. 174. Introduction to Numerical Methods. 175. Terms for Numerical Methods. 176. Finite Difference Formulas. 177. Finite Difference Methodology. 178. Grid Generation. 179. Richardson Extrapolation. 180. Stability: ODE Approximations. 181. Stability: Courant Criterion. 182. Stability: Von Neumann Test. 183. Testing Differential Equation Routines.

IV.B Numerical Methods for ODEs. 184. Analytic Continuation. 185. Boundary Value Problems: Box Method. 186. Boundary Value Problems: Shooting Method. 187. Continuation Method. 188. Continued Fractions. 189. Cosine Method. 190. Differential Algebraic Equations. 191. Eigenvalue/Eigenfunction Problems. 192. Euler's Forward Method. 193. Finite Element Method. 194. Hybrid Computer Methods. 195. Invariant Imbedding. 196. Multigrid Methods. 197. Neural Networks & Optimization. 198. Nonstandard Finite Difference Schemes. 199. ODEs with Highly Oscillatory Terms. 200. Parallel Computer Methods. 201. Predictor-Corrector Methods. 202. Probabilistic Methods. 203. Quantum computing. 204. Runge-Kutta Methods. 205. Stiff Equations. 206. Integrating Stochastic Equations. 207. Symplectic Integration. 208. System Linearization Via Koopman. 209. Using Wavelets. 210. Weighted Residual Methods.

IV.C Numerical Methods for PDEs. 211. Boundary Element Method. 212. Differential Quadrature. 213. Domain Decomposition. 214. Elliptic Equations: Finite Differences. 215. Elliptic Equations: Monte-Carlo Method. 216. Elliptic Equations: Relaxation. 217. Hyperbolic Equations: Method of Characteristics. 218. Hyperbolic Equations: Finite Differences. 219. Lattice Gas Dynamics. 220. Method of Lines. 221. Parabolic Equations: Explicit Method. 222. Parabolic Equations: Implicit Method. 223. Parabolic Equations: Monte-Carlo Method. 224. Pseudospectral Method.

V. Computer Languages and Systems. 225. Computer Languages and Packages. 226. Julia Programming Language. 227. Maple Computer Algebra System. 228. Mathematica Computer Algebra System. 229. MATLAB Programming Language. 230. Octave Programming Language. 231. Python Programming Language. 232. R Programming Language. 233. Sage Computer Algebra System.