Daniel Zwillinger, Vladimir Dobrushkin

Handbook of Differential Equations (eBook, PDF)

• Format: PDF

Produktbeschreibung

The book is a compilation of methods for solving and approximating differential equations. These include the most widely applicable methods for solving and approximating differential equations, as well as numerous methods. Topics include methods for ODEs, PDEs, stochastic DEs, and systems of such equations.

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Produktdetails

• Verlag: Taylor & Francis
• Seitenzahl: 736
• Erscheinungstermin: 30. Dezember 2021
• Englisch
• ISBN-13: 9781000468168
• Artikelnr.: 63128178

Autorenporträt

Daniel Zwillinger has more than 35 years of proven technical expertise in numerous areas of engineering and the physical sciences. He earned a Ph.D. in applied mathematics from the California Institute of Technology. He is the Editor of CRC Standard Mathematical Tables and Formulas, 33rd edition and also Table of Integrals, Series, and Products, Gradshteyn and Ryzhik. He serves as the Series Editor on the CRC Series of Advances in Applied Mathematics.

Vladimir A. Dobrushkin is a Professor at the Division of Applied Mathematics, Brown University. He holds a Ph.D. in Applied mathematics and Dr.Sc. in mechanical engineering. He is the author of three books for CRC Press, including Applied Differential Equations: The Primary Course, Applied Differential Equations with Boundary Value Problems, and Methods in Algorithmic Analysis.

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.