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Advances in Applied Mathematics

Handbook of Differential Equations

Dobrushkin, Vladimir (Brown University, RI, USA) & Zwillinger, Daniel (Rensselaer Polytechnic Institute, NY, USA)

Handbook of Differential Equations

Advances in Applied Mathematics

Handbook of Differential Equations

Advances in Applied Mathematics: Handbook of Differential Equations


Through the previous three editions, Handbook of Differential Equations has proven an invaluable reference for anyone working within the field of mathematics, including academics, students, scientists, and professional engineers.

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Beschrijving Advances in Applied Mathematics: Handbook of Differential Equations

Through the previous three editions, Handbook of Differential Equations has proven an invaluable reference for anyone working within the field of mathematics, including academics, students, scientists, and professional engineers.

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 ordinary differential equations, partial differential equations, stochastic differential equations, and systems of such equations.

Included for nearly every method are:

  • The types of equations to which the method is applicable

  • The idea behind the method

  • The procedure for carrying out the method

  • At least one simple example of the method

  • Any cautions that should be exercised

  • Notes for more advanced users

The fourth edition includes corrections, many supplied by readers, as well as many new methods and techniques. These new and corrected entries make necessary improvements in this edition.

Table of Contents

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.


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.

Advances in Applied Mathematics
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