Handbook of Differential Equations (Advances in Applied Mathematics)

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Introduction
How to Use This Book
I.A. Definitions and Concepts
	1. Definition of Terms
	2. Alternative Theorems
	3. Bifurcation Theory
	4. Chaos in Dynamical Systems
	5. Classi cation 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*
		48.1. Ordinary Differential Equations
		48.2. Partial Differential Equations
		48.3. Systems of Differential Equations
		48.4. Hamiltonians Representing Differential Equations
		48.5. The Laplacian in Different Coordinate Systems
		48.6. Parametrized Differential Equations at Specific Values
	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
	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. Backlund 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: StŁackel 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: Poincare 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: Ito Calculus
	153. Monge's Method
	154. Newton's Method
	155. Pade 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
		176.1. One Dimension: Rectilinear Grid
		176.2. Two Dimensions: Rectilinear Grid
		176.3. Two Dimensions: Irregular Grid
		176.4. Two Dimensions: Triangular Grid
		176.5. Numerical Schemes for the ODE: y' = f(x, y)
		176.6. Explicit Numerical Schemes for the PDE: aux + ut = 0
		176.7. Implicit Numerical Schemes for the PDE: aux + ut = S(x, t)
		176.8. Numerical Schemes for the PDE: F(u)x + ut = 0
		176.9. Numerical Schemes for the PDE: ux = utt
	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
Mathematical Nomenclature
Named Differential Equations
Index