# Handbook of Differential Equations

## 2nd Edition

**Authors:**Daniel Zwillinger

**eBook ISBN:**9781483263960

**Imprint:**Academic Press

**Published Date:**28th January 1992

**Page Count:**808

## Description

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 "natural" 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.

## Table of Contents

Preface

Introduction

How to Use This Book

I.A Definitions and Concepts

1 Definition of Terms

2 Alternative Theorems

3 Bifurcation Theory

4 A Caveat for Partial Differential Equations

5 Chaos in Dynamical Systems

6 Classification of Partial Differential Equations

7 Compatible Systems

8 Conservation Laws

9 Differential Resultants

10 Existence and Uniqueness Theorems

11 Fixed Point Existence Theorems

12 Hamilton-Jacobi Theory

13 Inverse Problems

14 Limit Cycles

15 Natural Boundary Conditions for a PDE

16 Normal Forms: Near-Identity Transformations

17 Self-Adjoint Eigenfunction Problems

18 Stability Theorems

19 Sturm-Liouville Theory

20 Variational Equations

21 Well-Posedness of Differential Equations

22 Wronskians and Fundamental Solutions

I.B Transformations

23 Canonical Forms

24 Canonical Transformations

25 Darboux Transformation

26 An Involutory Transformation

27 Liouville Transformation - 1

28 Liouville Transformation - 2

29 Reduction of Linear ODEs to a First Order System

30 Prüfer Transformation

31 Modified Prüfer Transformation

32 Transformations of Second Order Linear ODEs - 1

33 Transformations of Second Order Linear ODEs - 2

34 Transformation of an ODE to an Integral Equation

35 Miscellaneous ODE Transformations

36 Reduction of PDEs to a First Order System

37 Transforming Partial Differential Equations

38 Transformations of Partial Differential Equations

II Exact Analytical Methods

39 Introduction to Exact Analytical Methods

40 Look Up Technique

41 Look Up ODE Forms

II.A Exact Methods for ODEs

42 An N-th Order Equation

43 Use of the Adjoint Equation

44 Autonomous Equations

45 Bernoulli Equation

46 Clairaut's Equation

47 Computer-Aided Solution

48 Constant Coefficient Linear Equations

49 Contact Transformation

50 Delay Equations

51 Dependent Variable Missing

52 Differentiation Method

53 Differential Equations with Discontinuities

54 Eigenfunction Expansions

54 Eigenfunction Expansions

55 Equidimensional-in-x Equations

56 Equidimensional-in-y Equations

57 Euler Equations

58 Exact First Order Equations

59 Exact Second Order Equations

60 Exact N-th Order Equations

61 Factoring Equations

62 Factoring Operators

62 Factoring Operators

63 Factorization Method

64 Fokker-Planck Equation

65 Fractional Differential Equations

66 Free Boundary Problems

66 Free Boundary Problems

67 Generating Functions

68 Green's Functions

68 Green's Functions

69 Homogeneous Equations

70 Method of Images

71 Integrable Combinations

72 Integral Representations: Laplace's Method

71 Integrable Combinations

72 Integral Representations: Laplace's Method

73 Integral Transforms: Finite Intervals

74 Integral Transforms: Infinite Intervals

74 Integral Transforms: Infinite Intervals

75 Integrating Factors

76 Interchanging Dependent and Independent Variables

77 Lagrange's Equation

78 Lie Groups: ODEs

79 Operational Calculus

76 Interchanging Dependent and Independent Variables

77 Lagrange's Equation

78 Lie Groups: ODEs

79 Operational Calculus

80 Pfaffian Differential Equations

81 Reduction of Order

82 Riccati Equation

83 Matrix Riccati Equations

84 Scale Invariant Equations

85 Separable Equations

86 Series Solution

87 Equations Solvable for x

88 Equations Solvable for y

89 Superposition

87 Equations Solvable for x

88 Equations Solvable for y

89 Superposition

90 Method of Undetermined Coefficients

91 Variation of Parameters

92 Vector Ordinary Differential Equations

II.B Exact Methods for PDEs

93 Bäcklund Transformations

94 Method of Characteristics

95 Characteristic Strip Equations

96 Conformai Mappings

97 Method of Descent

98 Diagonalization of a Linear System of PDEs

99 Duhamel's Principle

100 Exact Equations

101 Hodograph Transformation

102 Inverse Scattering

103 Jacobi's Method

104 Legendre Transformation

105 Lie Groups: PDEs

106 Poisson Formula

107 Riemann's Method

108 Separation of Variables

109 Similarity Methods

110 Exact Solutions to the Wave Equation

111 Wiener-Hopf Technique

III Approximate Analytical Methods

112 Introduction to Approximate Analysis

113 Chaplygin's Method

114 Collocation

115 Dominant Balance

116 Equation Splitting

117 Floquet Theory

118 Graphical Analysis: The Phase Plane

119 Graphical Analysis: The Tangent Field

120 Harmonic Balance

121 Homogenization

122 Integral Methods

123 Interval Analysis

124 Least Squares Method

125 Lyapunov Functions

126 Equivalent Linearization and Nonlinearization

127 Maximum Principles

128 McGarvey Iteration Technique

129 Moment Equations: Closure

130 Moment Equations: Itô Calculus

131 Monge's Method

132 Newton's Method

133 Padé Approximants

134 Perturbation Method: Method of Averaging

135 Perturbation Method: Boundary Layer Method

136 Perturbation Method: Functional Iteration

137 Perturbation Method: Multiple Scales

138 Perturbation Method: Regular Perturbation

139 Perturbation Method: Strained Coordinates

140 Picard Iteration

141 Reversion Method

142 Singular Solutions

143 Soliton Type Solutions

144 Stochastic Limit Theorems

145 Taylor Series Solutions

146 Variational Method: Eigenvalue Approximation

147 Variational Method: Rayleigh-Ritz

148 WKB Method

IV.A Numerical Methods: Concepts

149 Introduction to Numerical Methods

150 Definition of Terms for Numerical Methods

151 Available Software

152 Finite Difference Methodology

153 Finite Difference Formulas

154 Excerpts from GAMS

155 Grid Generation

156 Richardson Extrapolation

157 Stability: ODE Approximations

158 Stability: Courant Criterion

159 Stability: Von Neumann Test

IV.B Numerical Methods for ODEs

160 Analytic Continuation

91 Variation of Parameters

92 Vector Ordinary Differential Equations

II.B Exact Methods for PDEs

93 Bäcklund Transformations

94 Method of Characteristics

95 Characteristic Strip Equations

96 Conformai Mappings

97 Method of Descent

98 Diagonalization of a Linear System of PDEs

99 Duhamel's Principle

100 Exact Equations

101 Hodograph Transformation

102 Inverse Scattering

103 Jacobi's Method

104 Legendre Transformation

105 Lie Groups: PDEs

106 Poisson Formula

107 Riemann's Method

108 Separation of Variables

109 Similarity Methods

110 Exact Solutions to the Wave Equation

111 Wiener-Hopf Technique

III Approximate Analytical Methods

112 Introduction to Approximate Analysis

113 Chaplygin's Method

114 Collocation

115 Dominant Balance

116 Equation Splitting

117 Floquet Theory

118 Graphical Analysis: The Phase Plane

119 Graphical Analysis: The Tangent Field

120 Harmonic Balance

121 Homogenization

122 Integral Methods

123 Interval Analysis

124 Least Squares Method

125 Lyapunov Functions

126 Equivalent Linearization and Nonlinearization

127 Maximum Principles

128 McGarvey Iteration Technique

129 Moment Equations: Closure

130 Moment Equations: Itô Calculus

131 Monge's Method

132 Newton's Method

133 Padé Approximants

134 Perturbation Method: Method of Averaging

135 Perturbation Method: Boundary Layer Method

136 Perturbation Method: Functional Iteration

137 Perturbation Method: Multiple Scales

138 Perturbation Method: Regular Perturbation

139 Perturbation Method: Strained Coordinates

140 Picard Iteration

141 Reversion Method

142 Singular Solutions

143 Soliton Type Solutions

144 Stochastic Limit Theorems

145 Taylor Series Solutions

146 Variational Method: Eigenvalue Approximation

147 Variational Method: Rayleigh-Ritz

148 WKB Method

IV.A Numerical Methods: Concepts

149 Introduction to Numerical Methods

150 Definition of Terms for Numerical Methods

151 Available Software

152 Finite Difference Methodology

153 Finite Difference Formulas

154 Excerpts from GAMS

155 Grid Generation

156 Richardson Extrapolation

157 Stability: ODE Approximations

158 Stability: Courant Criterion

159 Stability: Von Neumann Test

IV.B Numerical Methods for ODEs

160 Analytic Continuation

161 Boundary Value Problems: Box Method

162 Boundary Value Problems: Shooting Method

163 Continuation Method

163 Continuation Method

164 Continued Fractions

165 Cosine Method

166 Differential Algebraic Equations

167 Eigenvalue/Eigenfunction Problems

168 Euler's Forward Method

169 Finite Element Method

166 Differential Algebraic Equations

167 Eigenvalue/Eigenfunction Problems

168 Euler's Forward Method

169 Finite Element Method

170 Hybrid Computer Methods

171 Invariant Imbedding

171 Invariant Imbedding

172 Multigrid Methods

173 Parallel Computer Methods

174 Predictor-Corrector Methods

175 Runge-Kutta Methods

176 Stiff Equations

173 Parallel Computer Methods

174 Predictor-Corrector Methods

175 Runge-Kutta Methods

176 Stiff Equations

177 Integrating Stochastic Equations

178 Weighted Residual Methods*

IV.C Numerical Methods for PDEs

179 Boundary Element Method

180 Differential Quadrature

181 Domain Decomposition

182 Elliptic Equations: Finite Differences

183 Elliptic Equations: Monte Carlo Method

184 Elliptic Equations: Relaxation

185 Hyperbolic Equations: Method of Characteristics

186 Hyperbolic Equations: Finite Differences

187 Lattice Gas Dynamics

188 Method of Lines

189 Parabolic Equations: Explicit Method

190 Parabolic Equations: Implicit Method

191 Parabolic Equations: Monte Carlo Method

192 Pseudo-Spectral Method

Mathematical Nomenclature

Differential Equation Index

Index

## Details

- No. of pages:
- 808

- Language:
- English

- Copyright:
- © Academic Press 1992

- Published:
- 28th January 1992

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483263960

## About the Author

### Daniel Zwillinger

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.

### Affiliations and Expertise

Rensselaer Polytechnic Institute, Troy, NY, USA