# Handbook of Differential Equations

## 1st Edition

**Authors:**Daniel Zwillinger

**eBook ISBN:**9781483220963

**Imprint:**Academic Press

**Published Date:**28th January 1988

**Page Count:**694

## Description

Handbook of Differential Equations is a handy reference to many popular techniques for solving and approximating differential equations, including exact analytical methods, approximate analytical methods, and numerical methods. Topics covered range from transformations and constant coefficient linear equations to finite and infinite intervals, along with conformal mappings and the perturbation method.

Comprised of 180 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 Classification of Partial Differential Equations

6 Compatible Systems

7 Conservation Laws

8 Differential Resultants

9 Fixed Point Existence Theorems

10 Hamilton-Jacobi Theory

11 Limit Cycles

12 Natural Boundary Conditions for a PDE

13 Self-Adjoint Eigenfunction Problems

14 Sturm-Liouville Theory

15 Variational Equations

16 Well-Posedness of Differential Equations

17 Wronskians and Fundamental Solutions

I.B Transformations

18 Canonical Forms

19 Canonical Transformations

20 Darboux Transformation

21 An Involutory Transformation

22 Liouville Transformation - 1

23 Liouville Transformation - 2

24 Reduction of Linear ODEs to a First Order System

25 Transformations of Second Order Linear ODEs - 1

26 Transformations of Second Order Linear ODEs - 2

27 Transformation of an ODE to an Integral Equation

28 Miscellaneous ODE Transformations

29 Reduction of PDEs to a First Order System

30 Transforming Partial DifFerential Equations

31 Transformations of Partial DifFerential Equations

II Exact Analytical Methods

32 Introduction to Exact Analytical Methods

33 Look Up Technique

II.A Exact Methods for ODEs

34 An N-th Order Equation

35 Use of the Adjoint Equation

36 Autonomous Equations

37 Bernoulli Equation

38 Clairaut's Equation

39 Computer-Aided Solution

40 Constant Coefficient Linear Equations

41 Contact Transformation

42 Delay Equations

43 Dependent Variable Missing

44 Differentiation Method

45 Differential Equations with Discontinuities

36 Autonomous Equations

37 Bernoulli Equation

38 Clairaut's Equation

39 Computer-Aided Solution

40 Constant Coefficient Linear Equations

41 Contact Transformation

42 Delay Equations

43 Dependent Variable Missing

44 Differentiation Method

45 Differential Equations with Discontinuities

46 Eigenfunction Expansions

47 Equidimensional-in-x Equations

48 Equidimensional-in-y Equations

49 Euler Equations

50 Exact First Order Equations

51 Exact Second Order Equations

52 Exact N-th Order Equations

53 Factoring Equations

47 Equidimensional-in-x Equations

48 Equidimensional-in-y Equations

49 Euler Equations

50 Exact First Order Equations

51 Exact Second Order Equations

52 Exact N-th Order Equations

53 Factoring Equations

54 Factoring Operators

55 Factorization Method

56 Fokker-Planck Equation

57 Fractional Differential Equations

55 Factorization Method

56 Fokker-Planck Equation

57 Fractional Differential Equations

58 Free Boundary Problems

59 Generating Functions

59 Generating Functions

60 Green's Functions

61 Homogeneous Equations

62 Method of Images

61 Homogeneous Equations

62 Method of Images

63 Integrable Combinations

64 Integral Representations: Laplace's Method

65 Integral Transforms: Finite Intervals

65 Integral Transforms: Finite Intervals

66 Integral Transforms: Infinite Intervals

67 Integrating Factors

67 Integrating Factors

68 Interchanging Dependent and Independent Variables

69 Lagrange's Equation

70 Lie Groups: ODEs

71 Operational Calculus

72 PfafSan Differential Equations

73 Prüfer Substitution

74 Reduction of Order

75 Riccati Equation - 1

76 Riccati Equation - 2

77 Matrix Riccati Equations

78 Scale Invariant Equations

79 Separable Equations

80 Series Solution

72 PfafSan Differential Equations

73 Prüfer Substitution

74 Reduction of Order

75 Riccati Equation - 1

76 Riccati Equation - 2

77 Matrix Riccati Equations

78 Scale Invariant Equations

79 Separable Equations

80 Series Solution

81 Equations Solvable for x

82 Equations Solvable for y

83 Superposition

84 Method of Undetermined Coefficients

84 Method of Undetermined Coefficients

85 Variation of Parameters

86 Vector Ordinary Differential Equations

II.B Exact Methods for PDEs

87 Bäcklund Transformations

88 Method of Characteristics

89 Characteristic Strip Equations

90 Conformai Mappings

91 Method of Descent

92 Diagonalization of a Linear System of PDEs

93 Duhamel's Principle

94 Hodograph Transformation

95 Inverse Scattering

96 Jacobi's Method

97 Legendre Transformation

98 Lie Groups: PDEs

99 Poisson Formula

100 Riemann's Method

101 Separation of Variables

102 Similarity Methods

103 Exact Solutions to the Wave Equation

104 Wiener-Hopf Technique

III Approximate Analytical Methods

105 Introduction to Approximate Analysis

106 Chaplygin's Method

107 Collocation

108 Dominant Balance

109 Equation Splitting

110 Equivalent Linearization

111 Equivalent Nonlinearization

112 Floquet Theory

113 Graphical Analysis: The Phase Plane

114 Graphical Analysis: The Tangent Field

115 Harmonic Balance

116 Homogenization

117 Integral Methods

118 Interval Analysis

119 Least Squares Method

120 Liapunov Functions

121 Maximum Principles

122 McGarvey Iteration Technique

123 Moment Equations: Closure

124 Moment Equations: Itô Calculus

125 Monge's Method

126 Newton's Method

127 Padé Approximants

128 Perturbation Method: Method of Averaging

129 Perturbation Method: Boundary Layer Method

130 Perturbation Method: Functional Iteration

131 Perturbation Method: Multiple Scales

132 Perturbation Method: Regular Perturbation

133 Perturbation Method: Strained Coordinates

134 Picard Iteration

135 Modified Prüfer Substitution

136 Reversion Method

137 Singular Solutions

138 Soliton Type Solutions

139 Stochastic Limit Theorems

140 Taylor Series Solutions

141 Variational Method: Eigenvalue Approximation

142 Variational Method: Rayleigh-Ritz

143 WKB Method

IV.A Numerical Methods: Concepts

144 Introduction to Numerical Methods

145 Definition of Terms for Numerical Methods

146 Courant-Priedrichs-Lewy Consistency Criterion

147 Finite Difference Schemes for ODEs

148 Richardson Extrapolation

149 Software Libraries

150 Von Neumann Test

IV.B Numerical Methods for ODEs

151 Analytic Continuation

152 Boundary Value Problems: Box Method

153 Boundary Value Problems: Shooting Method

154 Continuation Method

152 Boundary Value Problems: Box Method

153 Boundary Value Problems: Shooting Method

154 Continuation Method

155 Continued Fractions

156 Cosine Method

157 Differential Algebraic Equations

158 Finite Element Method

157 Differential Algebraic Equations

158 Finite Element Method

159 Forward Euler's Method

160 Hybrid Computer Methods

161 Invariant Imbedding

161 Invariant Imbedding

162 Predictor-Corrector Methods

163 Runge-Kutta Methods

164 Stiff Equations

165 Integrating Stochastic Equations

166 Numerical Method for Sturm-Liouville Problems

167 Weighted Residual Methods

165 Integrating Stochastic Equations

166 Numerical Method for Sturm-Liouville Problems

167 Weighted Residual Methods

IV.C Numerical Methods for PDEs

168 Boundary Element Method

169 Differential Quadrature

170 Elliptic Equations: Finite Differences

171 Elliptic Equations: Monte Carlo Method

172 Elliptic Equations: Relaxation

173 Hyperbolic Equations: Method of Characteristics

174 Hyperbolic Equations: Finite Differences

175 Method of Lines

176 Parabolic Equations: Explicit Method

177 Parabolic Equations: Implicit Method

178 Parabolic Equations: Monte Carlo Method

179 Pseudo-Spectral Method

180 Schwarz's Method

Mathematical Nomenclature

Differential Equation Index

Index

## Details

- No. of pages:
- 694

- Language:
- English

- Copyright:
- © Academic Press 1989

- Published:
- 28th January 1988

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483220963

## 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