This work meets the need for an affordable textbook that helps in understanding numerical solutions of ODE. Carefully structured by an experienced textbook author, it provides a survey of ODE for various applications, both classical and modern, including such special applications as relativistic systems. The examples are carefully explained and compiled into an algorithm, each of which is presented independent of a specific programming language. Each chapter is rounded off with exercises.

Donald Greenspan is Professor of Mathematics at the University of Texas, where he received the Distinguished Research Award in 1983. An experienced lecturer, he has authored 200 papers and 14 books, many of them textbooks on computational mathematics. His assignments included positions at Harvard, Stanford, Berkeley and Princeton.

Preface

ix

Euler's Method

1

(10)

Introduction

1

(1)

Euler's Method

1

(4)

Convergence of Euler's Method*

5

(3)

Remarks

8

(1)

Exercises

9

(2)

Runge-Kutta Methods

11

(26)

Introduction

11

(1)

A Runge-Kutta Formula

11

(4)

Higher-Order Runge-Kutta Formulas

15

(7)

Kutta's Fourth-Order Formula

22

(1)

Kutta's Formulas for Systems of First-Order Equations

23

(3)

Kutta's Formulas for Second-Order Differential Equations

26

(2)

Application -- The Nonlinear Pendulum

28

(3)

Application -- Impulsive Forces

31

(3)

Exercises

34

(3)

The Method of Taylor Expansions

37

(12)

Introduction

37

(1)

First-Order Problems

37

(3)

Systems of First-Order Equations

40

(1)

Second-Order Initial Value Problems

41

(2)

Application -- The van der Pol Oscillator

43

(2)

Exercises

45

(4)

Large Second-Order Systems with Application to Nano Systems

49

(28)

Introduction

49

(1)

The N-Body Problem

49

(1)

Classical Molecular Potentials

50

(2)

Molecular Mechanics

52

(1)

The Leap Frog Formulas

52

(1)

Equations of Motion for Argon Vapor

53

(1)

A Cavity Problem

54

(2)

Computational Considerations

56

(1)

Examples of Primary Vortex Generation

56

(3)

Examples of Turbulent Flow

59

(2)

Remark

61

(1)

Molecular Formulas for Air

62

(1)

A Cavity Problem

63

(1)

Initial Data

64

(1)

Examples of Primary Vortex Generation

65

(1)

Turbulent Flow

66

(4)

Colliding Microdrops of Water Vapor

70

(2)

Remarks

72

(2)

Exercises

74

(3)

Completely Conservative, Covariant Numerical Methodology

77

(34)

Introduction

77

(1)

Mathematical Considerations

77

(1)

Numerical Methodology

78

(1)

Conservation Laws

79

(3)

Covariance

82

(3)

Application -- A Spinning Top on a Smooth Horizontal Plane

85

(18)

Application -- Calogero and Toda Hamiltonian Systems

103

(5)

Remarks

108

(1)

Exercises

109

(2)

Instability

111

(22)

Introduction

111

(1)

Instability Analysis

111

(11)

Numerical Solution of Mildly Nonlinear Autonomous Systems

122

(8)

Exercises

130

(3)

Numerical Solution of Tridiagonal Linear Algebraic Systems and Related Nonlinear Systems

133

(10)

Introduction

133

(1)

Tridiagonal Systems

133

(3)

The Direct Method

136

(1)

The Newton-Lieberstein Method

137

(3)

Exercises

140

(3)

Approximate Solution of Boundary Value Problems

143

(16)

Introduction

143

(1)

Approximate Differentiation

143

(1)

Numerical Solution of Boundary Value Problems Using Difference Equations

144

(4)

Upwind Differencing

148

(2)

Mildly Nonlinear Boundary Value Problems

150

(2)

Theoretical Support*

152

(3)

Application -- Approximation of Airy Functions

155

(1)

Exercises

156

(3)

Special Relativistic Motion

159

(18)

Introduction

159

(1)

Inertial Frames

160

(1)

The Lorentz Transformation

161

(1)

Rod Contraction and Time Dilation

161

(2)

Relativistic Particle Motion

163

(1)

Covariance

163

(2)

Particle Motion

165

(1)

Numerical Methodology

166

(3)

Relativistic Harmonic Oscillation

169

(1)

Computational Covariance

170

(4)

Remarks

174

(1)

Exercises

175

(2)

Special Topics

177

(10)

Introduction

177

(1)

Solving Boundary Value Problems by Initial Value Techniques

177

(1)

Solving Initial Value Problems by Boundary Value Techniques

178

(1)

Predictor-Corrector Methods

179

(1)

Multistep Methods

180

(1)

Other Methods

180

(1)

Consistency*

181

(1)

Differential Eigenvalue Problems

182

(2)

Chaos*

184

(1)

Contact Mechanics

184

(3)

Appendix

A. Basic Matrix Operations

187

(4)

Solutions to Selected Exercises

191

(6)

References

197

(6)

Index

203

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