Numerical Solution of Ordinary Differential Equations

by Atkinson, Kendall; Han, Weimin; Stewart, David E.

• ISBN13: 9780470042946
• ISBN10: 047004294X
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-02-09
• Publisher: Wiley

Summary

This precise and highly readable book provides a complete and concise introduction to classical topics in the numerical solution of ordinary differential equations (ODEs). It contains many up-to-date references to both analytical and numerical ODE literature while offering new unifying views on different problem classes. A related Web site provides MATLABr programs that let the reader explore numerical methods experimentally, as well as Graphical User Interfaces (GUIs) to make experimental exploration easier. Written by well-known authors who are proven communicators and researchers, this is an ideal reference or text for students in mathematics, engineering, and the sciences.

Author Biography

Kendall E. Atkinson, PhD, is Professor Emeritus in the Departments of Mathematics and Computer Science at the University of Iowa. He has authored books and journal articles in his areas of research interest, which include the numerical solution of integral equations and boundary integral equation methods. Weimin Han, PhD, is Professor in the Department of Mathematics at the University of Iowa, where he is also Director of the interdisciplinary PhD Program in Applied Mathematical and Computational Science. Dr. Han currently focuses his research on the numerical solution of partial differential equations. David E. Stewart, PhD, is Professor and Associate Chair in the Department of Mathematics at the University of Iowa, where he is also the departmental Director of Undergraduate Studies. Dr. Stewart's research interests include numerical analysis, computational models of mechanics, scientific computing, and optimization.

Table of Contents

Introduction	p. 1
Theory of differential equations: An introduction	p. 3
General solvability theory	p. 7
Stability of the initial value problem	p. 8
Direction fields	p. 11
Problems	p. 13
Euler's method	p. 15
Definition of Euler's method	p. 16
Error analysis of Euler's method	p. 21
Asymptotic error analysis	p. 26
Richardson extrapolation	p. 28
Numerical stability	p. 29
Rounding error accumulation	p. 30
Problems	p. 32
Systems of differential equations	p. 37
Higher-order differential equations	p. 39
Numerical methods for systems	p. 42
Problems	p. 46
The backward Euler method and the trapezoidal method	p. 49
The backward Euler method	p. 51
The trapezoidal method	p. 56
Problems	p. 62
Taylor and Runge-Kutta methods	p. 67
Taylor methods	p. 68
Runge-Kutta methods	p. 70
A general framework for explicit Runge-Kutta methods	p. 73
Convergence, stability, and asymptotic error	p. 75
Error prediction and control	p. 78
Runge-Kutta-Fehlberg methods	p. 80
MATLAB codes	p. 82
Implicit Runge-Kutta methods	p. 86
Two-point collocation methods	p. 87
Problems	p. 89
Multistep methods	p. 95
Adams-Bashforth methods	p. 96
Adams-Moulton methods	p. 101
Computer codes	p. 104
Matlab Ode codes	p. 105
Problems	p. 106
General error analysis for multistep methods	p. 111
Truncation error	p. 112
Convergence	p. 115
A general error analysis	p. 117
Stability theory	p. 118
Convergence theory	p. 122
Relative stability and week stability	p. 122
Problems	p. 123
Stiff differential equations	p. 127
The method of lines for a parabolic equation	p. 131
MATLAB programs for the method of lines	p. 135
Backward differentiation formulas	p. 140
Stability regions for multistep methods	p. 141
Additional sources of difficulty	p. 143
A-stability and L-stability	p. 143
Time-varying problems and stability	p. 145
Solving the finite-difference method	p. 145
Computer codes	p. 146
Problems	p. 147
Implicit RK methods for stiff differential equations	p. 149
Families of implicit Runge-Kutta methods	p. 149
Stability of Runge-Kutta methods	p. 154
Order reduction	p. 156
Runge-Kutta methods for stiff equations in practice	p. 160
Problems	p. 161
Differential algebraic equations	p. 163
Initial conditions and drift	p. 165
DAEs as stiff differential equations	p. 168
Numerical issues: higher index problems	p. 169
Backward differentiation methods for DAEs	p. 173
Index 1 problems	p. 173
Index 2 problems	p. 174
Runge-Kutta methods for DAEs	p. 175
Index 1 problems	p. 176
Index 2 problems	p. 179
Index three problems from mechanics	p. 181
Runge-Kutta methods for mechanical index 3 systems	p. 183
Higher index DAEs	p. 184
Problems	p. 185
Two-point boundary value problems	p. 187
A finite-difference method	p. 188
Convergence	p. 190
A numerical example	p. 190
Boundary conditions involving the derivative	p. 194
Nonlinear two-point boundary value problems	p. 195
Finite difference methods	p. 197
Shooting methods	p. 201
Collocation methods	p. 204
Other methods and problems	p. 206
Problems	p. 206
Volterra integral equations	p. 211
Solvability theory	p. 212
Special equations	p. 214
Numerical methods	p. 215
The trapezoidal method	p. 216
Error for the trapezoidal method	p. 217
General schema for numerical methods	p. 219
Numerical methods: Theory	p. 223
Numerical stability	p. 225
Practical numerical stability	p. 227
Problems	p. 231
Taylor's Theorem	p. 235
Polynomial interpolation	p. 241
References	p. 245
Index	p. 250
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