Numerical Solution of Ordinary Differential Equations
by Atkinson, Kendall; Han, Weimin; Stewart, David E.Edition: 1st
Format: Hardcover
Pub. Date: 2009-02-09
Publisher(s): Wiley
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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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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