A concise introduction to numerical methodsand the mathematical
framework neededto understand their performance
Numerical Solution of Ordinary Differential Equations
presents a complete and easy-to-follow introduction to classical
topics in the numerical solution of ordinary differential
equations. The book’s approach not only explains the presented
mathematics, but also helps readers understand how these numerical
methods are used to solve real-world problems.
Unifying perspectives are provided throughout the text, bringing
together and categorizing different types of problems in order to
help readers comprehend the applications of ordinary differential
equations. In addition, the authors’ collective academic experience
ensures a coherent and accessible discussion of key topics,
including:
* Euler’s method
* Taylor and Runge-Kutta methods
* General error analysis for multi-step methods
* Stiff differential equations
* Differential algebraic equations
* Two-point boundary value problems
* Volterra integral equations
Each chapter features problem sets that enable readers to test
and build their knowledge of the presented methods, and a related
Web site features MATLAB® programs that facilitate the
exploration of numerical methods in greater depth. Detailed
references outline additional literature on both analytical and
numerical aspects of ordinary differential equations for further
exploration of individual topics.
Numerical Solution of Ordinary Differential Equations is
an excellent textbook for courses on the numerical solution of
differential equations at the upper-undergraduate and beginning
graduate levels. It also serves as a valuable reference for
researchers in the fields of mathematics and engineering.
สารบัญ
Preface.
Introduction.
1. Theory of differential equations: an introduction.
1.1 General solvability theory.
1.2 Stability of the initial value problem.
1.3 Direction fields.
Problems.
2. Euler’s method.
2.1 Euler’s method.
2.2 Error analysis of Euler’s method.
2.3 Asymptotic error analysis.
2.3.1 Richardson extrapolation.
2.4 Numerical stability.
2.4.1 Rounding error accumulation.
Problems.
3. Systems of differential equations.
3.1 Higher order differential equations.
3.2 Numerical methods for systems.
Problems.
4. The backward Euler method and the trapezoidal
method.
4.1 The backward Euler method.
4.2 The trapezoidal method.
Problems.
5. Taylor and Runge-Kutta methods.
5.1 Taylor methods.
5.2 Runge-Kutta methods.
5.3 Convergence, stability, and asymptotic error.
5.4 Runge-Kutta-Fehlberg methods.
5.5 Matlab codes.
5.6 Implicit Runge-Kutta methods.
Problems.
6. Multistep methods.
6.1 Adams-Bashforth methods.
6.2 Adams-Moulton methods.
6.3 Computer codes.
Problems.
7. General error analysis for multistep methods.
7.1 Truncation error.
7.2 Convergence.
7.3 A general error analysis.
Problems.
8. Stiff differential equations.
8.1 The method of lines for a parabolic equation.
8.2 Backward differentiation formulas.
8.3 Stability regions for multistep methods.
8.4 Additional sources of difficulty.
8.5 Solving the finite difference method.
8.6 Computer codes.
Problems.
9. Implicit RK methods for stiff differential
equations.
9.1 Families of implicit Runge-Kutta methods.
9.2 Stability of Runge-Kutta methods.
9.3 Order reduction.
9.4 Runge-Kutta methods for stiff equations in practice.
Problems.
10. Differential algebraic equations.
10.1 Initial conditions and drift.
10.2 DAEs as stiff differential equations.
10.3 Numerical issues: higher index problems.
10.4 Backward differentiation methods for DAEs.
10.5 Runge-Kutta methods for DAEs.
10.6 Index three problems from mechanics.
10.7 Higher index DAEs.
Problems.
11. Two-point boundary value problems.
11.1 A finite difference method.
11.2 Nonlinear two-point boundary value problems.
Problems.
12. Volterra integral equations.
12.1 Solvability theory.
12.2 Numerical methods.
12.3 Numerical methods – Theory.
Problems.
Appendix A. Taylor’s theorem.
Appendix B. Polynomial interpolation.
Bibliography.
Index.
เกี่ยวกับผู้แต่ง
Kendall E. Atkinson, Ph D, 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, Ph D, is Professor in the Department of Mathematics at the University of Iowa, where he is also Director of the interdisciplinary Ph D Program in Applied Mathematical and Computational Science. Dr. Han currently focuses his research on the numerical solution of partial differential equations. David E. Stewart, Ph D, 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.
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