Numerical Solution of Ordinary Differential Equations
February 2009, ©2009
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:
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.
1. Theory of differential equations: an introduction.
1.1 General solvability theory.
1.2 Stability of the initial value problem.
1.3 Direction fields.
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.
3. Systems of differential equations.
3.1 Higher order differential equations.
3.2 Numerical methods for systems.
4. The backward Euler method and the trapezoidal method.
4.1 The backward Euler method.
4.2 The trapezoidal method.
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.
6. Multistep methods.
6.1 Adams-Bashforth methods.
6.2 Adams-Moulton methods.
6.3 Computer codes.
7. General error analysis for multistep methods.
7.1 Truncation error.
7.3 A general error analysis.
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.
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.
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.
11. Two-point boundary value problems.
11.1 A finite difference method.
11.2 Nonlinear two-point boundary value problems.
12. Volterra integral equations.
12.1 Solvability theory.
12.2 Numerical methods.
12.3 Numerical methods - Theory.
Appendix A. Taylor’s theorem.
Appendix B. Polynomial interpolation.
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.
Contains many up-to-date references to both analytical and numerical ODE literature
Offers new unifying views on different problem classes
Related website provides MATLAB® programs that allow the reader to explore numerical methods experimentally
Related website also includes Graphical User Interfaces (GUIs) to make experimental exploration even easier
Written by well-known authors who have proven to be effective communicators and outstanding researchers
Offers complete and extensive topic coverage to allow instructors increased freedom for class structure. Also allows the interested student to pursue further topics of interest.