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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.
Contents
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.
