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.
- 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.