Aims & Learning Objectives:
Aims: To provide an introduction to the numerical solution of differential equations and how they arise in applications. To provide background linear Algebra.
The students should become familiar with the software packages Matlab and Maple, should learn computational methods for solving differential equations, should see how differential equations can be applied to a wide variety of modl problems, and should have a background knowledge of numerical linear algebra and how problems are formulated and solved using Matlab.
1. Introduction to Maple and Matlab and their facilities: basic matrix manipulation, eigenvalue calculation, FFT analysis, special functions, solution of simultaneous linear and nonlinear equations, simple optimization. Basic graphics, data handling, use of toolboxes. Problem formulation and solution using Matlab.
2. Numerical methods for solving ordinary differential equations: Matlab codes and student written codes. Convergence and Stability. Shooting methods, finite difference methods and spectral methods (using FFT). Sample case studies chosen from: the two body problem, the three body problem, combustion, nonlinear control theory, the Lorenz equations, power electronics, Sturn-Liouville theory, eigenvalues, and orthogonal basis expansions.