# Partial Differential Equations and their Approximation module (MA42003)

** 15 **

** MA42003 **

Mathematical models in most areas of science and engineering generate partial differential equations (PDEs) that cannot usually be solved exactly or reduced to simpler ordinary differential equations. Some typical examples include chemical reactions, population genetics, power station cooling systems, and airflow around cars and aircraft.

In this module, you will be given a broad introduction to PDEs that includes classification into different types, classical solution methods, qualitative properties and, for many problems that cannot be solved exactly, techniques for constructing approximate solutions. The module will not tackle specific applications but aims to provide a sound basis by focussing on model examples. This module will develop your analytical and numerical analysis skills for simulations of scientific and engineering problems and prepare you to explore core areas of applied and computational mathematics.

Topics include:

- Classification of PDEs
- Elliptic PDEs, maximum principles, separation of variables
- Finite difference methods of elliptic PDEs, local truncation error, discrete maximum principles
- Parabolic PDEs, maximum principles, the heat equation
- Finite difference methods of parabolic PDEs, consistency, convergence and stability
- Hyperbolic PDEs, systems of first-order equations, method of characteristics
- Finite difference methods of hyperbolic PDEs.

## Courses

This module is available on following courses: