# Partial Differential Equations and their Approximation module (MA42003)

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Credits

15

Module code

MA42003

### About the module

This module gives a broad introduction to PDEs that includes classification into different types, classical solution methods, qualitative properties and, for the majority of problems that cannot be solved exactly, provides techniques for constructing approximate solutions. This module is mandatory for Level 4 students taking the BSc or MSci in Mathematical Biology, and may optionally be taken in combination with other modules by students taking the BSc or MMath in Mathematics or any of the other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

### Prerequisites

Students taking this module must have taken the module MA41003, or equivalent.

### Indicative Content

• #### First and Second Order PDEs

Basic Theory; examples of fundamental solutions Second order linear PDEs; classification, characteristics; dAlemberts solution of the one-dimensional wave equation.

• #### Boundary Value Problems for PDEs

Finite-difference methods for second order problems (Poisson's equation): the treatment of boundary conditions and curved boundaries in two dimensions.

• #### Initial Value Problems for PDEs

Parabolic and Hyperbolic equations: Fundamental solutions. General discussion of basic qualitative properties such as dissipation (energy inequalities) and characteristics. Construction of numerical methods: two-level methods and brief reference to three-level methods (if time permits). Local truncation errors. Stability and choice of norm: Maximum norm, L2 norm via von Neumann's method. Application to hyperbolic systems. The Method of Lines.

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

### Credit Rating

This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.