# Module Information

Module Identifier
MA34110
Module Title
Partial Differential Equations
2015/2016
Co-ordinator
Semester
Semester 1
Pre-Requisite
Pre-Requisite
Pre-Requisite
External Examiners
• Dr Andrew Hazel (Reader - University of Manchester)

Other Staff

#### Course Delivery

Delivery Type Delivery length / details
Lecture 33 x 1 Hour Lectures

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   100%
Supplementary Exam 2 Hours   100%

### Learning Outcomes

On completion of this module, a student should be able to:
1. solve simple linear partial differential equations;
2. illustrate with suitable examples the occurrence of such equations in physics and industry;
3. interpret the meaning of mathematical solutions of partial differential equations in the appropriate context.

### Brief description

Many mathematical problems arising in the physical sciences, engineering, and technology, may be formulated in terms of partial differential equations. In attempting to solve such problems, one must be aware of the various types of partial differential equation which exist, and of the different boundary conditions associated with each type. These factors determine which method of solution one should use.

### Aims

To teach the student how to recognise the type of a partial differential equation, and how to choose and implement an appropriate method of solution.

### Content

1. Fundamentals: definitions and examples, simple partial differential equations.
2. First order equations: the method of characteristics.
3. Boundary conditions: Dirichlet, Neumann, Robin, well-posedness and ill-posedness.
4. Second order equations: classification, reduction to canonical forms.
5. The wave equation: general solution, Cauchy problem, reflection principle, Duhamel principle, bounded string, energy and uniqueness.
6. The heat equation: maximum principle, uniqueness, separation of variables, properties of solutions, the fundamential solution.
7. Fourier transform methods: definition, properties, derivative theorem, convolution theorem, consideration of applicable functional spaces, application to heat equation Cauchy problem.

### Notes

This module is at CQFW Level 6