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# Partial Differential Equations - MAT00120M

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• Department: Mathematics
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2024-25
• See module specification for other years: 2023-24

## Module summary

A partial differential equation (PDE) is a differential equation that contains an unknown function of several variables and its partial derivatives. PDEs are used to describe a wide range of natural processes. Examples include fluid mechanics, elasticity theory, electrodynamics, quantum mechanics, etc. The aim of this module is to introduce basic properties of PDEs and basic analytical and numerical techniques to solve them.

## Related modules

• None

### Prohibited combinations

Pre-requisite module; Vector & Complex Calculus

This module is taught at both H- and M-level. You can only take the module once

M-level students will have 4 hours of extra lectures and 1 extra seminar which will be used to teach more advanced topics.

## Module will run

Occurrence Teaching period
A Semester 1 2024-25

## Module aims

A partial differential equation (PDE) is a differential equation that contains an unknown function of several variables and its partial derivatives. PDEs are used to describe a wide range of natural processes. Examples include fluid mechanics, elasticity theory, electrodynamics, quantum mechanics, etc. The aim of this module is to introduce basic properties of PDEs and basic analytical and numerical techniques to solve them.

## Module learning outcomes

By the end of the module, students will be able to:

1. Solve simple first-order PDEs.

2. Determine the type of a second order PDE.

3. Use analytical techniques for solving classical PDEs such as the wave equation, the heat equation and the Laplace and Poisson equations.

4. Analyse the error and stability of finite-difference methods for solving PDEs.

5. Obtain numerical solutions of simple PDEs with the help of MATLAB.

6. Solve the inhomogeneous wave and heat equations.

7. Find periodic solutions of the heat equation.

## Module content

Syllabus

1. Introduction: what a PDE is, first-order linear PDEs, initial and boundary conditions, well-posed problems, types of second-order PDEs.

2. Heat (diffusion) equation: maximum principle, heat equation on the whole line and on the half-line.

3. Wave equation: d'Alembert’s formula, causality and energy, reflection of waves.

4. Laplace equation: maximum principle, Poisson’s formula, rectangular domain.

5. Finite-differences, truncation error, convergence and stability.

6. Explicit and implicit finite-difference schemes for parabolic PDEs. The alternating- direction method.

7. Inhomogeneous heat and wave equations.

8. Periodic solutions of heat equation.

## Indicative assessment

Closed/in-person Exam (Centrally scheduled) 90
Essay/coursework 10

None

### Indicative reassessment

Closed/in-person Exam (Centrally scheduled) 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

1. W. A. Strauss, Partial Differential Equations. An Introduction. New York: Wiley, 1992 (1st ed.), 2008 (2nd ed.) (Library catalogue S 7.383 STR).

2. W.F. Ames, Numerical Methods for Partial Differential Equations. New York: Academic Press, 1977 (Library catalogue S 7.383 AME).

3. M.H. Holmes, Introduction to Numerical Methods in Differential Equations, 2007, Springer (Electronic copy available via the University library).

The information on this page is indicative of the module that is currently on offer. The University constantly explores ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.