## MATH2431 Fourier Series, Partial Differential Equations and Transforms.

### 10 creditsClass Size: 200

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

### Pre-requisites

MATH1400, MATH1410, MATH2420, or equivalent.

This module is approved as an Elective

### Objectives

To discuss Fourier series and Fourier and Laplace transforms and their application to the solution of classical Partial Differential Equations of mathematical physics. On completion of this module, students should be able to: a) obtain the whole or half range Fourier series of a simple function; b) apply the method of separation of variables to the solution of boundary and initial value problems for the classical PDEs of mathematical physics in terms of Cartesian co-ordinates. c) obtain the Fourier transforms of simple functions and apply Fourier transforms to the solution of classical PDEs. d) obtain the Laplace transforms of simple functions and apply Laplace transforms to the solution of initial value problems for linear ODEs with constant coefficients.

### Syllabus

Many real world situations can be modelled by one of three partial differential equations:

1. Laplace's equation, which describes e.g. the steady flow of heat or electric charge in a metal or the behaviour of the gravitational potential in the solar system. 2. The heat (or diffusion equation), which describes e.g. the unsteady flow of heat in a metal or the dispersal of cigarette smoke through a room. 3. The wave equation, which describes e.g. waves on the surface of the sea or vibrations of a plucked guitar string. This module discusses these equations and methods for their solution. In particular, use is made of the remarkable result of Fourier that almost any periodic function (i.e. one whose graph endlessly repeats the same pattern) can be represented as a sum of sines and cosines, called its Fourier series. An analogous representation for non-periodic functions is provided by the Fourier transform and the closely related Laplace transform.

The topics covered are: Fourier series (whole and half range) with examples. Solution of boundary and initial value problems for the heat equation. Laplace's equation and the wave equation (in terms of Cartesian co-ordinates) by separation of variables.

Fourier and Laplace transforms, definitions and basic properties (including the convolution theorem), simple examples, applications to the solution of linear PDEs and ODEs.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures (22 hours) and examples classes (11 hours).

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

2 hour written examination at end of semester (85%), coursework (15%).