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

### 10 creditsClass Size: 200

Module manager: Dr O. Chalykh
Email: oleg@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

### Pre-requisite qualifications

MATH2360 or MATH2420, or equivalent.

### This module is mutually exclusive with

 MATH2375 Linear Differential Equations and Transforms

This module is approved as an Elective

### Module summary

Many real world situations can be modelled by partial differential equations. 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.

### 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

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.

### Teaching methods

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

 Delivery type Number Length hours Student hours Example Class 11 1.00 11.00 Lecture 22 1.00 22.00 Private study hours 67.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

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

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated