The University of Southampton
Courses

# MATH3084 Integral Transform Methods

## Module Overview

Many classes of problems are difficult to solve in their original domain. An integral transform maps the problem from its original domain into a new domain in which solution is easier. The solution is then mapped back to the original domain with the inverse of the integral transform. This module will provide a systematic mathematical treatment of the theory of integral transforms and its varied applications in applied mathematics and engineering.

### Aims and Objectives

#### Module Aims

This module will provide a systematic mathematic treatment of the theory of integral transforms and its varied applications in applied mathematics and engineering.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Be confident in the use of complex variable theory and contour integration
• Be able to demonstrate knowledge of a range of applications of these methods
• Understand how integral transforms can be used to solve a variety of differential equations

### Syllabus

• Analyticity of complex functions; Taylor and Laurent series • Contour integration of functions and multifunctions • Integration contours, including semi-circles, segments, box contours and keyhole contours • Use of complex methods for evaluation of real integrals • Laplace transforms and inverse Laplace transforms • Application of Laplace transforms to PDEs and to closed loop circuits (Nyquist stability theory) • Fourier transforms and their applications to PDEs • Fourier sine and cosine transforms and their applications to PDEs • Other examples of integral transforms (e.g. Hankel) and their applications • Fredholm theory (if time permits) • Higher-dimensional examples of integral transforms and their uses • Sturm-Liouville theory • Relation of integral transform methods with separation of variables for finite domain problems

### Learning and Teaching

#### Teaching and learning methods

The lecturer will provide a structured week-by-week study programme, based largely on the notes provided. Each week there will be three hours of lectures. There will be tutorials every other week, each lasting one hour. The tutorial classes will be used to study problems illustrating the lecture material. Students should spend their private study time studying the lecture notes and working through these problem sets.

TypeHours
Teaching60
Independent Study90
Total study time150

C Wylie and L C Barrett. Advanced engineering mathematics.

M R Spiegel. Theory and problems of complex variables (Schaum).

H A Priestley. An introduction to complex analysis.

Other. The module is based on lecture notes which are provided, so no book purchase is required but reference to any of the suggested texts below is recommended. Books which can be found in the university library are listed with their library number; those which are not in the university library are listed with their ISBN number

M D Greenberg. Advanced engineering mathematics.

L Debnath. Integral transforms and their applications.

G Stephenson and P M Radmore. Advanced mathematical methods for engineering and science students.

R V Churchill and J W Brown. Complex variables and applications.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Exam 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

Prerequisites: MATH2038 or MATH2047 or MATH2048

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.