Integral Transform Methods | MATH3084 | University of Southampton

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.

Linked modules

Prerequisites: MATH2038 OR MATH2047 OR MATH2048

Aims and Objectives

Learning Outcomes

Having successfully completed this module you will be able to:

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
Be confident in the use of complex variable theory and contour integration

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.

Study time
Type	Hours
Independent Study	90
Teaching	60
Total study time	150

Resources & Reading list

General Resources

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

Textbooks

M D Greenberg. Advanced engineering mathematics. Cambridge University Press.

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

E Kreyszig. Advanced engineering mathematics. Wiley.

C Wylie and L C Barrett. Advanced engineering mathematics. McGraw Hill.

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

H A Priestley. An introduction to complex analysis. Oxford University Press.

L Debnath. Integral transforms and their applications. Chapman and Hall.

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

Assessment

Summative

This is how we’ll formally assess what you have learned in this module.

Breakdown
Method	Percentage contribution
Coursework	50%
Written assessment	50%

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Breakdown
Method	Percentage contribution
Written assessment	100%

Repeat Information

Repeat type: Internal & External