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
Learning Outcomes
Knowledge and Understanding
Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:
- 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.
Subject Specific Intellectual and Research Skills
Having successfully completed this module you will be able to:
- 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
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 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.
| Type | Hours |
|---|---|
| Independent Study | 120 |
| Teaching | 60 |
| Total study time | 180 |
Resources & Reading list
Textbooks
R V Churchill and J W Brown. Complex Variables and Applications. McGraw Hill.
M D Greenberg. Advanced Engineering Mathematics. Cambridge University Press.
H A Priestley. An introduction to complex analysis. Oxford University Press.
E Kreyszig. Advanced Engineering Mathematics. Wiley.
C Wylie and L C Barrett. Advanced Engineering Mathematics. McGraw Hill.
L Debnath. Integral transforms and their applications. Chapman and Hall.
Assessment
Summative
This is how we’ll formally assess what you have learned in this module.
| Method | Percentage contribution |
|---|---|
| Examination | 60% |
| Coursework | 40% |
Referral
This is how we’ll assess you if you don’t meet the criteria to pass this module.
| Method | Percentage contribution |
|---|---|
| Written assessment | 100% |
Repeat
An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.
| Method | Percentage contribution |
|---|---|
| Examination | 100% |
Repeat Information
Repeat type: Internal & External