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
Learning Outcomes
Having successfully completed this module you will be able to:
- Be confident in the use of complex variable theory and contour integration
- Understand how integral transforms can be used to solve a variety of differential equations
- Be able to demonstrate knowledge of a range of applications of these methods
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.
| Type | Hours |
|---|---|
| Teaching | 60 |
| Independent Study | 90 |
| 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
G Stephenson and P M Radmore. Advanced mathematical methods for engineering and science students. Cambridge University Press.
M R Spiegel. Theory and problems of complex variables (Schaum). McGraw Hill..
M D Greenberg. Advanced engineering mathematics. Cambridge University Press.
C Wylie and L C Barrett. Advanced engineering mathematics. McGraw Hill.
R V Churchill and J W Brown. Complex variables and applications. McGraw Hill..
L Debnath. Integral transforms and their applications. Chapman and Hall.
E Kreyszig. Advanced engineering mathematics. Wiley.
H A Priestley. An introduction to complex analysis. Oxford University Press.
Assessment
Summative
This is how we’ll formally assess what you have learned in this module.
| Method | Percentage contribution |
|---|---|
| Coursework | 40% |
| Examination | 60% |
Referral
This is how we’ll assess you if you don’t meet the criteria to pass this module.
| Method | Percentage contribution |
|---|---|
| Examination | 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