Integral Transform Methods

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

• 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

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.

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.

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..

E Kreyszig. Advanced engineering mathematics. Wiley.

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

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

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

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

Summative

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

Referral

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

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.

Repeat Information

Repeat type: Internal & External