## MA4XJ-Integral Equations

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Spring term module
Pre-requisites: MA2DE Differential Equations and MA2RCA Real and Complex Analysis or MA3RCA Real and Complex Analysis
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3XJ Integral Equations
Current from: 2021/2

Module Convenor: Prof Simon Chandler-Wilde

Type of module:

Summary module description:

This module covers the theory, application and solution of integral equations, with an emphasis on applications that are part of research across the School, at Reading (for example wave scattering of water waves, of acoustic and electromagnetic waves by atmospheric particles, etc.).

Aims:

To introduce the theory, application and solution of integral equations, with some emphasis on aspects relevant to the large research effort in this area in mathematics and meteorology.

Assessable learning outcomes:

By the end of the module students are expected to be able to:

• formulate integral equations as problems in a Banach space;

• apply approximation techniques for solving integral equations and be able to draw conclusions about their accuracy;

• formulate simple wave-scattering problems as integral equations;

• formulate two-dimensional boundary value problems in potential theory as boundary inte gral equations, and solve these equations numerically.

This module will be assessed to a greater depth than the excluded module MA3XJ.

Outline content:

In applied mathematics many physical problems are best formulated as integral equations. In this course, a general introduction to the key issues is followed by a discussion of widely used approximation techniques, and this leads on to a detailed examination of real-world wave-scattering problems, arising in our research in mathematics, and in applications in metetorology. The main components of the module are:

• Classification of integral equations.
< br /> • Exact solution of degenerate kernel Fredholm integral equations;

• Boundedness of integral operators with continuous and weakly singular kernels, and computation of the norm;

• Questions of uniqueness and existence of solution (in part tackled by functional analysis methods): the Fredholm alternative and Neumann series;

• Numerical methods for Fredholm and Volterra integral equations, namely degenerate kernel app roximations and Trapezium rule time-stepping;

• Applications of integral equation methods to wave scattering: the Lippmann Schwinger integral equation and application in atmospheric particle scattering;

• The numerical analysis of the trapezium rule method for Volterra integral equations via Gronwall inequalities;

• Boundary integral equations in potential theory in 2D and their numerical solution by a simple Nystrom method. (This last topic studied largely in the written assignment.)

Brief description of teaching and learning methods:
Lectures supported by problem sheets.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 4 Project Supervision 2 Guided independent study: 74 Total hours by term 100 Total hours for module 100

Summative Assessment Methods:
 Method Percentage Written exam 70 Written assignment including essay 20 Set exercise 10

Summative assessment- Examinations:
2 hours.

Summative assessment- Coursework and in-class tests:

One examination, plus assignments.

Formative assessment methods:
Problem sheets.

Penalties for late submission:

The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.
The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:

A mark of 50% overall.

Reassessment arrangements:

One examination paper of 2 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (70% exam, 30% coursework).