MA3XJ-Integral Equations

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:6
Terms in which taught: Spring term module
Pre-requisites: MA2DE Differential Equations or MA2ODE Ordinary Differential Equations and MA1RA1 Real Analysis I or MA2RA1 Real Analysis I
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Module version for: 2017/8

Module Convenor: Prof Simon Chandler-Wilde

Summary module description:

This module in concerned with 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 students to 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 one-dimensional wave-scattering problems as integral equations.

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 meteorology. The main components of the module are:

•    Classification of integral equations.

•    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 approximations 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.

Brief description of teaching and learning methods:

Lectures, with some supported by course notes, problem sheets.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 4 Guided independent study 76 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written exam 85 Set exercise 15

Other information on summative assessment:

Formative assessment methods:

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the 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.

Length of examination:

2 hours.

Requirements for a pass:

A mark of 40% 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 (85% exam, 15% coursework).