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MATH3422 / MATH5003M / MATH5004M Project / Assignment in Applied Mathematics: Integral Equations

Project supervisor: Professor Mark A Kelmanson

News! This was the only Applied Mathematics project in Britain and Ireland to reach the finals of the 2006 Science, Engineering and Technology Awards.

Project description

Introduction - Many problems arising in the applied sciences can be formed in two distinct but connected ways, namely as differential equations (DEs) or integral equations (IEs). In the former approach, the boundary conditions must be applied externally, i.e. after a DE has been solved. However, in the latter approach, the boundary conditions are incorporated within the IE formulation and, on occasion, this has distinct advantages over the DE approach, particularly with regard to solving the problem as an infinite series.

Motivation - From the above description, IEs therefore comprise a rather succinct approach to problem solving, an approach which is rarely (these days) encountered at undergraduate level. As such, the project should constitute new and exciting ground which requires little in the way of specific prerequisites, although a fastidious approach towards detailed algorithms will be essential. This project should therefore appeal to students who enjoy a good blend of mathematical analysis and theory, coupled with the prospect of meticulous numerical and algorithmic solution methods. It should also provide a solid foundation for possible postgraduate work.

Overview - This project will consider simple analytical and numerical solution techniques for IEs in one variable.The project will primarily be concerned with linear IEs, although non-linear IEs may be considered if time permits.

Content - Specific subject matter will include a selection (varying from year to year) from the following broad topics: classification of IEs; connection between IEs and DEs; Green's functions; convolution IEs; Fredholm equations; Volterra equations; method of successive approximations; singular kernels; Hilbert space; degenerate kernels; variational principles, and; numerical quadrature for bounded and singular IEs.

Project requirements and booklist

Pre-requisites - Although there are no formal pre-requisites, it will be advantageous to have taken MATH2600 Numerical Analysis at level 2 and to take MATH3474 Numerical Methods at level 3, since these modules consolidate some of the mathematical ideas of the project.

Regulations - General information about the MATH3422 15-credit project and MATH5003M 30-credit and MATH5004M 40-credit assignments can be found here, with further details of rules and regulations for the project and assignments. Broadly speaking, the project should culminate in a professionally prepared 35-to-60-page scientific report of high quality. Note that consultation time with your supervisor is deliberately limited to one hour per week, since the object of the exercise is for you do demonstrate a degree of independence.

What is required of you - As well as reading the selected topics guided by your supervisor, you will, as the project proceeds, be required to learn the algebraic manipulator Maple to assist with the analytical solution of IEs. You may also be required to learn a basic programming language such as  Fortran in order to undertake numerical solution of IEs when no exact solution can be found. You will certainly be required to learn the text-processing language LaTeX in order to prepare your final report. Locally available details on all of these aspects can be found here (NB this page cannot be viewed from off-campus).  

Booklist - The idea is that you should read widely, so that this link (Updated for session 2010-11; includes latest library acquisitions) gives only a partial list.

Page created by Professor M A Kelmanson on 12th June 2005; last updated 21st September 2010.