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.
