Final year project in Applied Mathematics:
Sharp Error Estimates for the Numerical Solution of Fredholm Integral Equations of the Second Kind
(2005/2006)
| Supervisor: | Prof. Mark A Kelmanson | |
| Project–Homepage: | @ Uni Leeds |
Abstract
Several factors affecting the accuracy of the numerical solution of Fredholm integral equations of the second kind (FIE2s) are investigated. In the existing literature, generous error bounds are given in terms of the solution of the FIE2 and/or its derivatives. In this project, we obtain novel sharp error bounds in terms of explicit series expansions in integer powers of the numerical mesh size. Our explicit error analysis is developed in several directions. First, we investigate the effect of using increasingly accurate Lagrange functions to interpolate the solution of simple degenerate–kernel FIE2s, whose solutions we can obtain explicitly. Second, we are able to calibrate several numerical projection methods and to corroborate the accuracy of our theoretical error predictions. Third, we extend the analysis to higher–dimensional degenerate kernels with the aim of using these, in the infinite–sum limit, to obtain explicit error analyses of FIE2s with non–degenerate kernels. In this way, we make a novel contribution to the prediction of sharp error bounds for the numerical solution of quite general FIE2s whose explicit closed–form solution is unknown.