Error Analyses for Nyström Methods for Solving Fredholm Integral and Integro-Differential Equations

This thesis concerns the development and implementation of novel error analyses for ubiquitous Nyström-type methods used in approximating the solution in 1-D of both Fredholm integral- and integro-differential equations of the second-kind, (FIEs) and (FIDEs). The distinctive contribution of the present work is that it offers a new systematic procedure for predicting, to spectral accuracy, error bounds in the numerical solution of FIEs and FIDEs when the solution is, as in most practical applications, a priori unknown.
The classic Legendre-based Nyström method is extended through Lagrange interpolation to admit solution of FIEs by collocation on any nodal distribution, in particular, those that are optimal for not only integration but also differentiation. This offers a coupled extension of optimal-error methods for FIEs into those for FIDEs. The so-called FIDE-Nyström method developed herein motivates yet another approach in which (demonstrably ill-conditioned) numerical differentiation is bypassed by reformulating FIDEs as hybrid Volterra-Fredholm integral equations (VFIEs). A novel approach is used to solve the resulting VFIEs that utilises Lagrange interpolation and Gaussian quadrature for the Volterra and Fredholm components respectively.
All error bounds implemented for the above numerical methods are obtained from novel, often complex extensions of an established but hitherto-unimplemented theoretical Nyström-error framework. The bounds are computed using only the available computed numerical solution, making the methods of practical value in, e.g., engineering applications. For each method presented, the errors in the numerical solution converge (sometimes exponentially) to zero with N, the number of discrete collocation nodes; this rate of convergence is additionally confirmed via large-N asymptotic estimates. In many cases these bounds are spectrally accurate approximations of the true computed errors; in those cases that the bounds are not, the non-applicability of the theory can be predicted either a priori from the kernel or a posteriori from the numerical solution.