An exponentially convergent Volterra-Fredholm method for integro-differential equations

Extending the authors’ recent work [15] on the explicit computation of error bounds for Nystrom solvers applied to one-dimensional Fredholm integro-differential equations (FIDEs), presented herein is a study of the errors incurred by first transforming (as in, e.g., [21]) the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE). The VFIE is solved via a novel approach that utilises N-node Gauss-Legendre interpolation and quadrature for its Volterra and Fredholm components respectively: this results in numerical solutions whose error converges to zero exponentially with N, the rate of convergence being confirmed via large-

N asymptotics. Not only is the exponential rate inherently far superior

to the algebraic rate achieved in [21], but also it is demonstrated, via diverse test problems, to improve dramatically on even the exponential rate achieved in [15] via direct Nystrom discretisation of the original FIDE; this improvement is confirmed theoretically.