Error analysis of a spectrally accurate Volterra-transformation method for solving 1-D Fredholm integro-differential equations

Spectrally accurate a priori error estimates for Nyström-method approximate solutions of one-dimensional Fredholm integro-differential equations (FIDEs) are obtained indirectly by transforming the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE), which is solved via a novel approach that utilises N-node Gauss-Legendre interpolation and quadrature for its Volterra and Fredholm components respectively. Errors in the numerical solutions of the VFIE converge to zero exponentially with N, the rate of convergence being confirmed via large-N asymptotics. Not only is the exponential rate far superior to the algebraic rate achieved in previous literature [29] but also it is demonstrated, via diverse test problems, to improve dramatically on even the exponential rate achieved in the approach [21] of direct Nyström discretisation of the original FIDE; this improvement is confirmed theoretically.