A priori Nyström-method error bounds in approximate solutions of 1-D Fredholm integro-differential equations

Abstract

A novel procedure is proposed for the a priori computation of error bounds for the ubiquitous Nyström solver applied to one-dimensional Fredholm integro-differential equations. The distinctive feature of the new approach is that the bounds are computed not only to spectral accuracy, but also explicitly, and in terms of only the numerical solution itself. Details are given of both the error analysis and its numerical implementation, and a corroborative asymptotic theory is developed in order to yield independent predictions of the convergence rates expected from Nyström discretisations of increasing order. All theory is first convincingly validated on a proof-of-concept continuous-kernel test problem whose solution is a priori known. The method is then applied to a novel integro-differential-equation formulation of a static, fourth-order, Euler-Bernoulli beam-deflection boundary-value problem in which the flexural rigidity varies along the beam, and for which no exact solution is attainable; in this case, validation of the resulting discontinuous-kernel approach is achieved using an asymptotic solution derived on the (realistic) assumption that variations in the cross-section of the beam occur on spatial scales an order of magnitude less than the beam’s length and width. Potential limitations of the new approach are discussed.