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

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

“…by the latter process, which is considerably less accurate than the former because of the ill-conditioning of its inherent differentiation matrix. Accordingly, an approach independent of [21] is presently pursued in which the need for numerical differentiation is circumvented by first transforming the FIDE (as in, e.g., [29]) into a Volterra-Fredholm integal equation (VFIE); though the solution of this can be approximated in a number of ways (see, e.g., [26,16,13,9,33]), a different approach, first explored by the authors in [19], is adopted herein.…”

“…In §3 the VFIE is solved numerically to spectral order in N , the degree of the highest-order orthogonal polynomial used in the approximation of the VFIE solution. This approach obviates the need for the numerical differentiation matrices, used in a companion paper [21], the ill-conditioning of which is reviewed and analysed in §3.1. In §4 is presented a novel error analysis, for the VFIE numerical solution procedure, whose distinctive aspect is computation of the error in the numerical solution of the original FIDE explicitly in terms of the numerical approximation of the derivate that results from the VFIE reformulation.…”

“…For completeness, circumvention of the use of numerical differentiation (as employed in [21]) is now discussed. The action of the differentiation operator D in (2) is approximated by the operator D N , with D N u N in the pre-transformed system from which (17) was derived.…”

“…By (65) the error u − u N in the exact solution u of FIDE (2) is predicted explicitly in terms of the numerical solution v N of approximate VFIE (19), where u ′ N = v N . By contrast, in the independent approach adopted in [21] (hereafter referred to as case 0), the error in u is given explicitly not by v N but by…”

“…The present work is therefore motivated on two fronts: to develop not only a novel numerical method that converges exponentially in the dimension N of the discrete numerical method, but also an explicit error analysis that is implementable and yields errors in terms of only the computed numerical solution. In [21], the authors develop a novel approach for achieving these two goals, but the method developed thereinbased on a combination of numerical quadrature and numerical differentiation-has a global error dictated…”