On the Solution of Discretely Given Fredholm Integral Equations Over Lines

Abstract

The purpose of this paper is the study of the solution of Fredholm integral equations given over lines, whose parametrization is not explicitly known. We rather assume these lines known only at some points, and we apply quadrature collocation methods. The latter are based on Newton-Cotes type formulae. A preceding investigation of the direct problem, [1], i. e. of the evaluation line integrals, shows unexpected results. The convergence is indeed higher than expected on the basis of interpolation theory, in certain cases. In this study we extend the results to show that they hold for the inverse problem as well. One possible situation where this investigation could have a positive impact is in the case of planar boundary integral equations, when the boundary is known, but an explicitly differentiable parametrization for it is not available. Another application of these results could be in computer graphics.