Adaptive spectral solution method for Fredholm integral equations of the second kind

Abstract

The purpose of this paper is to develop and analyse an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits a localized rapid variations, steep gradients, or steep front. The strategy of the adaptive procedure is to transform the given equation into an equivalent one with a sufficiently smooth behavior in order to ensure the convergence of the Legendre spectral collocation method without dividing the domain of the integral equation, as usual, into the sub intervals. Furthermore, the existence and uniqueness of the solution are discussed. Error analysis is carried out in both L ∞ -norm and L 2 -norm. Finally, several numerical examples are provided to show that the proposed method is preferable to its classical predecessor and some other existing approaches.