Supergeometric Convergence of Spectral Collocation Methods for Weakly Singular Volterra and Fredholm Integral Equations with Smooth Solutions

Abstract

A spectral collocation method is proposed to solve Volterra or Fredholm integral equations with weakly singular kernels and corresponding integro-differential equations by virtue of some identities. For a class of functions that satisfy certain regularity conditions on a bounded domain, we obtain geometric or supergeometric convergence rate for both types of equations. Numerical results confirm our theoretical analysis.