Supergeometric convergence of spectral collocation methods for weakly singular Volterra and fredholm integral equations with smooth solutions

A spectral collocation method is proposed to solve Volterra or Predholm 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.