A spectral collocation method for eigenvalue problems of compact integral operators

Abstract

We propose and analyze a new spectral collocation method to solve eigenvalue problems of compact integral operators, particularly, piecewise smooth operator kernels and weakly singular operator kernels of the form 1/|t − s| µ , 0 < μ < 1. We prove that the convergence rate of eigenvalue approximation depends upon the smoothness of the corresponding eigenfunctions for piecewise smooth kernels. On the other hand, we can numerically obtain a higher rate of convergence for the above weakly singular kernel for some μ's even if the eigenfunction is not smooth. Numerical experiments confirm our theoretical results. 2010 AMS Mathematics subject classification. Primary 47A10, 47A58, 65J99, 65MR20.