A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels

Abstract

In this paper, we propose a hybrid spectral method for a type of nonlocal problems, nonlinear Volterra integral equations (VIEs) of the second kind. The main idea is to use the shifted generalized Log orthogonal functions (GLOFs) as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals. This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions. The well-posedness and the related error estimates will be provided. Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.