A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays

摘要

In this paper, we develop a hybrid spectral method for the nonlinear second-kind Volterra integral equations (VIEs) with weakly singular kernel and vanishing delays. Our main strategy is to divide the original interval into subintervals, to employ the shifted generalized log orthogonal functions (GLOFs) as the basis on the first interval, to take the classical shifted Legendre polynomials as the basis on other intervals. We analyze the existence and uniqueness of the numerical scheme, and derive the corresponding error estimates. A series of examples demonstrate the effectiveness of the proposed method.