A multi-domain hybrid spectral collocation method for nonlinear Volterra integral equations with weakly singular kernel

Abstract

In this paper, we propose a multi-domain hybrid spectral collocation method for the nonlinear Volterra integral equations (VIEs), whose solutions exhibit weak singularities at the endpoint t=0. In order to efficiently approximate the weakly singular solutions, we divide the interval [0,T] into N subintervals and use the Gauss points of the generalized log orthogonal functions as the collocation points to approximate the weakly singular integral term in the first subinterval. In the remaining subintervals, we use Legendre and Jacobi Gauss points as the collocation points to approximate the corresponding integral terms. We also provide a rigorous hp-version convergence analysis for the hybrid spectral collocation method under L2-norm. A series of examples demonstrate our method is particularly suitable for solutions that have weak singularities at one endpoint.