We propose and analyze the spectral method for the partial integro-differential equations(PIDE) with a weakly singular kernel. The space discretization is based on the spectral methods using Jacobi orthogonal polynomials. We prove the unconditional stability and obtain the optimal error bounds which depend on the degree of polynomial and the Sobolev regularity of the solution. Moreover, we analyze the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points for the Jacobi weights.
We obtain stability and convergence results for two different time discretization of PIDE, based on the equally spaced and graded mesh backward Euler schemes, respectively. Especially, graded mesh backward Euler schemes give us second order convergence rate for the time discretization for partial integro-differential equations with weakly singular kernels.
