Spectral Petrov-Galerkin methods using generalized log orthogonal functions for two types of weakly singular Volterra integro-differential equations

In this paper, we study highly accurate discretizations of two classes of weakly singular Volterra integro-differential equations (VIDEs). First, we consider ordinary VIDEs. A spectral Petrov–Galerkin method using generalized log orthogonal functions is proposed, and its convergence is investigated. After that, a space-time spectral method for partial VIDEs, which combines the aforementioned spectral Petrov–Galerkin method in time with a spectral Legendre–Galerkin method in space, is further developed. The well-posedness and convergence of the fully discrete scheme are studied. The results show that, under certain conditions, both schemes can achieve exponential convergence for some problems with solutions that exhibit weak singularities at the initial time. Finally, several numerical experiments are given to validate the efficiency of the methods.