A STOCHASTIC COLLOCATION METHOD FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS AND RANDOM INPUTS

In this paper we propose a stochastic collocation method to solve the Volterra integro-differential equations with weakly singular kernels, random coefficients, and forcing terms. The input data are assumed to depend on a finite number of random variables. The method consists of the hp-versions of the continuous Galerkin and discontinuous Galerkin time-stepping schemes in time and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, which naturally leads to the solution of uncoupled deterministic problems. We establish a priori error estimates that are completely explicit with respect to all the discretization parameters. In particular, we show that exponential rates of convergence can be achieved in both the temporal direction and the probability space for solutions with start-up singularities by using geometrically refined time-steps and linearly increasing polynomial degrees. Numerical experiments are provided to illustrate the theoretical results.