High-order methods for Volterra integral equations with weak singularities

We consider the numerical solution of some classes of linear Volterra integral equations with singularities. We apply to them a smoothing transformation so that the exact solution of the resulting equation does not contain any singularities in its derivatives up to a certain order. After that the regularized equation is solved by a piecewise polynomial collocation method on a mildly graded or uniform grid. Global convergence estimates are derived and some superconvergence results are given.