Smoothing and spline collocation for weakly singular Volterra integro-differential equations

The numerical solution of linear weakly singular Volterra integro-differential equations is discussed. Using an integral equation reformulation of the initial-value problem, we apply to it 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.