Smoothing Transformation and Piecewise Polynomial Collocation for Weakly Singular Volterra Integral Equations

We discuss a possibility to construct high-order numerical algorithms on uniform or mildly graded grids for solving linear Volterra integral equations of the second kind with weakly singular or other nonsmooth kernels. We first regularize the solution of integral equation by introducing a suitable new independent variable and then solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid.