Numerical methods for 2D weakly singular Volterra integral equations of the second kind

The paper is dedicated to the numerical solution of 2D weakly singular Volterra integral equations with two different kernels. In order to weaken a singularity influence on the numerical computations we transform these equations in both cases into equivalent equations. The piecewise polynomial approximation of the exact solution is then applied. For numerical computing of weakly singular integrals we construct the special Gauss-type cubature formula based on a nonuniform grid. Also we suggest the practical mesh which is less nonuniform than standard graded mesh tk = (k/N)^{r /(1–α)} T, and at the same time it gives an equivalent approximation error for the functions from C^r,α (0, T ]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)