On solving 2D weakly singular Volterra integral equations of the second kind

In this paper, we present a semi-analytical approach to solving two-dimensional Volterra integral equations of the second kind which include weakly singular kernels. This approach is based on two-dimensional Laplace transform expansion method and on bivariate rational approximants. More specifically, the proposed approach consists in expanding the two-dimensional Laplace transform of the unknown function for large values and inverting it term by term in order to obtain a double convergent series expansion of the solution. Then, using a few coefficients from the obtained convergent series expansion, we provide the bivariate rational function representation of the unknown function of these integral equations through the application of bivariate homogeneous Padé approximants. Furthermore, several numerical examples have been provided in order to show the efficiency of our study.