Numerical solution of two-dimensional weakly singular Volterra integral equations of the first kind via bivariate rational approximants

We present a numerical method for solving the two-dimensional weakly singular Volterra integral equations of the first kind. The integral equation is first converted into algebraic form using the two-dimensional Laplace transform. We then derived the series expansion for large values, which is inverted term by term to provide the convergent series expansion of the solution for small values. The asymptotic expansion of the solution is extracted from the series expansions of the two-dimensional Laplace transform around the singularity points. The bivariate homogeneous two-point Padé approximants are used to improve convergence. Numerical results are provided to illustrate the accuracy of the method.