Nyström Method to Solve Two-Dimensional Volterra Integral Equation with Discontinuous Kernel

Abstract

In this paper, a linear two-dimensional Volterra integral equation of the second kind with the discontinuous kernel is considered. The conditions for ensuring the existence of a unique continuous solution are mentioned. The product Nystrom method, as a well-known method of solving singular integral equations, is presented. Therefore, the Nystrom method is applied to the linear Volterra integral equation with the discontinuous kernel to convert it to a linear algebraic system. Some formulas are expanded in two dimensions. Weights’ functions of the Nystrom method are obtained for kernels of logarithmic and Carleman types. Some numerical applications are presented to show the efficiency and accuracy of the proposed method. Maple18 is used to compute numerical solutions. The estimated error is calculated in each case. The Nystrom method is useful and effective in treating the two-dimensional singular Volterra integral equation. Finally, we conclude that the time factor and the parameter v have a clear effect on the results.