Numerical solution of two-dimensional nonlinear Volterra integral equations of the first kind using operational matrices of hybrid functions

In this paper, operational matrices of two-dimensional hybrid of block-pulse functions and Legendre polynomials are employed to approximate the solutions of two-dimensional nonlinear Volterra integral equations of the first kind (2DNVIEF). To this aim, we first convert 2DNVIEF to the two-dimensional linear Volterra integral equations of the second kind (2DLVIES). Then the operational matrices of integration, together with the product operational matrix, can be applied to reduce the 2DLVIES to a system of algebraic equations. The convergence of the proposed method is studied, and an error estimation under the L-2-norm is provided. Finally, some numerical examples are prepared to clarify the accuracy and efficiency of the numerical method especially in dealing with functions that are not continuous in the whole interval.