Application of hybrid functions operational matrices in the numerical solution of two-dimensional nonlinear integral equations

Abstract

In this paper, an effective numerical technique based on the two-dimensional hybrid of Block-pulse functions and Legendre polynomials is presented to approximate the solution of a class of two-dimensional nonlinear Volterra integral equations of the second kind. The main idea of this work is to develop the operational matrices of hybrid functions into two-dimensional mode. These operational matrices are then applied to convert two-dimensional nonlinear integral equations into a system of nonlinear algebraic equations that can be solved numerically by Newton's method. An approximation of the error bound for the proposed method will be presented by proving some theorems. Finally, some numerical examples are considered to confirm the applicability and efficiency of the method, especially in cases where the solution is not sufficiently smooth.