Solving Nonlinear Two-Dimensional Volterra Integral Equations of the First-kind Using the Bivariate Shifted Legendre Functions

In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations of the first-kind is proposed. This problem is transformed to a nonlinear two-dimensional Volterra integral equation of the second-kind. The properties of the bivariate shifted Legendre functions are presented. The operational matrices of integration together with the product operational matrix are utilized to reduce the solution of the second-kind equation to the solution of a system of linear algebraic equations. Finally, a system of nonlinear algebraic equations is obtained to give an approximate solution of the main problem. Also, numerical examples are included to demonstrate the validity and applicability of the method.