solving nonlinear two-dimensional volterra integral equations of the first-kind using 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 transformedto a nonlinear two-dimensional volterra integral equation of the second-kind. the properties ofthe bivariate shifted legendre functions are presented. the operational matrices of integrationtogether 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 themethod.