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.