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This paper studies the method for establishing an approximate solution of nonlinear two dimensional Volterra integral equations (NLTD-VIE). The Newton-Kantorovich (NK) suppositions are employed to modify NLTD-VIE to the sequence of linear two dimensional Volterra integral equation (LTD-VIE). The properties of the two dimensional Gauss-Legenre (GL) quadrature fromula are used to abridge the sequence of LTD-VIE to the solution of the linear algebraic system. The existence and uniqueness of the approximate solution is demonstrated, and an illustrative example is provided to show the precision and authenticity of the method.

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Newton-Kantorovich method is applied to obtain an approximate solution for a system of nonlinear Volterra integral equations which describes a large class of problems in ecology, economics, medicine and other fields. The system of nonlinear integral equations is reduced to find the roots of nonlinear integral operator. This nonlinear integral operator is solved by the Newton-Kantorovich method with initial guess and this procedure is continued by iteration method to find the unknown functions. Finally, numerical examples are provided to show the validity and the efficiency of the method presented.