Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

The Gauss product quadrature rules and collocation method are applied to reduce the second-kind nonlinear two-dimensional Fredholm integral equations (FIE) to a nonlinear system of equations. The convergence of the proposed numerical method is proved under certain conditions on the kernel of the integral equation. An iterative method for approximating the solution of the obtained nonlinear system is provided and its convergence is proved. Also, some numerical examples are presented to show the efficiency and accuracy of the proposed method.

► We consider the second-kind nonlinear two-dimensional Fredholm integral equations.
► We use the Gauss quadrature rules and collocation method to discretize the problem.
► An iterative method for the solution of the obtained nonlinear system is provided.
► The convergence of the proposed numerical method is proved.
► High precision of the results is obtained with a small number of quadrature points.