A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis

Abstract

Regarding the efficient previous method for approximation of one-dimensional functions integrals through Bernstein polynomials in Amirfakhrian (2011), this paper presents a development of this method for approximation of two-dimensional functions integrals for the first time. Then, by combining this approximation with Bernstein collocation method for numerical solution of two-dimensional nonlinear Fredholm integral equations, the kernels double integrals of integral equations will be approximated. Combination of two-dimensional functions numerical integration method with numerical solution of integral equations method (in both methods, Bernstein polynomials were used) will result in increase of convergence speed and accuracy of the method. The convergence analysis of the method is completely presented. Finally, numerical examples are presented to illustrate the efficiency and superiority of our method in comparing it with other methods.