Approximate solution of the nonlinear Fredholm integral equation of the second kind

This article deals with the numerical treatment of nonlinear Fredholm integral equations of the second kind. The equation treated in this paper has particular kernel, in sense that it is composed of the product between two parts: a weakly singular part not depending on the solution and a nonlinear Frechet differentiable part depending on our solution. The approximate solution proposed in this work is defined as an iterative sequence of Newton-Kantorovich type. To construct this solution, we use three numerical methods: the Newton-Kantorovich method to linearize our problem, the method of regularization with convolution and Fourier series expansion. It needs to obtain a finite rank sequence and "Hat functions projection'' to deal with nonlinear term in the Newton-Kantorovich construction. We prove that this particular Newton-like sequence converges perfectly to the exact solution. In addition, we construct some numerical example to demonstrate its effectiveness in practice. The obtained numerical results confirm the accuracy of the theoretical results.