Construction of the generalized iterative methods used for solution of the Fredholm integral equation

In this paper, we consider the Fredholm integral equations of the second kind and construct a new iterative scheme associated to the Nyström method, which was elaborated by Atkinson to approximate the solution over a large interval. Primarily, we demonstrate the inability to generalize the Atkinson iterative methods. Then, we describe our modified generalization in detail and discuss its advantages such as convergence of the iterative solution to the exact solution in the sense norm of the Banach space $C^0[a, b]$. Finally, we give a numerical examples to illustrate the accuracy and reliability of our generalization.