Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods

Abstract

The paper presents the modifications of the variational iteration method (MVIM), Laplace Adomian decomposition method (MLADM), and the homotopy perturbation method (MHPM) for solving the nonlinear Fredholm integro-differential equation of the second kind. In these methods a series is created, the summation of which gives the solution of the discussed equation. Conditions ensuring convergence of this series are presented in the paper. An example illustrating the usage of the investigated methods is presented as well and the results reveal that the proposed methods are very effective, able, and simple. comparison between our proposed methods with the exact solution and some traditional methods is presented during a numerical example. The results reveal that (MHPM) and (MLADM) lead to an exact solution and (MVIM) leads to a limited solution. The uniqueness of the solutions and the convergence of the proposed methods are also proved.