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This paper presents the modi cations of the variational iteration method (MVIM), the Laplace Adomian decomposition method (MLADM), and the homotopy perturbation method (MHPM) for solving the nonlinear Fredholm integro-di erential equation of the second kind. In these techniques, a series is established, the summation of which gives the solution of the discussed equation. The conditions ensuring convergence of this series are presented. Some examples to illustrate the investigated methods are presented as well, and the results reveal that the proposed methods are very e ective. Moreover, we present the comparison between our proposed methods with the exact solution and some traditional methods during numerical examples. The results show that (MHPM) and (MLADM) lead to an exact solution, whereas (MVIM) leads to limited solutions. Finally, the uniqueness of solutions and the convergence of the proposed methods are also proved.

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