This paper presents the modications of the variational iteration method (MVIM), the Laplace Adomian decomposition method (MLADM), and the homotopy perturbation method (MHPM) for solving the nonlinear Fredholm integro-dierential 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 eective. 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.

In this paper, we apply the homotopy perturbation method (HPM), modied homotopy perturbation method (MHPM), variational iteration method (VIM), Adomian decomposition method (ADM), and modied Adomian decomposition method (MADM) to solve nonlinear mixed Volterra-Fredholm integral equations and its system. We investigate the approximate solution of this equation and its system via proposed methods. The validity and eciency of these methods are demonstrated through various numerical examples that illustrate the eciency, accuracy, and simplicity of the proposed methods. Moreover, the convergence and uniqueness of the solution of the suggested methods are conrmed and compared with the exact solutions.