A Method to Estimate the Solution of a Weakly Singular Non-linear Integro-differential Equations by Applying the Homotopy Methods

In this paper, the weakly singular nonlinear integro-differential equation is solved by using the homotopy perturbation and homotopy analysis methods . the approximation solution of this equation is calculated in the form of a series which its components are computed easily . the existence and uniqueness of the solution and the convergence of the proposed method are proved. a numerical example is studied to demonstrate the accuracy of the presented method