Analytic approximate solution for some integral equations by optimal homotopy analysis transform method

The main aim of this paper is to propose a new and simple algorithm namely homotopy analysis transform method (HATM), to obtain approximate analytical solutions of integral equations. Integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions also we show that the proposed method is very efficient and computationally attractive. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter $\hslash$ to guarantee the convergence of the obtained series solution