Fredholm integral equations of 1st and 2nd kinds are of practical importance and have wide range of applications. The present paper, deals mainly with system of non-linear Fredholm equations of the 2nd kind. In the present paper, the homotopy perturbation technique in different version from normal version is applied. The new version of the perturbation method confirms the simplicity and efficiency of the proposed method compared with other approximate solutions; also it confirms that this method is a suitable method for solving any nonlinear Fredholm Integral Equations of 2ndKind and / or systems of nonlinear Fredholm integral equations of 2nd kind. In the present paper, a new version of the homotopy perturbation technique is applied to solve system of nonlinear Fredholm integral equations. The new version based on the idea of considering the solution as a sum of an infinite series which is very rapid convergence to the accurate solution. The results due to the present version of the homotopy perturbation technique gave promises for further developing other issues of the homotopy perturbation method. The results due to the present method are compared with Adomain decomposition method.

The current paper is a trial to investigate the Eigen functions appears through mathematical treatment of 2nd kind of Fredholm integral equations by making use a new developed an Inverse Iterative Numerical Scheme (IINS). To test the applicability and accuracy of the proposed IINS, two numerical examples were solved and compared with previous results. The discussion of the results gave a good coincides up to some decimal places of results with the available ones.