A new vision of the He's homotopy perturbation method

This paper proposes a reliable modification of the homotopy perturbation method which can serve as a promising tool for solving a large class of differential equations. It may be concluded that the homotopy methodology is very powerful and efficient in finding analytical as well as numerical solutions for wide classes of linear and nonlinear differential equations. It provides more realistic series solutions that converge very rapidly in real physical problems. It is worth noting that the major advantage of He's homotopy perturbation method is that the perturbation equation can be freely constructed in many ways by homotopy in topology and the initial approximation can also be freely selected and yield solutions in convergent series forms with easily computable terms, and in some cases, provide exact solutions in one iteration. In contrast to the traditional perturbation methods, it does not require a small parameter in the system. Therefore, taking advantage of these points, we propose a reliable modification of He's homotopy perturbation method. Indeed, this constructs an initial trial-function without unknown parameters, which is called the modified homotopy perturbation method. Some of the linear and nonlinear and integral equations are examined by the modified method to illustrate the effectiveness and convenience of this method, and in all cases, the modified technique performed excellently.