Abstract
Although attempts have been made to solve time-dependent differential equations using homotopy perturbation method (HPM), none of the researchers have provided a universal homotopy equation. In this paper, going one step forward, we intend to make some guidelines for beginners who want to use the homotopy perturbation technique for solving their equations. These guidelines are based on the L part of the homotopy equation and the initial guess. Afterwards, for solving time-dependent differential equations, we suggest a universal L and v0 in the homotopy equation. Examples assuring the efficiency and convenience of the suggested homotopy equation are comparatively presented.
1. Introduction
In recent years, the homotopy perturbation method (HPM), first proposed by Dr. Ji Huan He [1,2], has successfully been applied to solve many types of linear and nonlinear functional equations. This method, which is a combination of homotopy in topology and classic perturbation techniques, provides us with a convenient way to obtain analytic or approximate solutions for a wide variety of problems arising in different fields.
7. Conclusions
In this paper, we proposed some guidelines for beginners who intend to solve their problems using the homotopy perturbation method. In the sequel we comparatively reviewed procedures which are used by researchers, through two examples. Then we presented a simple way to choose L and v0 when we use the homotopy perturbation method to solve time-dependent differential equations. In most cases, our simple choice yields exact an solution or at least very good approximations. Although there are examples that show our choice isn’t as good as other choices, it still produces convergent series that makes it a reliable one in solving a wide class of functional equations.
