Homotopy Perturbation Method for a Reliable Analytic Treatment of some Evolution Equations

A new perturbation method called the homotopy perturbation method (HPM) [1 – 4] was proposed by Ji-Huan He in 1999, which is a coupling of the traditional perturbation method and homotopy in topology. The traditional perturbation methods are based on assuming a small parameter, and the approximate solutions obtained by those methods, in most cases, are valid only for small values of the small parameter even though many of nonlinear problems having strong nonlinearity have no small parameters at all. The homotopy perturbation method, without demanding a small parameter in the equations, deforms continuously to a simple problem which is easily solved. In this method, the solution is given in an infinite series, which usually converges rapidly to an accurate solution. The approximations obtained by the HPM are universally valid for small parameters, but also for very large parameters. This new method was further developed and improved by He, and a considerable amount of research has been conducted in applying this method to various kinds of linear and nonlinear problems, such as nonlinear oscillators with discontinuities and conservative nonlinear oscillators [5 – 8], nonlinear wave equations [9], limit cycle and bifurcations [10 – 13], nonlinear boundary value problems [14 – 15], asymptotology [16], integro-differential equation [17 – 19], non-Newtonian flow [20 – 21], systems of differential equations and stiff systems [22 – 24], differentialdifference equations [25 – 26], nonlinear Korteweg-de