Application of He's Homotopy Perturbation Method to Sumudu Transform

TL;DR: In this article, an application of He's homotopy perturbation method is proposed to compute Sumudu transform, in contrast of usual methods which need integration, requires simple differentiation.

Abstract: In this work, an application of He’s homotopy perturbation method is proposed to compute Sumudu transform. The method, in contrast of usual methods which need integration, requires simple differentiation. The results reveal that the method is very effective and simple. Keywords: Perturbation methods; Homotopy perturbation method; Sumudu transform 1. Introduction Various perturbation methods [1,2] have been widely applied by scientists and engineers to solve nonlinear problems. The traditional perturbation techniques are based on the existence of small parameter. These techniques are so powerful that sometimes small parameter is introduced artificially into a problem having no parameter and then finally set equal to unity to recover the solution of the original problem. He [3-6] proposed a new method called homotopy perturbation method (HPM) in 1998. The HPM, in fact, is a coupling of the traditional perturbation method and homotopy in topology. The HPM method, without demanding a small parameter in equations, deforms continuously to a simple problem which is easily solved. This method yields a very rapid convergence of the solution series in most cases, usually only a few iterations leading to very accurate solutions. This new method was further developed and improved by He and applied to non-linear oscillators with discontinuity [7], asymptotology [8], non-linear wave equations [9], bifurcation of nonlinear problems [10], limit cycle [11], delay-differential equations [[12], and boundary values problems [13]. He’s method is a universal one which can solve various types of non-linear problems. For example, it was applied to the non-Newtonian flow by Siddiqui et al. [14-15], to Volterra’s integro-differential equation by El-Shahed [16], to Helmholtz equation and fifth-order KdV equation by Rafei et al. [17], to nonlinear oscillator by Cai et al. [18], and to compute the Laplace transform by Abbasbandy [19]. A complete review on HPM’s applications is given in [20-21]. Sumudu transform was probably first time introduced by Watugala in his work [22]. Its simple formulation and direct applications to ordinary differential equations immediately sparked interest in this new tool. This new transform was further developed and applied to many problems by various workers. Asiru [23,24] applied to integro-differential equations, Watugala [25,26] extended the transform to two variables with emphasis on solution to partial differential equations and applications to engineering control problem, and its fundamentals properties were established by Belgacem et al. [27-29]. The Sumudu transform has very special and useful properties and can