Homotopy perturbation method to solve non-linear differential equations

Abstract

The thesis contains a survey of the perturbation method [cf. Nayfeh, 1981] and generalization of the perturbation method for the solution of linear and Nonlinear differential equations. The generalization of perturbationmethod has been first introduced by Ji-Huan He (1999) called Homotopy Perturbation Method (HPM). According to Cheniguel and Reghioua (2013), Biazar and Eslami (2011), Mechee et al. (2017), etc., HPM is one of the new and excellent methods for solving the nonlinear differential equation. It is well known that the perturbation theory is based on an assumption of an equation (in the form of power series) with a small parameter. A perfect choice of small parameter leads to the excellent result. However, if the choice of the small parameter is not suitable then the solution is going to be a bad asset. In such cases, the HPM can find the accurate approximate solution of the differential equation. This method does not depend on the small parameter in the assumed equation. The HPM is a combination of homotopy and perturbation method which provides an advantageous way to obtain an analytical or approximate solution of the differential equations. Chapter 1, contains the literature survey of perturbation, homotopy and generalized homotopy perturbation method. In Chapter 2, we have discussed the basic theory of perturbation method and its application from the books of Nayfeh (1981) and Liao (1995). The Chapter 3, started with the basic idea of homotopy [cf. Ji-Huan He, 1999] and extended to the study of the HPM[He, 2005, 2006; Hemeda, 2012]. In this chapter, we have discussed Inviscid Burgers Nonlinear problem and nonhomogeneous Advection Nonlinear problem using the HPM. In Chapter 4, we survey the literature of the generalization of the homotopy perturbation method (GHPM) from the article of Hector, (2014). The last Chapter contains the conclusions and future plan. Keywords: Homotopy, Homotopy perturbation method, Linear, Nonhomogeneous, Nonlinear, ODEs, PDEs, Perturbation method, Power series, Topology.