Optimal Homotopy Analysis Method

Abstract

In this chapter, we describe and compare the different optimal approaches of the homotopy analysis method (HAM). A generalized optimal HAM is proposed, which logically contains the basic optimal HAM with only one convergence-control parameter and also the optimal HAM with an infinite number of parameters. It is found that approximations given by the optimal HAMs converge fast in general. Especially , the basic optimal HAM mostly gives good enough approximations. Thus, the optimal HAMs with a couple of convergence-control parameters are strongly suggested in practice. 3.1 Introduction Nonlinear equations are much more difficult to solve than linear ones, especially by means of analytic methods. In general, there are two standards for a satisfactory analytic method of nonlinear equations: (a) It can always provide analytic approximations efficiently. (b) It can ensure that analytic approximations are accurate enough for all physical parameters. Using above two standards as criterion, let us compare different analytic techniques for nonlinear problems. Perturbation techniques (Lindstedt, 1882; Cole, 1992; Von Dyke, 1975; Mur-dock, 1991; Kevorkian and Cole, 1995; Nayfeh, 2000) are widely applied in science and engineering. Perturbation techniques are based on small/large physical parameters (called perturbation quantities) in governing equations or initial/boundary conditions. In general, perturbation approximations are expressed in series of perturbation quantities, and a nonlinear equation is replaced by an infinite number of linear (sometimes even nonlinear) sub-problems, which are completely determined by the type of the original governing equation and especially by the place where perturbation quantities appear. Perturbation methods are simple, and easy to understand.