Systematic Descriptions and Related Theorems

Abstract

In this chapter, the homotopy analysis method (HAM) is systematically described in details as a whole. Mathematical theorems related to the so-called homotopy-derivative operator and deformation equations are proved, which are helpful to gain high-order approximations. Some theorems of convergence are proved, and the methods to control and accelerate convergence are generally described. A few of open questions are discussed. 4.1 Brief frame of the homotopy analysis method In Chapter 2, the basic ideas of the homotopy analysis method (HAM) (Liao Liao and Tan, 2007; Li et al., 2010; Xu et al., 2010) are described by means of two simple examples. In this chapter, we systematically describe the HAM in a general way. The starting-point of the homotopy analysis method is to construct the so-called zeroth-order deformation equation. Given an original nonlinear equation N [u(x,t)] = 0 denoted by E 1 E E , which has at least one solution u(x,t), where N denotes a nonlinear operator, x is a vector of all spatial independent-variables, t denotes the temporal independent-variable, respectively. Assume that we can choose an initial equation E 0 E E whose solution u 0 (x,t) is easy to know, and that we can construct such a homotopy (Armstrong, 1983; Sen, 1983) of equations˜Eequations˜ equations˜E (q) : E 0 E E ∼ E 1 E E that, as the homotopy-parameter q ∈ [0, 1] increases from 0 to 1, ˜ E (q) deforms (or varies) continuously from the initial equation E 0 E E to the original equation E 1 E E , while its solution exists for q ∈ [0, 1] and besides varies continuously from the known solution u 0 (x,t) of the initial equation E 0 E E to the unknown solution u(x,t) of the original equation E 1 , i.e. N [u(x,t)] = 0. Such kind of homotopy of equations is called the zeroth-order