Comparison between the homotopy analysis method and homotopy perturbation method

In this paper, we show that the so-called "homotopy perturbation method" is only a special case of the homotopy analysis method. Both methods are in principle based on Taylor series with respect to an embedding parameter. Besides, both can give very good approximations by means of a few terms, if initial guess and auxiliary linear operator are good enough. The difference is that, "the homotopy perturbation method" had to use a good enough initial guess, but this is not absolutely necessary for the homotopy analysis method. This is mainly because the homotopy analysis method contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region and rate of solution series. So, the homotopy analysis method is more general. Besides, the update of the concept of the "analytical solution" is discussed. (c) 2004 Elsevier Inc. All rights reserved.