Homotopy analysis method

The Homotopy analysis method (HAM) is proposed by Shi-Jun Liao (1992) who is currently a professor in Shanghai Jiao Tong University. The method aims to solve the nonlinear ODE/PDE. It distinguishes itself from other analytical methods in the following four aspects. First, it is a series expansion method but it is independent of small physical parameters at all. Thus it is applicable for not only weakly but also strongly nonlinear problems. Secondly, the HAM is a unified method for the Lyapunov artificial small parameter method, the delta-expansion method and the Adomian decomposition method. We argue that the so-called ``homotopy perturbation method (HPM) which is proposed six years latter than the HAM is no newer than the HAM. Thirdly, the HAM provides us with a simple way to ensure the convergence of the solution; also it provides us with great freedom to choose the base function of the desired solution. Fourthly, the HAM can be combined with many other mathematical methods, such as the numerical methods, the series expansion method, the integral transform methods and so forth.