BEYOND PERTURBATION: THE BASIC CONCEPTS OF THE HOMOTOPY ANALYSIS METHOD AND ITS APPLICATIONS

A new and rather general analytic method for strongly nonlinear problems, namely the homotopyanalysis method (HAM), is reviewed. Different from perturbationtechniques, the homotopy analysis method is totally independent ofsmall physical parameters, and thus is suitable for mostnonlinear problems. Besides, different from all other analytictechniques, it provides us a simple way to ensure the convergenceof solution series, so that one can always get accurate enoughanalytic approximations. Furthermore, different from all other analyticmethods, it provides us a great freedom to choose base functions ofsolution series, so that a nonlinear problem may be approximated more effectively. The homotopyanalysis method provides us a completely new way and a differentapproach to solve nonlinear problems, especially those withoutsmall physical parameters. In this review paper, the basic conceptsof the homotopy analysis method and its applications in nonlinearmechanics, physics, chemistry, biology, finance, engineering,computational mathematics and so on are discussed, together withits difference and relationship to perturbation techniques,Lyapunov artificial small parameter method, $\delta$-expansionmethod, Adomian decomposition method, and the so-called homotopyperturbation method.