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 homotopy analysis method (HAM),is reviewed.Different from perturbation techniques,the homotopy analysis method is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.Besides, different from all other analytic techniques,it provides us a simple way to ensure the convergence of solution series,so that one can always get accurate enough analytic approximations.Furthermore,different from all other analytic methods,it provides us a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The homotopy analysis method provides us a completely new way and a different approach to solve nonlinear problems,especially those without small physical parameters.In this review paper,the basic concepts of the homotopy analysis method and its applications in nonlinear mechanics,physics,chemistry,biology,finance,engineering,computational mathematics and so on are discussed,together with its difference and relationship to perturbation techniques,Lyapunov artificial small parameter method,δ-expansion method,Adomian decomposition method,and the so-called homotopy perturbation method.