Homotopy analysis method - A new analytic approach for highly nonlinear problems

Abstract

In this talk, we briefly describe the basic ideas and applications of the homotopy analysis method (HAM), an analytic technique for highly nonlinear problems. Compared to other analytic approximation methods, the HAM has some advantages. First, unlike perturbation techniques, the HAM has nothing to do with any small/large physical parameters so that it works for more problems, especially for those without small/large physical parameters. Besides, unlike all other methods, the HAM provides us a simple way to guarantee the convergence of solution series. In addition, the HAM provides us great freedom to choose equation-type and solution expression of high-order equations so that it is easy to obtain approximations at rather high order. Due to these advantages, the HAM have been successfully applied to solve lots of nonlinear problems in science, engineering, finance and so on. By means of the HAM, some classical problems have been solved with much better results. Especially, some new concepts have been proposed and some new solutions have been found by means of the HAM, mainly because a truly new method always brings us something new/different!