Solving Nonlinear Partial Differential Equations By Homotopy Analysis Method

In the early 1990 s, based on the application of homotopy idea in topology theory, Shi-jun Liao proposed the homotopy analysis method(HAM) for the first time. Different from the traditional analytic method, the homotopy analysis method does not depend on small parameters.At the same time, The method enjoys great freedom in choosing different basis function to represent the solution of nonlinear problems and it has the function to regulate the region and the rate of convergence of the series. Therefore, the homotopy analysis method is an important method to solve the nonlinear problem.In this paper, we mainly introduce the basic ideas of the homotopy analysis method.According to this method, we can obtain the approximate solutions of the Ostrovsky equation,KPP equation and STO equation with the initial value. In the process of solving the first two equations, we change the partial differential equations into ordinary differential equations by means of traveling wave transformation. When solving STO equation with the initial value,we can get different form of solution due to the difference of basis function. In this paper, we calculate the approximate periodic solutions and approximate solitary wave solution of nonlinear equations. With the aid of computer algebra system Mathematica, we can make the approximate solutions for the error analysis. The results further illustrate the applicability and superiority of the homotopy analysis method.