Three Methods For Seeking Asymptotic Solutions Of Nonlinear PDEs

In this paper, we focus on the method research on the process of seeking solutions for partial differential equations(PDEs, for short). Firstly, a transformation-assumption method is introduced for solving nonlinear partial differential equation with coefficients variable both in the space and time coordinates. Nonlinear Schr(o|¨)dinger equation with variable coefficients and Sine-Gorden equation with variable coefficients are employed to illuminate the transformation-assumption method. And many solutions for some specific situations of the above mentioned equations are obtained. Although in some cases only the approximate solutions are received for the corresponding constant coefficients ordinary differential equations instead of the analytic solutions, this method still supply us a new way to deal with the nonlinear equations with variable coefficients.Secondly, we introduce a modified Adomian decomposition method, which is used to solve inhomogeneous nonlinear partial differential equation with boundary conditions. At the light of Adomian decomposition method(ADM), we propose the modified Adomian decomposition method, which is different in dealing with the nonlinear term, compared with the ADM. The process of seeking the solution by the modified Adomian decomposition method is much faster and simpler than ADM in this paper, which is shown through an example of a boundary value problem.Thirdly, we implement the Bessel function expansion method by introducing Bessel equation as an assistant equation and changing the form of the anaze to the solutions, when the coefficients of the Bessel equation changes, we could get more solutions expressed by Ei, erf, Airy function, Whittaker function, and so on, not only limited on Bessel function. By this method, we obtain the exact solutions of KdV-Burgers equation and mKdV-Burgers equation with source modification successfully.Finally, the conclusions obtained from this paper are concluded and the prospect of the development for solving the nonlinear PDEs is discussed.