用于求解某些变系数非线性偏微分方程的变换假设法

In this paper, we introduce a transformation-assumption method for solving nonlinear partial differential equation with coefficients variable both in the space and time coordinates. Many works have been done in the nonlinear equations with variable coefficients, but in most of the works, the discussion are limited on the spacial cases which the coefficients are variable only in one coordinate, the space or the time coordinate. A transformation-assumption method, which is used to transform the nonlinear partial differential equations into an ordinary differential equation with constant coefficients under the assumption between the variable coefficients and transformation of independent variables, is proposed in this paper. We employ nonlinear Schrodinger equation with variable coefficients, and Sine-Gorden equation with variable coefficients to illuminate this method, and obtain many solutions for some specific situations. Although in some cases only the approximate solutions are obtained 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.