一个解特定二阶驻定方程的新方法——特征伴随方程法

A new method for solving specific second order stationary equations:characteristic adjoint equation method

本文对文献[1]中提出的一类二阶驻定方程G″/G=∑^(m)_(j=0)P_(j)(G′/G)^(j)的求法进行探讨,提出了全新的求解方法——特征伴随方程法,通过该求法得到这类方程当m取不同非负整数(本文仅讨论m=2)时的通解,并相应给出该方程作为求解非线性偏微分方程的辅助方程时所需要的G′/G的解析表达式;同时给出一个作为常微分方程中驻定方程的应用实例,以及该方程作为非线性偏微分方程的辅助方程时利用扩展G′/G展开法的求解例子,通过该实例给出方程的精确行波解。

The characteristic adjoint equation method is presented for solving a type of second-order m stationary equation,which is denoted as G″/G=∑^(m)_(j=0)P_(j)(G′/G)^(j),m=j=00,1,2,⋯,and was initially proposed in paper[1].With the new method,the general solution of the equation can be obtained when m is equal to different non-negative integers(we only discuss m=2 in this text).The analytical expression G′/G of the equation can also be obtained when the second-order stationary equation is used as an auxiliary equation for solving nonlinear partial differential equations.Two application examples are shown.In the first case,the second-order stationary equation is used as a stationary equation in the ordinary differential equation;in the second example,it is regarded as an auxiliary equation for solving the nonlinear partial differential equation by extending G′/G using the expansion method,which obtains the corresponding exact traveling wave solution.