用试探函数法求KdV-Burgers方程的精确解析解

Seeking Exact Analytical Solution to KdV-Burgers Equation in Trial Function Method

利用两种试探函数法,即先作变换后选取试探函数的方法和直接选取试探函数的方法,将一个难于求解的非线性偏微分方程化为一组易于求解的非线性代数方程,然后用待定系数法确定相应的常数,最后简洁地求得了KdV-Burgers方程的精确解析解,两种方法所求得的解完全相同,且与已有文献所得结果一致.本方法可望进一步推广用于求解其他非线性偏微分方程.

By applying two trial function approaches, one that introduces a transformation before selecting trial function and one that directly selects trial function, a nonlinear partial differential equation, which is hard to solve by using the regular technique, could be reduced to a set of nonlinear algebraic equations which could be easily solved, and the associated coefficients could be easily determined by making use of the method of undetermined coefficients. As a result, the exact analytical solution to the KdV-Burgers equation was successfully derived by means of these two approaches. The results obtained in these two approaches were the same, in very good agreement with those obtained in existing literatures. These two approaches can be generalized to construct the solutions of other nonlinear partial differential equations.