KdV-Burgers方程的级数解

THE SERIES SOLUTIONS OF KdV-BURGERS EQUATION

给出ＫｄＶ－Ｂｕｒｇｅｒｓ方程的有界行波解的精确级数解，采用Ａｄｏｍｉａｎ算子分解法分别求得二个区域＜０和＞０的级数解，然后利用对接连续条件构成整体级数解．所得级数解能精确满足对接连续条件，并由此得到确定级数的系数递推公式，无需解非线性高阶代数方程组．与某些精确解及其它方法比较，计算简捷且在对接点处是收敛的．对某些非线性波动现象的研究，可作为计算和分析的数学依据．

This paper presents an accurate series solution of traveling wave for KdV-Burgers equation. From deterministy analysis we know that the traveling wave solution is either monotone wave solution or oscillation wave solution. In Section 2, the monotone wave solution is given by Adomian's operator decomposition method. We firstly find left series solution as < 0 and right series solution as > 0, then the integral series solution and recurrence relations of coefficients in the series airs obtained by the connected continuous conditions at = 0. Further, the oscillation wave solution is also provided in Section 3. In particularly, we can yet find the series solution where left and right series have degrees of different approximation, and recurrence relations of coefficients are given correspondly. In Section 4, several numerical examples are presented. The calculated results show that the method is brief and valid. Generally, approximate series solution in less than ten terms can better approach to analytic solution, and characteristics of the traveling wave can be obtained by these examples. According to the method of the paper, the integral series solution which only need to find left and right series can be obtained by the given formulas. The method comparing with the paher [7] need not find middle series solution and need not solve nonlinear algebraic equations at two connected points. The series solution satisfies exactly connected continuous conditions and is convergence, and can be applied to analyze and compute some nonlinear wave problems.