四阶Burgers方程的非线性边值-初值问题  

The Nonlinear Boundary-initial Value Problem for the Fourth Order Burgers Equation

作　　者：张金良[1] 王飞 王明亮[1,3] ZHANG JINLIANG;WANG FEI;WANG MINGLIANG(School of Mathematics&Statistics,Henan University of Science&Technologu,Luoyang 471000,China;Yellow River Conseroancy Technical Institute,Kaiferng 475004,China;School of Mathematics&Statistcs,Lanzhou University,Lanzhou 730000,China)

机构地区：[1]河南科技大学数学与统计学院,洛阳471000 [2]黄河水利职业技术学院,开封475004 [3]兰州大学数学与统计学院,兰州730000

出　　处：《应用数学学报》2020年第6期1029-1041,共13页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(51675161)资助项目。

摘　　要：本文以变阻尼四阶Burgers方程为例,来说明简化的齐次平衡法是构造某些非线性发展方程非线性边值-初值问题非常有用的方法.首先借助于简化的齐次平衡法,推导出变阻尼四阶Burgers方程与四阶线性方程解之间的非线性变换及其逆变换;由此导出一个新的半无限直线上非线性边值-初值问题;最后讨论了五个特殊的变阻尼四阶Burgers方程,得到了这些非线性边值-初值问题的精确解,特别地,得到了柱(球)四阶Burgers方程非线性边值-初值问题的衰减解.The major target of this paper is to show that the simplified homogeneous balance method is very useful to construct new nonlinear boundary-initial value problems for some nonlinear evoluiton equations.The fourth order Burgers equation with general variable damping is taken as an example to illustrate this problem.Firstly,the nonlinear transformation and its inversion between the fourth order Burgers equation with general variable damping and the fourth order linear equation are derived;and then a new nonlinear boundary-initial value problem on the half infinite straight line is put forward for the first time;lastly,five special fourth order Burgers equations are studied,and the exact solutions to these nonlinear boundary-initial value problems are obtained,especially,the decay mode solutions of nonlinear boundary-initial value problems for the cylindrical(spherical)fourth order Burgers equation are derived.

关 键 词：齐次平衡法 四阶Burgers方程 柱(球)四阶Burgers方程 非线性变换 非线性边值-初值问题 衰减解 

分 类 号：O175.2[理学—数学]