李群分析和幂级数方法在非线性偏微分方程求解中的应用

发布时间：2018-02-11 15:55

本文关键词： 非线性偏微分方程经典李群方法 /G　出处：《聊城大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文主要利用李群分析理论和方法对下面的三个非线性偏微分方程进行了研究:推广的Burgers方程、非线性LC电路方程和一类高阶非线性波方程.首先,求解以上方程的李点对称,然后利用求得的对称约化原方程,最后求解得到了一些新的精确解.在第一章中,根据/G'G展开法以及相关理论研究了推广的Burgers方程,并对方程的解是否存在进行了讨论,最后方程所有可能情形下的解被求出.在第二章中,根据李群分析理论及方法求得了非线性LC电路方程的李对称,并且根据求得的对称得到了该方程的一些精确解,包括幂级数解等等.在第三章中,根据李群分析理论及方法得到了一类高阶非线性波方程的李对称,并且求出了它的最优系统和精确解,最后给出了方程的一些守恒定律.综上所述,本文的思路是在非线性偏微分方程的求解当中应用李群分析理论.从而求得非线性偏微分方程的精确解.最后,利用方程的向量场和相关定理,求得了高阶波方程的守恒律。
[Abstract]:In this paper, the following three nonlinear partial differential equations are studied by using Li Qun's analysis theory and method: generalized Burgers equation, nonlinear LC circuit equation and a class of high order nonlinear wave equations. Then, some new exact solutions are obtained by using the symmetric reductive primitive equation. In the first chapter, the generalized Burgers equation is studied according to the method of / GG expansion and related theories, and the existence of the solution is discussed. In the second chapter, the lie symmetry of the nonlinear LC circuit equation is obtained according to Li Qun's analysis theory and method, and some exact solutions of the equation are obtained according to the obtained symmetry. In chapter 3, according to Li Qun's analysis theory and method, we obtain the lie symmetry of a class of high order nonlinear wave equations, and obtain its optimal system and exact solution. Finally, some conservation laws of the equation are given. In summary, the idea of this paper is to apply Li Qun's analysis theory to the solution of nonlinear partial differential equation, so as to obtain the exact solution of the nonlinear partial differential equation. The conservation law of higher order wave equation is obtained by using vector field and correlation theorem of the equation.
【学位授予单位】：聊城大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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