Exp函数法的应用及两个非线性方程的对称

发布时间：2018-01-28 22:30

本文关键词： 非线性演化方程 Exp函数法弹性杆波动方程辅助方程法对称 Lie代数结构精确解　出处：《内蒙古师范大学》2015年硕士论文　论文类型：学位论文

【摘要】：非线性演化方程在数学和物理领域都占有非常重要的地位,因此研究非线性演化方程的求解方法就变得至关重要.尽管人们在这方面已经做了很多研究,但由于非线性演化方程的的复杂性,至今没有统一的一般的精确求解方法.但值得庆幸的是,在孤立子理论中还是存在着一些构造精确解行之有效的方法,例如Jacobi椭圆函数法]31[?、齐次平衡法]5,4[、完全近似法]6[、试探函数法]97[?、双曲函数展开法1,10[]1、约化摄动法]1412[?、辅助方程法]1715[?、F-展开法]19,18[、Exp函数法]2320[?,双线性变换法]24[,达布变换法]2825[?等.本文主要运用Exp函数法和辅助方程法求解非线性方程,并研究非线性方程的对称及其Lie代数结构.第一章是绪论部分,主要介绍孤立子理论的发展和非线性演化方程求解与对称的研究状况,最后还介绍了本文的主要工作.第二章第一节介绍了Exp函数法求解的基本思路;第二节,第三节分别对文献[10]和文献[19]中的两个非线性弹性杆波动方程运用Exp函数法并借助Mathematica数学软件进行求解,最后给出这两个弹性杆波动方程的精确解.第三章第一节介绍了辅助方程法的求解过程;第二节,第三节,第四节分别通过行波解假设将文献[29]中的三个五阶非线性方程化为常微分方程并借助辅助方程法和Mathematica软件,最终得到这三个五阶非线性方程更丰富的精确解.第四章第一节介绍了研究非线性方程对称的待定系数法和Lie代数结构的确定方法;第二节,第三节通过直接假设法给出文献[30]的两个非线性方程的一些简单对称及其这些对称所构成的李代数结构,并利用对称约化方法给出对应的精确约化的常微分方程.第五章对研究生期间所做的工作做了总结,并对未来的工作提出规划.
[Abstract]:Nonlinear evolution equations play a very important role in the field of mathematics and physics, so it is very important to study the solving methods of nonlinear evolution equations, although a lot of research has been done in this field. However, due to the complexity of nonlinear evolution equations, there is no uniform and general exact solution method. But fortunately, there are still some effective methods to construct exact solutions in soliton theory. For example, Jacobi elliptic function method] 31. [? , homogeneous equilibrium method] 5. [, complete approximation] 6. [, heuristic function method] 97. [? Hyperbolic function expansion method. [] 1, reduced perturbation method] 1412. [? , auxiliary equation method] 1715. [? F- expansion method. [Exp function method] 2320. [? , bilinear transformation method] 24. [, Darboux transformation] 2825. [? In this paper, we mainly use Exp function method and auxiliary equation method to solve nonlinear equation, and study the symmetry of nonlinear equation and its Lie algebraic structure. The first chapter is the introduction part. The development of soliton theory and the research status of solving nonlinear evolution equations and symmetry are introduced. Finally, the main work of this paper is introduced. In the first section of chapter 2, the basic idea of solving Exp function method is introduced. Section II, section III, respectively. [10] and documentation. [Two nonlinear elastic rod wave equations are solved by Exp function method and Mathematica software. Finally, the exact solutions of the two elastic rod wave equations are given. In the first section of chapter 3, the process of solving the auxiliary equation method is introduced. Section 2, section 3, section 4th, by means of the travelling wave solution hypothesis, respectively. [The three fifth-order nonlinear equations are transformed into ordinary differential equations with the aid of the auxiliary equation method and Mathematica software. Finally, the more abundant exact solutions of the three fifth order nonlinear equations are obtained. In the first section of Chapter 4th, the undetermined coefficient method for studying the symmetry of nonlinear equations and the method for determining the Lie algebraic structure are introduced. Section II, section III gives the literature by direct hypotheses. [Some simple symmetries of two nonlinear equations and their lie algebraic structures. In Chapter 5th, the work done during graduate school is summarized, and the future work is planned.
【学位授予单位】：内蒙古师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.7

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