本文關鍵詞： 非線性演化方程 Exp函數(shù)法 彈性桿波動方程 輔助方程法 對稱 Lie代數(shù)結構 精確解　出處：《內(nèi)蒙古師范大學》2015年碩士論文　論文類型：學位論文

【摘要】：非線性演化方程在數(shù)學和物理領域都占有非常重要的地位,因此研究非線性演化方程的求解方法就變得至關重要.盡管人們在這方面已經(jīng)做了很多研究,但由于非線性演化方程的的復雜性,至今沒有統(tǒng)一的一般的精確求解方法.但值得慶幸的是,在孤立子理論中還是存在著一些構造精確解行之有效的方法,例如Jacobi橢圓函數(shù)法]31[?、齊次平衡法]5,4[、完全近似法]6[、試探函數(shù)法]97[?、雙曲函數(shù)展開法1,10[]1、約化攝動法]1412[?、輔助方程法]1715[?、F-展開法]19,18[、Exp函數(shù)法]2320[?,雙線性變換法]24[,達布變換法]2825[?等.本文主要運用Exp函數(shù)法和輔助方程法求解非線性方程,并研究非線性方程的對稱及其Lie代數(shù)結構.第一章是緒論部分,主要介紹孤立子理論的發(fā)展和非線性演化方程求解與對稱的研究狀況,最后還介紹了本文的主要工作.第二章第一節(jié)介紹了Exp函數(shù)法求解的基本思路;第二節(jié),第三節(jié)分別對文獻[10]和文獻[19]中的兩個非線性彈性桿波動方程運用Exp函數(shù)法并借助Mathematica數(shù)學軟件進行求解,最后給出這兩個彈性桿波動方程的精確解.第三章第一節(jié)介紹了輔助方程法的求解過程;第二節(jié),第三節(jié),第四節(jié)分別通過行波解假設將文獻[29]中的三個五階非線性方程化為常微分方程并借助輔助方程法和Mathematica軟件,最終得到這三個五階非線性方程更豐富的精確解.第四章第一節(jié)介紹了研究非線性方程對稱的待定系數(shù)法和Lie代數(shù)結構的確定方法;第二節(jié),第三節(jié)通過直接假設法給出文獻[30]的兩個非線性方程的一些簡單對稱及其這些對稱所構成的李代數(shù)結構,并利用對稱約化方法給出對應的精確約化的常微分方程.第五章對研究生期間所做的工作做了總結,并對未來的工作提出規(guī)劃.
[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.
【學位授予單位】：內(nèi)蒙古師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O241.7

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本文編號：1471724