两类发展方程精确解的研究

发布时间：2018-02-13 03:29

本文关键词： 精确解李对称 Tanh方法最简函数法幂级数解法广义(G　出处：《江苏大学》2017年硕士论文　论文类型：学位论文

【摘要】：目前,非线性科学研究已经成为科学研究领域的焦点之一。在不同的研究领域中,会遇到各种不同类型的非线性方程,对如何求解这些不同类型的非线性方程也成为了该领域研究的关键。近年来,随着数学机械化的广泛应用,出现了大量的求解非线性方程的新方法,这些方法有效地推动了非线性系统的发展。本文研究了两类方程的精确解,即非线性变系数Sharma-Tasso-Olver(STO)方程和分数阶非线性Klein-Gordon方程。本文共分为五个章节。第一章,在这一章主要介绍了变系数非线性方程及分数阶偏微分方程的研究背景以及现有的研究方法等,最后给出了全文各章的研究内容。第二章,在这一章首先介绍了STO方程的研究背景以及已有的求解方法和结果,其次基于李点对称方法对变系数STO方程进行对称分析,得到了其对称约化方程,最后利用tanh法、最简函数法和幂级数解法得到了所有的约化方程的精确解,进而得到变系数STO方程的精确解。第三章,利用广义(G'/G)展开法再次讨论了变系数STO方程,得到了不同于第二章结果的新的精确解。第四章,在这一章中,利用复杂变换结合三种不同的方法得到了改进的Riemann-Liouville分数阶非线性Klein-Gordon方程的精确解。第五章,对全文进行总结。
[Abstract]:At present, nonlinear scientific research has become one of the focuses of scientific research. In different research fields, there are various types of nonlinear equations. In recent years, with the wide application of mathematical mechanization, a large number of new methods for solving nonlinear equations have emerged. These methods have effectively promoted the development of nonlinear systems. In this paper, we study the exact solutions of two kinds of equations, namely, Sharma-Tasso-Olversto equation and fractional nonlinear Klein-Gordon equation. This paper is divided into five chapters. In this chapter, the research background and existing research methods of variable coefficient nonlinear equations and fractional partial differential equations are introduced. In this chapter, the research background of STO equation, the existing methods and results are introduced. Secondly, based on the lie point symmetry method, the symmetric reduction equation of STO equation with variable coefficients is obtained. Finally, the tanh method is used. The exact solutions of all the reduced equations are obtained by the simplest function method and the power series method, and then the exact solutions of the variable coefficient STO equation are obtained. In chapter 3, the STO equation with variable coefficients is discussed again by using the generalized GG / G expansion method. In this chapter, the exact solutions of the improved Riemann-Liouville fractional order nonlinear Klein-Gordon equation are obtained by using complex transformation combined with three different methods. Chapter 5th summarizes the full text.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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