基于对称群方法研究几类非线性偏微分方程（组）的不变解

发布时间：2018-06-26 03:21

本文选题：Lie对称 + 最优系统　；参考：《内蒙古工业大学》2017年硕士论文

【摘要】：随着近代物理和数学的发展,物理学中的非线性现象、问题受到越来越多人的关注.许多非线性问题的研究可以被归结为对非线性偏微分方程(以下简称PDE)的研究.求解非线性PDE是十分必要的,方法也有很多.Lie对称方法是一个较普适性而行之有效的方法,是研究非线性PDE不变解的基础.本文基于符号计算系统MATHEMATICA,研究了一类非线性偏微分方程组(PDEs)和两类非线性PDE的经典Lie对称、条件对称、近似对称、对称分类、一维最优系统、相似约化及不变解的构造.第一章简述了本文的研究背景及意义,并简单介绍了经典Lie对称、条件对称、近似Lie对称的方法.第二章利用经典Lie对称的方法研究了一类含两个任意函数的非线PDEs,获得对称分类.对其中的两种情况做进一步分析,构造一维最优系统,并利用最优系统中的元素对PDEs相似约化,求不变解.另外,利用条件对称的方法研究了PDEs的一种特殊情况,并利用条件对称对该方程组进行相似约化、求不变解.第三章研究了一类非线性渗流方程vt=k(v_x)v_(xx).当k(v_x)=e~x和k(v_x)=(v_x)n时,分别对这两种情况的PDE进行研究.构造一维最优系统,对PDE进行相似约化,进而求不变解.此外,还分别利用条件对称研究了这两种情况的PDE,并利用条件对称对方程进行相似约化,进而求不变解.第四章利用Baikov,Gazizov和Ibragimov提出的近似对称方法,研究了扰动Boussinesq方程.构造了一维最优系统,分析方程的近似不变解,并给出了一些近似不变量.第五章对本文的研究内容进行总结,展望需要进一步研究的内容.
[Abstract]:With the development of modern physics and mathematics, more and more people pay attention to nonlinear phenomena in physics. Many studies on nonlinear problems can be attributed to the study of nonlinear partial differential equations (PDE). It is very necessary to solve nonlinear PDE, and there are many methods. Lie symmetry method is a more universal and effective method, which is the basis of studying nonlinear PDE invariant solution. In this paper, the classical lie symmetry, conditional symmetry, approximate symmetry, symmetric classification, one-dimensional optimal system, similarity reduction and invariant solution of a class of nonlinear partial differential equations (PDEs) and two classes of nonlinear PDE are studied based on the symbolic computing system MATHEMATICA. In the first chapter, the research background and significance of this paper are briefly introduced, and the classical lie symmetry, conditional symmetry and approximate lie symmetry are briefly introduced. In chapter 2, we use the classical lie symmetry method to study a class of nonlinear PDEs with two arbitrary functions, and obtain symmetric classification. The two cases are further analyzed to construct the one-dimensional optimal system, and the elements in the optimal system are used to reduce the PDEs similarity to obtain the invariant solution. In addition, a special case of PDEs is studied by using the method of conditional symmetry, and the equations are reduced similarly by using conditional symmetry, and the invariant solution is obtained. In chapter 3, we study a class of nonlinear seepage equation vt=k (VX) v _ (xx). When k (VX) and k (VX) = (vSX) n, the PDE of these two cases are studied respectively. The one-dimensional optimal system is constructed, and the PDE is reduced by similarity, and the invariant solution is obtained. In addition, the PDE of these two cases is studied by using conditional symmetry, and the equation is reduced similarly by using conditional symmetry, and the invariant solution is obtained. In chapter 4, the perturbation Boussinesq equation is studied by using the approximate symmetry method proposed by Baikovan Gazizov and Ibragimov. The one-dimensional optimal system is constructed, the approximate invariant solution of the equation is analyzed, and some approximate invariants are given. The fifth chapter summarizes the research content of this paper, and looks forward to further research content.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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