几何不变流的最优系统及群不变解
发布时间：2018-07-13 13:03
【摘要】：几何不变流的研究来源于图像处理和晶体增长等方面,有着广泛的应用.本文运用Lie对称群方法系统地研究了两个曲线流 中心仿射不变流和双曲型仿射不变流的群不变解问题.全文共有四章:第一章:介绍了研究背景及相关预备知识,给出了本文的主要工作.第二章:研究了中心仿射不变流对应的非线性偏微分方程的群不变解问题.首先利用Lie群理论,给出了方程的对称群.接着应用Olsiannikov和Olver的思想方法,得到了一个最优系统及其约化方程,最后讨论了相应的群不变解.第三章:我们研究了双曲型仿射不变流对应的非线性偏微分方程,计算了方程的李点对称群,并构造其一个最优系统,最后利用最优系统对非线性偏微分方程进行对称约化,得到相应的约化方程和一些群不变解.第四章:对全文进行总结.
[Abstract]:The research of geometric invariant flow comes from image processing and crystal growth and has been widely used. In this paper, we systematically study the problem of group invariant solutions for two central affine invariants and hyperbolic affine invariants by using lie symmetric group method. There are four chapters in this paper: chapter 1: introduce the research background and related preparatory knowledge, and give the main work of this paper. In chapter 2, the problem of group invariant solutions of nonlinear partial differential equations corresponding to central affine invariant flow is studied. Firstly, the symmetric group of the equation is given by using lie group theory. Then, by using Olsiannikov and Olver's method, an optimal system and its reduced equations are obtained. Finally, the corresponding group invariant solutions are discussed. In chapter 3, we study the nonlinear partial differential equation corresponding to hyperbolic affine invariant flow, calculate the lie point symmetric group of the equation, and construct an optimal system. Finally, we use the optimal system to reduce the nonlinear partial differential equation symmetrically. The corresponding reduction equation and some group invariant solutions are obtained. Chapter four: summarize the full text.
【学位授予单位】：宁波大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2119466
[Abstract]:The research of geometric invariant flow comes from image processing and crystal growth and has been widely used. In this paper, we systematically study the problem of group invariant solutions for two central affine invariants and hyperbolic affine invariants by using lie symmetric group method. There are four chapters in this paper: chapter 1: introduce the research background and related preparatory knowledge, and give the main work of this paper. In chapter 2, the problem of group invariant solutions of nonlinear partial differential equations corresponding to central affine invariant flow is studied. Firstly, the symmetric group of the equation is given by using lie group theory. Then, by using Olsiannikov and Olver's method, an optimal system and its reduced equations are obtained. Finally, the corresponding group invariant solutions are discussed. In chapter 3, we study the nonlinear partial differential equation corresponding to hyperbolic affine invariant flow, calculate the lie point symmetric group of the equation, and construct an optimal system. Finally, we use the optimal system to reduce the nonlinear partial differential equation symmetrically. The corresponding reduction equation and some group invariant solutions are obtained. Chapter four: summarize the full text.
【学位授予单位】：宁波大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
【参考文献】
相关期刊论文 前1条
1 ;Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Ampère Equation[J];Chinese Annals of Mathematics(Series B);2012年02期
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