李群方法在求解几类偏微分方程中的应用

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	李群方法在求解几类偏微分方程中的应用
其他题名	The application of Lie group to the solution of several classical partial differential equations
	何再明
导师	游亚戈
	2006-06-14
学位授予单位	中国科学院广州能源研究所
学位授予地点	广州能源研究所
学位名称	硕士
关键词	李群无穷小生成元偏微分方程科学计算软件
摘要	传统能源供应日益紧张，开发可再生能源已成为当务之急。海洋能是众多可再生能源的一种，波浪能利用是现今各国海洋能开发研究的重点，水动力学数值计算和非线性偏微分方程求解则是研究波浪能利用的工具。在众多求解偏微分方程方法中，李群给我们提供了一种如何构造函数和自变量的变换方法，实现原微分方程的约化降维，求解此降维方程便获得原方程的解，它使人们在运用变换求解偏微分方程时变得有章可循。本文的主要工作是，通过对求解偏微分方程的数值方法现状大致介绍和分析其自身局限性和缺陷，以李群理论的基础为始详细介绍了几类李群对称方法，之后以Mathematica这一科学计算软件为平台，通过对热传导方程的经典和非经典对称方法的推导，演示其推导方法和结果。通过对其特征方程的常数赋予特值，获得其群不变量，再带回原方程实现降维约化，并求解该降维方程，从而获得热传导方程的解，然后对Burgers方程，Boussinesq方程，KdV方程不同方法求解来获得其更丰富的解。本文的研究意义在于应用李群方法求解偏微分方程的精确解，同时利用科学计算软件进行符号计算推导，扩展了李群方法应用的适用性，为水动力学方程的求解提供了参考，同时对几类偏微分方程的求解也增加了其解的丰富性，且为波浪能的基础研究提供了理论铺垫，并对进一步的工程实际应用具有一定的参考价值。关键词：李群无穷小生成元偏微分方程科学计算软件
其他摘要	Today, energy plays a very important role in the development and civilization of the social. On the other hand, studying and developing the clear energy is the only way we should to do. Ocean energy is one kind of new and renewable energy, and is being studied in many countries. The research of the nonlinear partial differential equations (PDEs) is the basis of the conversion of wave energy. In the methods of solving the PDEs, Lie’s theory is applicable to construct the transformation between the dependent and independent variables, and the symmetry group allows us to reduce the order of the equation, and gives us a rule that we can gain the transformation. In this thesis, we introduce the methods of solving the PDEs numerically, and point out the disadvantage and the limitation firstly. In chapter 2, beginning with the basic knowledge of groups, we introduce the classical symmetries, non-classical symmetries, generalized symmetries, approximate symmetries and direct methods. In chapter 3, the heat equations are calculated firstly by classical and non-classical symmetries, through the constant of characteristic equation given special values, we gained the determining equations and infinitesimal invariance, so we can get the solutions by replace the original equation and calculate the reduce equation . And then the Burgers equations, Boussinesq equations, KdV equations are considered to access their rich solutions. The significance of this thesis is that the scientific analysis software is employed in the symbolic computation at the same time the Lie group symmetry is applied in the solution to the PDEs, which improves the applicability of the Lie group theory and increases the abundance solutions of several classical PDEs. The works provide some useful idea in the solution to the PDEs in hydrodynamics, accumulate some theory foundation for the basis research of wave energy, and give some useful help for the practical design in the ocean emerging. Key words: Lie Group, infinitesimal generator, partial differential equations (PDEs), scientific analysis software.
页数	58
语种	中文
文献类型	学位论文
条目标识符	http://ir.giec.ac.cn/handle/344007/4047
专题	中国科学院广州能源研究所
推荐引用方式 GB/T 7714	何再明. 李群方法在求解几类偏微分方程中的应用[D]. 广州能源研究所. 中国科学院广州能源研究所,2006.

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