若干偏微分方程的无网格重心插值配点法的研究

发布时间：2018-10-16 11:40

【摘要】：无网格重心插值配点法是一种高精度的数值计算方法,是依赖微分方程的强形式的配点方法,未知函数的近似函数用离散节点上的重心型插值表示.无网格重心插值配点法包括两种,分别是重心Lagrange插值配点法和重心有理插值配点法.在以前,中外数学研究者已经对无网格重心插值配点法做过大量的研究,但是,在他们的研究中,用此方法处理具体偏微分方程和方程组的研究却很少.由于此方法来源于工科,也并没有发现关于此方法解微分方程的理论分析.那么,本文将以此为出发点,将无网格重心插值配点法应用在KdV方程,KdV-Burgers方程和奇异摄动延迟偏微分方程等方程中.并突破无网格重心插值配点法解微分方程的理论分析,包括收敛性,稳定性.这篇文章的重点是,将无网格重心插值配点法应用在KdV方程,KdV-Burgers方程和奇异摄动延迟偏微分方程等方程中.突破无网格重心插值配点法解微分方程的理论分析.文章结合数值算例,来讨论无网格重心插值配点法算法的精确性,计算量等问题.第一章,介绍无网格重心插值配点法的发展,阐述此方法的研究现状和研究意义.第二章,介绍用重心插值配点法处理微分方程的具体步骤,边界条件的施加和直接线性迭代法.第三章,介绍运用无网格重心插值配点法求解具体的KdV方程,KdV-Burgers方程和奇异摄动延迟偏微分方程.第四章,讨论无网格重心插值配点法解微分方程的理论分析,包括收敛性,稳定性.第五章,总结本文研究的内容,并且对这种方法的进一步研究提出一些建议和想法.
[Abstract]:Meshless barycentric interpolation collocation method is a high-precision numerical method, which depends on the strong form of differential equations. The approximate functions of unknown functions are expressed by the barycentric interpolation on discrete nodes. Meshless barycenter interpolation collocation method includes two kinds, barycenter Lagrange interpolation collocation method and barycentric rational interpolation collocation method. In the past, Chinese and foreign mathematical researchers have done a lot of research on meshless barycenter interpolation collocation method, but in their research, there are few researches on how to deal with specific partial differential equations and equations. Since the method is derived from engineering, there is no theoretical analysis on the solution of differential equations. In this paper, the meshless barycenter interpolation collocation method is applied to KdV equation, KdV-Burgers equation and singular perturbed delay partial differential equation. It also breaks through the theoretical analysis of meshless barycenter interpolation collocation method for solving differential equations, including convergence and stability. The emphasis of this paper is to apply the meshless barycentric interpolation collocation method to KdV equation, KdV-Burgers equation and singular perturbation delay partial differential equation. Breaking through the theoretical analysis of meshless barycenter interpolation collocation method for solving differential equations. In this paper, the accuracy and computational complexity of the meshless barycenter interpolation collocation algorithm are discussed with numerical examples. In chapter 1, the development of meshless barycenter interpolation collocation method is introduced, and the research status and significance of this method are described. In the second chapter, the concrete steps, the application of boundary conditions and the direct linear iterative method for the differential equation are introduced by using the centroid interpolation collocation method. In chapter 3, the meshless barycenter interpolation collocation method is used to solve the specific KdV equation, KdV-Burgers equation and singular perturbation delay partial differential equation. In chapter 4, the theoretical analysis of meshless barycenter interpolation collocation method for differential equations is discussed, including convergence and stability. The fifth chapter summarizes the content of this paper, and puts forward some suggestions and ideas for further study of this method.
【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】：O241.82

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