一类时滞抛物型方程的紧差分格式研究

发布时间：2018-05-03 20:45

  本文选题：时滞抛物型方程 + 紧差分格式　； 参考：《延边大学》2017年硕士论文

【摘要】：在自然界中,时滞现象普遍存在且无法避免,这也是影响系统稳定性及其性能的主要原因之一,时滞微分方程在理学、工学等众多领域中都有着广泛应用.过去,人们在研究天体力学、物理学、动力系统等学科中的问题时,总认为所考虑的系统服从这样一个规律,即系统将来的状态仅由系统当前的状态决定并用相应的模型加以刻画.然而,随着人们对许多自然现象有了更深入的分析后发现,现实世界中,系统的状态除了依赖当前发展状态也依赖过去的发展系统.在多数情况下,若用忽略时滞的方法来降低问题的难度,会给系统带来比较大的负面影响,但也正因为有时滞项,其理论的分析难度较大,想获得其精确解的解析表达式是很困难的.所以,我们在解决实际问题的时候,时滞微分方程精确解的得出一般都用其数值解来替代.这一研究弥补了理论上的不足,同时具有重要的现实意义.本文阐述了如何构造时滞抛物型方程的紧差分格式,同时也介绍了其对应的数值格式理论分析.第一章主要讲述了专家学者们对有关时滞微分方程的数值方法研究的多年进展状况,以及有关时滞微分方程研究的背景和意义,并且说明了本文的主要研究内容及意义.第二章主要用了差分离散的方法为一维非线性时滞抛物型方程的初边值问题构造出一个紧差分格式,同时用能量分析法证明了其在该格式下解的存在唯一性、无条件稳定性和在L∞范数下阶数为O(T2+ 4 的收敛性.最后,用一个数值算例说明该格式具有可行性.第三章阐述了如何构造二维时滞抛物型方程初边值问题的紧差分格式,这里,我们用交替方向的技巧来提高计算效率,并对紧差分格式进行求解,接着研究了解的先验估计式和稳定性.最后,用一个数值算例说明该格式具有可行性.
[Abstract]:In nature, the phenomenon of delay exists widely and cannot be avoided, which is one of the main reasons that affect the stability and performance of systems. Delay differential equations are widely used in many fields, such as science, engineering and so on. In the past, when people studied the problems in astromechanics, physics, dynamical systems, and other subjects, they always thought that the system under consideration was based on such a rule. That is, the future state of the system is only determined by the current state of the system and described by the corresponding model. However, with more in-depth analysis of many natural phenomena, it is found that in the real world, the state of the system depends not only on the current state of development, but also on the development system of the past. In most cases, if we use the method of neglecting time delay to reduce the difficulty of the problem, it will bring more negative effects to the system, but it is also difficult to analyze the theory because of the delay term. It is difficult to obtain an analytical expression of its exact solution. Therefore, when we solve practical problems, the exact solutions of delay differential equations are generally replaced by their numerical solutions. This research makes up for the deficiency in theory and has important practical significance at the same time. This paper describes how to construct a compact difference scheme for the parabolic equation with time delay, and also introduces its corresponding numerical scheme theory analysis. In the first chapter, the progress of the numerical methods of delay differential equations, the background and significance of the research on delay differential equations are described, and the main contents and significance of this paper are explained. In the second chapter, a compact difference scheme is constructed for the initial boundary value problem of one-dimensional nonlinear parabolic equations with delay by using the method of difference discretization, and the existence and uniqueness of its solution under the scheme are proved by energy analysis. Unconditional stability and convergence of order O(T2 4 under L 鈭，

本文编号：1840023