延迟型热传导方程两步法的收敛性与稳定性分析

发布时间：2018-01-08 06:12

本文关键词：延迟型热传导方程两步法的收敛性与稳定性分析　出处：《哈尔滨工业大学》2015年硕士论文　论文类型：学位论文

【摘要】：在现实生活中,学者们发现许多问题不仅仅和它们当前的状态有关,而且和它们过去某一时刻的状态也有着非常密切的关系。因此在用数学模型描述这类问题时,需要加入一个刻画过去状态的量,通常把这个量称作为延迟项,本文主要研究三类热传导方程的两步法。延迟型热传导方程是偏微分方程的重要组成部分,它可以应用到许多领域中。例如电动力学、生态学、自动控制、非线性电力系统、金融等领域,它的出现极大地促进了社会的进步。热传导方程的理论解一般不易求出,人们通常用数值方法求解这类方程。用于求解热传导方程的数值方法有很多,例如边界元法,有限差分法、有限元法。本文用有限差分法对延迟型热传导方程进行数值计算,主要工作可分为两部分。第一部分,推导出双延迟项热传导方程理论解渐近稳定的条件,在延迟型热传导方程理论解渐近稳定的前提下,用两步法求解该方程,并给出差分格式和数值算法渐近稳定的定义,用谱半径方法分析该数值算法的渐近稳定性。最后通过数值算例检验自己的结论。第二部分,在延迟型热传导方程理论解渐近稳定的前提下,用蛙跳方法求解该方程,给出求解方程的差分格式和数值算法渐近稳定的定义,并分析蛙跳方法的收敛性,最后通过数值算例检验自己的结论。
[Abstract]:In real life, scholars have found that many problems are not just about their current state. It is also closely related to their past state at some time, so when describing this kind of problem with mathematical model, we need to add a quantity that depicts the past state, which is usually referred to as the delay term. This paper mainly studies the two-step method of three kinds of heat conduction equations. The delayed heat conduction equation is an important part of partial differential equation, which can be applied in many fields, such as electrodynamics, ecology, automatic control. The emergence of nonlinear power system, finance and other fields has greatly promoted the progress of society. The theoretical solution of heat conduction equation is generally difficult to find. Numerical methods are usually used to solve this kind of equations. There are many numerical methods for solving heat conduction equations, such as boundary element method and finite difference method. Finite element method. In this paper, the delay heat conduction equation is numerically calculated by finite difference method. The main work can be divided into two parts. In the first part, the asymptotic stability conditions for the theoretical solution of the double delay heat conduction equation are derived. On the premise of the asymptotic stability of the theoretical solution of the delayed heat conduction equation, the two-step method is used to solve the equation, and the definitions of the difference scheme and the numerical algorithm for asymptotic stability are given. The asymptotic stability of the numerical algorithm is analyzed by spectral radius method. Finally, the results are verified by numerical examples. The second part, on the premise of asymptotic stability of the theoretical solution of the delayed heat conduction equation. The difference scheme of solving the equation and the definition of asymptotic stability of numerical algorithm are given, and the convergence of the leapfrog method is analyzed. Finally, the conclusion is verified by numerical examples.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

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