两类时滞偏微分方程的差分法

发布时间：2018-03-18 02:08

本文选题：时滞微分方程　切入点：数值解　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】：本文对两类时滞偏微分方程的差分方法展开研究,并且进行了理论分析。由于带时滞量的微分方程的复杂性,大多数时滞微分方程都不能求得显式的解析表达式。因此,此类方程的数值解法在理论和实际应用中都有着重要意义。近几十年来,对时滞微分方程的数值处理的研究在国际上掀起了高潮,许多求解时滞微分方程的数值方法也陆续被提了出来,如Runge-Kutta法、?法、线性多步法等。对时滞微分方程的数值处理在自动控制、土木工程及环境科学等诸多领域中扮演着越来越重要的角色。本文首先对差分法的基础知识与应用进行简单的介绍。其次,针对一类含有小延迟量的偏微分方程初边值问题,通过Taylor展开式的思想将其转化为不带延迟量的偏微分方程,然后再构造差分格式,并用已经非常成形的没有延迟项的偏微分方程的知识对格式进行理论分析。数值算例中通过与其他方法数值解的比较,得到本文中所介绍的数值方法具有较好的适用性,有更高的数值精度。最后,对另一种类型的时滞抛物型方程构造了Crank-Nicolson格式,经理论分析发现它是一种无条件稳定的差分格式。同样的,通过数值算例对稳定性进行了验证。
[Abstract]:In this paper, the difference method for two kinds of partial differential equations with delay is studied, and the theoretical analysis is made. Because of the complexity of differential equations with delay, most delay differential equations can not obtain explicit analytical expressions. The numerical solution of this kind of equations is of great significance both in theory and in practice. In recent decades, the research on numerical treatment of delay differential equations has aroused a high tide in the world. Many numerical methods for solving delay differential equations have been proposed one after another, such as Runge-Kutta method? Method, linear multistep method, etc. The numerical processing of delay differential equation is controlled automatically, Civil engineering and environmental science play a more and more important role in many fields. In this paper, the basic knowledge and application of difference method are introduced briefly. Secondly, for a class of initial boundary value problems of partial differential equations with small delay, By using the idea of Taylor expansion, it is transformed into partial differential equation with no delay, and then the difference scheme is constructed. The scheme is theoretically analyzed by using the knowledge of partial differential equations with no delay term which has been formed very well. By comparing the numerical solutions with other numerical solutions, the numerical method presented in this paper has good applicability. Finally, Crank-Nicolson scheme is constructed for another type of delay parabolic equation, which is found to be an unconditionally stable difference scheme by theoretical analysis. Similarly, the stability 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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