摘要
由于延迟积分微分方程(DIDEs)在很多领域都突显出重要性,因此近年来出现了从多方面对它是研究。比如将某些方法应用到延迟积分微分方程(DIDEs)中,来研究其收敛性及稳定性等。而波形松弛法(WR)是二十世纪八十年代被提出的一种动态迭代方法,由于WR方法能将复杂系统进行解耦,并能使解耦后的系统保持原系统的一些特性,还能进行并行计算等优点,使得越来越多的学者开始关注此方法。本文主要是将WR方法应用到延迟积分微分方程(DIDEs)中,来讨论其收敛性。
首先,对延迟微分方程的数值方法进行了回顾,同时介绍了WR方法的相关理论以及目前为止国内外在应用WR方法方面的进展情况。
其次,给出延迟积分微分方程(DIDEs)的模型,并将WR方法应用到此方程中,从而得到了连续时间WR方法,并证明了连续时间WR方法的收敛性。同时还给出了扰动时间WR迭代的收敛性及解的存在性及唯一性的证明。
再次,为了得到离散时间WR迭代,本文用Runge-Kutta方法对连续时间WR迭代进行离散化,并证明了离散时间WR方法的收敛性。令外,通过给出数值算例及Matlab仿真,对收敛性进行了模拟,从而进一步验证了理论分析的正确性。
最后,对全文进行了总结,对WR方法应用到其他延迟微分方程的研究成果进行了展望。
Due to the importance of delay integro-differential equations (DIDEs) in many fields, so in recent years much of research to it appeared. For instance, some methods are applied to delay integro-differential equations (DIDEs), to study its convergence and stability, etc. And waveform relaxation methods (WR), a dynamic iterative methods , are proposed in 1980s, because WR methods can decouple complex system, and make the decoupling systems maintain the original system’s some properties, it can make parallel computation etc, so more and more scholars begin to pay close attention to this method. In this paper ,WR methods are used to delay integro-differential equations (DIDEs), to discuss its convergence.
First of all, delay differential equations’numerical methods are reviewed, meanwhile WR methods are introduced, and the progress at home and abroad of related theory of WR methods are introduced.
Secondly, delay integro-differential equations (DIDEs) model is given, and WR methods are used to this equation, thus getting continuous time WR methods, and prove the continuous time WR methods’convergence. Meanwhile perturbed continuous time WR iteration’s convergence and the existence and uniqueness of the solution is discussed.
Thirdly, in order to get the discrete time WR iteration, we put Runge - Kutta methods to continuous time WR iteration, and prove the discrete time WR methods’convergence. Given that, through the numerical examples and the simulation of Matlab, the validity of the theoretical analysis is further demonstrated.
Finally, summarized to the full text, and WR methods are applied to other delay differential equations’achievements is prospected.
引文
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