中立型延迟积分微分方程数值方法的稳定性

【作者】 刘建国；

【导师】 甘四清；

【作者基本信息】 中南大学 ， 计算数学， 2008， 硕士

【摘要】 延迟积分微分方程广泛出现于物理、工程、生物、医学、航天航空及经济等领域,其算法理论研究具有毋庸置疑的重要性,近年来逐渐引起众多学者的极大关注.中立型延迟积分微分方程是一类重要的延迟积分微分方程.本文研究中立型延迟积分微分方程及数值方法的渐近稳定性.近年来,有许多关于延迟积分微分方程数值方法的稳定性结果.然而,关于中立型延迟积分微分方程Runge-Kutta方法、多步Runge-Kutta方法及延迟积分微分方程Pouzet型线性多步方法的稳定性结论甚少.因此,研究这三类数值方法的稳定性具有重要意义.本文较系统地讨论这三类数值方法的稳定性.所获主要结果如下:(1)讨论了Runge-Kutta方法求解中立型延迟积分微分方程的渐近稳定性,证明了A-稳定的Runge-Kutta方法能够保持原线性系统的渐近稳定性.(2)讨论了多步Runge-Kutta方法求解中立型延迟积分微分方程的渐近稳定性,证明了A-稳定的多步Runge-Kutta方法能够保持原线性系统的渐近稳定性.(3)讨论了Pouzet型线性多步方法求解延迟积分微分方程的渐近稳定性,证明了A-稳定且强零-稳定的Pouzet型线性多步方法能够保持原线性系统的渐近稳定性.(4)通过数值例子和数值试验,对线性多步法及Runge-Kutta方法的稳定性进行了测试,测试结果进一步验证了本文所获理论结果的正确性.更多还原

【Abstract】 Delay integro-differential equations (DIDEs) arise widely in the fields of Physics, Engineering, Biology, Medical Science, Aviation, Economics and so on. The theory of computational methods is very important in DIDEs. Recently, many scholars have pay careful attention to it. Neutral delay integro-differential equations (NDIDEs) is special subclass of DIDEs. In this paper, we research the asymptotic stability of NDIDEs and numerical methods for NDIDEs.Recently, there are many results of the stability of numerical methods for DIDEs. But, there are few results of the stability of the Runge-Kutta methods and the multistep Runge-Kutta methods of NDIDE s and the linear multistep methods of Pouzet type for DIDEs. Therefore, it is significant to study the stability of these three classes of numerical methods. In this article, we systematically study the stability of these three classes of numerical methods. The main results in the thesis are as follows:(1) We investigate the asymptotic stability of the Runge-Kutta methods of NDIDEs. It is proven that A-stable Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.(2) We study the asymptotic stability of the multistep Runge-Kutta methods of NDIDEs. It is shown that A-stable multistep Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.(3) We analyze the asymptotic stability of the linear multistep methods of Pouzet type for DIDEs. It is proven that A-stable and strongly zero-stable linear multistep methods of Pouzet type can preserve the asymptotic stability of the underlying linear systems.(4) Numerical examples and numerical experiments are given for checking the stability of the linear multistep methods and the Runge-Kutt a methods, which confirm the theoretical results obtained in this paper.更多还原