【作者】 高波；

【导师】 赵景军；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2013， 硕士

【摘要】 延迟微分方程是一类特殊的泛函微分方程，广泛存在在科学研究中。由于延迟微分方程对事物的刻画更全面，科研人员也把注意力集中在对延迟微分方程的理论研究上。对延迟微分方程的理论的研究一个重要方面是对其稳定性的研究。本文主要研究的就是两类延迟微分方程：一类双延迟的微分方程和一类多延迟中立型Volterra积分微分方程。首先，简要介绍了延迟微分方程相关背景，回顾一些延迟微分方程数值方法稳定性结果，特别是块边值方法的研究现状。然后，在第二章中介绍了块边值方法的构造过程和几种具体形式，应用块边值方法到一类双延迟微分方程，通过特征值的理论和拉格朗日插值的思想，给出块边值方法要保持渐近稳定性的必要条件，在章末给出数值实验验证所得结论。最后，在第三章中简要介绍了块边值方法的可约化求积公式，并利用块边值方法对中立型Volterra积分微分方程进行离散，其中对积分项采用可约化求积公式进行离散，进而得到其特征方程，应用特征方程的理论得到系数矩阵在奇异时块边值方法要保持渐近稳定性的必要条件，同时给出在非奇异时块边值方法要保持渐近稳定性的充要条件。同第二章一样，在章末给出数值实验验证了所得到的稳定性结论的正确性。更多还原

【Abstract】 Delay differential equations, widespread in scientific research, are special classesof functional differential equations. Due to the delay differential equations fullerdepiction of things, researchers are focusing on the study of the theory of delaydifferential equations. Stability of delay differential equations is one of the mostimportant aspect of theoretical study.In this paper we study two types of delay differential equations: one class ofdifferential equations with double delays and one class of neutral Volterra integraldifferential equations with many delays.First, we give a brief introduction of delay differential equations and relatedbackground, and recalled some numerical stability results, especial on the researchstatus of such equations using the block boundary value methods.Then, in the second chapter, we describe the construction process of the blockboundary value methods, and several specific forms. And then we apply block boundaryvalue method to a class of double delay differential equations to get the necessarycondition to keep the asymptotic stability of the methods, though the eigenvalue theoryand Lagrange interpolation. A numerical experiment is carried to verify the correctnessof the conclusion.After that, in the third chapter, we give a brief introduction of reducible quadratureformula based on the block boundary value methods, and use the formula to discreteneutral Volterra integral differential equations, and then get its characteristic equation.By the characteristic theory, we have the necessary condition for the block boundaryvalue methods to hold asymptotic stability of the equations when coefficient matrix ofthe equations is singular, and at the same time, we give the necessary and sufficientconditions for such equations when the coefficient matrix is non-singular. The same asthe second chapter, a numerical experiment is carried to verify the conclusion.更多还原