二阶延迟微分方程论文关于几类二阶延迟微分方程数值解及其稳定性的研究.docVIP

二阶延迟微分方程论文：关于几类二阶延迟微分方程数值解及其稳定性的研究 【中文摘要】本文主要研究二阶延迟微分方程的数值解及其稳定性。全文由四章组成。第一章主要介绍了延迟微分方程的以及课题的现实意义。第二章主要讨论了二阶延迟微分方程周期解的存在唯一性及数值解法。首先,根据一个引理给出并且证明了方程存在唯一周期解的充分条件,然后利用牛顿法研究了周期数值解。第三章主要讨论了二阶滞后型微分方程的理论解解和数值解的稳定性。本章主要包括两个方面：一方面,由二阶延迟微分方程的特征方程,给出其渐近稳定的充要条件；另一方面,给出数值解的单支θ-方法的稳定性质,证明了当θ=1时数值解是稳定的。第四章主要讨论了中立型方程的理论解和数值解的稳定性。首先,利用特征根分析方法,获得了理论解稳定的充要条件；其次,在理论解稳定的基础之上,考虑方程单支θ-方法的稳定性质,证明当θ=1时,单支θ-方法是稳定的。 【英文摘要】This paper is concerned with the numerical and stability of solutions for several second-order delay differential equations, which is composed of four parts.In the first chapter, the research background and practical significance of the topics are simply summarized.The second chapter is concerned with the numerical solutions and stability of second-order delay differential equations. The sufficient and necessary condition is given under which the existence and uniqueness of periodic solutions is guaranteed. The numerical solutions and its stability are considered by using Newton method.In the third chapter, the analytical solutions and numerical solutions of second-order delay differential equation are considered by one-legθ-methods. A necessary and sufficient condition is given by analyzing eigenvalues of characteristic equation. It is proved that the numerical method is stable whenθ=1.The four chapter, we deal with the neutral equation as follow The eigenvalues of linear characteristic equation are analyzed in detail. The numerical solutions by one-legθ- methods are constructed and its stability is obtained. Specially, it is proved that the numerical method is stable whenθ=1. 【关键词】二阶延迟微分方程 周期解 牛顿法 特征方程 单支θ-方法 稳定性 【英文关键词】Second-order delay differential equation Periodic solution Newton method Characteristic equation Stability One-legθ-methods 【目录】关于几类二阶延迟微分方程数值解及其稳定性的研究 摘要 3-4 Abstract 4 第一章 绪论 6-14 1.1 研究背景 6-8 1.2 延迟微分方程稳定性理论研究现状 8-12 1.2.1 一阶延迟微分方程稳定性 8-10 1.2.2 二阶延迟微分方程稳定性 10-12 1.3 本文的主要研究内容 12-14 第二章 二阶延迟微分方程周期解的存在唯一性及数值解法 14-25 2.1 引言 14