非线性脉冲时滞偏微分方程与分数阶方程解的振动性质

发布时间：2018-03-20 10:17

本文选题：时滞　切入点：脉冲　出处：《中国科学院研究生院(武汉物理与数学研究所)》2015年博士论文　论文类型：学位论文

【摘要】：振动是一种带有普遍意义的物质运动形式,是系统的主要动力学性质之一。微分方程的振动理论在控制工程、机械振动、力学等领域都有广泛的应用。由G. Sturm建立的二阶线性微分方程解的零点分布的比较定理和分离定理,为微分方程振动性理论的研究奠定了基础。一个半世纪以来,微分方程的振动理论得到了迅猛的发展,有大批数学工作者从事这方面的研究,取得了一系列丰硕的研究成果。而时滞(偏)微分方程和脉冲(偏)微分方程振动理论是微分方程定性理论研究的一个重要组成部分.时滞和脉冲的存在使系统能更精确地反映事物的变化规律,同时也使得系统的振动性分析变得更加困难。时滞脉冲(偏)微分方程的振动性研究是近几十年来微分方程领域兴起的一个新的热点,并且受到人们的日益关注。另一方面,分数阶微积分理论(包含分数阶微分方程、分数阶积分方程、分数阶微分积分方程以及数学物理方程中的一些特殊的函数)作为一种全新的数学研究分支,在流体力学、多孔结构、扩散系统、动力系统的控制理论等领域都有重要的应用。由于分数阶微分方程在很多方面的理论研究才刚刚起步,如关于分数阶微分方程的振动理论尚很不完善。本文主要研究了非线性时滞脉冲偏微分方程及方程组解的振动性质,以及分数阶微分方程解的振动性及分数阶偏微分方程解的强迫振动性,推广并改进了文献中的相关结果。主要内容如下：第一章为综述,简要回顾了时滞脉冲偏微分方程(组)和分数阶常(偏)微分方程等的振动理论的研究背景和发展状况,同时介绍了本文的主要工作。第二章研究了非线性脉冲时滞偏微分方程及方程组解的振动性质,利用推广的Riccati变换,通过积分平均值方法,将含脉冲的时滞偏微分方程及方程组的振动性问题转化为含脉冲的时滞常微分不等式不存在最终正解或最终负解的问题,得到了方程及方程组的解产生振动的充分条件,建立了方程振动的一些新的准则。第三章通过引入一类H(t,s)型函数,利用推广的Riccati变换和辅助函数,结合积分平均值方法和Holder不等式,讨论了带阻尼项的脉冲时滞偏微分方程解的振动性质,得到了相关条件下解产生振动一些新的准则,推广并改进了已有的结果。第四章先介绍了与分数阶微分方程有关的一些概念,利用分数阶微积分的特点和性质,研究了一类分数阶常微分方程解振动性质及一类分数阶偏微分方程解的强迫振动性质,得到了方程的解振动及强迫振动的充分条件,这些结论可以看做是分数阶微分方程振动性研究新的补充。第五章对本文的研究内容和主要结果进行了归纳和总结,并对今后的研究工作进行了展望。
[Abstract]:Vibration is a kind of material motion with universal meaning, and is one of the main dynamic properties of the system. The vibration theory of differential equation is used to control engineering and mechanical vibration. The comparison theorem and separation theorem of the 00:00 distribution of solutions of second order linear differential equations established by G. Sturm have laid a foundation for the study of oscillatory theory of differential equations for a century and a half. The vibration theory of differential equations has been developed rapidly, and a large number of mathematics workers are engaged in the research in this field. The oscillation theory of delay differential equation and impulsive differential equation is an important part of the qualitative theory of differential equation. The existence of delay and impulse makes the system. Can more accurately reflect the changing law of things, At the same time, it also makes it more difficult to analyze the oscillation of the system. The oscillatory study of delay impulsive differential equations is a new hot spot in the field of differential equations in recent decades, and has been paid more and more attention. On the other hand, Fractional calculus theory (including fractional differential equations, fractional integral equations, fractional differential integral equations and some special functions in mathematical physics equations) is a new branch of mathematical research in fluid mechanics. Porous structures, diffusion systems, control theory of dynamic systems and other fields have important applications. For example, the oscillatory theory of fractional differential equations is not perfect. In this paper, the oscillatory properties of nonlinear impulsive partial differential equations with delay and the solutions of equations are studied. The oscillations of solutions of fractional differential equations and forced oscillations of solutions of fractional partial differential equations are generalized and improved. The main contents are as follows: chapter 1 is a review. The background and development of oscillatory theory of impulsive partial differential equations (systems) and fractional order ordinary (partial) differential equations are briefly reviewed. In the second chapter, the oscillatory properties of nonlinear impulsive partial differential equations with delay and the solutions of equations are studied. By using the generalized Riccati transform, the method of integral average value is used to study the oscillatory properties of nonlinear impulsive partial differential equations with delay and equations. In this paper, the oscillatory problem of delay partial differential equations and equations with impulses is transformed into the problem that the delay ordinary differential inequalities with impulses do not have the final positive solution or the final negative solution, and the sufficient conditions for the oscillation of the solutions of the equations and equations are obtained. Some new criteria for the oscillation of the equation are established. In chapter 3, by introducing a class of functions of Hautts type, using the generalized Riccati transformation and auxiliary function, combining the method of integral mean value and Holder inequality, In this paper, the oscillatory properties of solutions of impulsive delay partial differential equations with damping term are discussed, and some new criteria for the oscillation of solutions are obtained. In Chapter 4th, some concepts related to fractional differential equations are introduced, and the characteristics and properties of fractional calculus are used. The oscillatory properties of solutions of a class of fractional ordinary differential equations and the forced oscillations of solutions of a class of fractional partial differential equations are studied. The sufficient conditions for the oscillation and forced oscillation of solutions of the equations are obtained. These conclusions can be regarded as a new supplement to the research on the oscillation of fractional differential equations. Chapter 5th summarizes and summarizes the research contents and main results of this paper and looks forward to the future research work.
【学位授予单位】：中国科学院研究生院(武汉物理与数学研究所)
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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