本文選題：時(shí)滯　切入點(diǎn)：脈沖　出處：《中國(guó)科學(xué)院研究生院(武漢物理與數(shù)學(xué)研究所)》2015年博士論文　論文類型：學(xué)位論文

【摘要】：振動(dòng)是一種帶有普遍意義的物質(zhì)運(yùn)動(dòng)形式,是系統(tǒng)的主要?jiǎng)恿W(xué)性質(zhì)之一。微分方程的振動(dòng)理論在控制工程、機(jī)械振動(dòng)、力學(xué)等領(lǐng)域都有廣泛的應(yīng)用。由G. Sturm建立的二階線性微分方程解的零點(diǎn)分布的比較定理和分離定理,為微分方程振動(dòng)性理論的研究奠定了基礎(chǔ)。一個(gè)半世紀(jì)以來(lái),微分方程的振動(dòng)理論得到了迅猛的發(fā)展,有大批數(shù)學(xué)工作者從事這方面的研究,取得了一系列豐碩的研究成果。而時(shí)滯(偏)微分方程和脈沖(偏)微分方程振動(dòng)理論是微分方程定性理論研究的一個(gè)重要組成部分.時(shí)滯和脈沖的存在使系統(tǒng)能更精確地反映事物的變化規(guī)律,同時(shí)也使得系統(tǒng)的振動(dòng)性分析變得更加困難。時(shí)滯脈沖(偏)微分方程的振動(dòng)性研究是近幾十年來(lái)微分方程領(lǐng)域興起的一個(gè)新的熱點(diǎn),并且受到人們的日益關(guān)注。另一方面,分?jǐn)?shù)階微積分理論(包含分?jǐn)?shù)階微分方程、分?jǐn)?shù)階積分方程、分?jǐn)?shù)階微分積分方程以及數(shù)學(xué)物理方程中的一些特殊的函數(shù))作為一種全新的數(shù)學(xué)研究分支,在流體力學(xué)、多孔結(jié)構(gòu)、擴(kuò)散系統(tǒng)、動(dòng)力系統(tǒng)的控制理論等領(lǐng)域都有重要的應(yīng)用。由于分?jǐn)?shù)階微分方程在很多方面的理論研究才剛剛起步,如關(guān)于分?jǐn)?shù)階微分方程的振動(dòng)理論尚很不完善。本文主要研究了非線性時(shí)滯脈沖偏微分方程及方程組解的振動(dòng)性質(zhì),以及分?jǐn)?shù)階微分方程解的振動(dòng)性及分?jǐn)?shù)階偏微分方程解的強(qiáng)迫振動(dòng)性,推廣并改進(jìn)了文獻(xiàn)中的相關(guān)結(jié)果。主要內(nèi)容如下：第一章為綜述,簡(jiǎn)要回顧了時(shí)滯脈沖偏微分方程(組)和分?jǐn)?shù)階常(偏)微分方程等的振動(dòng)理論的研究背景和發(fā)展?fàn)顩r,同時(shí)介紹了本文的主要工作。第二章研究了非線性脈沖時(shí)滯偏微分方程及方程組解的振動(dòng)性質(zhì),利用推廣的Riccati變換,通過(guò)積分平均值方法,將含脈沖的時(shí)滯偏微分方程及方程組的振動(dòng)性問(wèn)題轉(zhuǎn)化為含脈沖的時(shí)滯常微分不等式不存在最終正解或最終負(fù)解的問(wèn)題,得到了方程及方程組的解產(chǎn)生振動(dòng)的充分條件,建立了方程振動(dòng)的一些新的準(zhǔn)則。第三章通過(guò)引入一類H(t,s)型函數(shù),利用推廣的Riccati變換和輔助函數(shù),結(jié)合積分平均值方法和Holder不等式,討論了帶阻尼項(xiàng)的脈沖時(shí)滯偏微分方程解的振動(dòng)性質(zhì),得到了相關(guān)條件下解產(chǎn)生振動(dòng)一些新的準(zhǔn)則,推廣并改進(jìn)了已有的結(jié)果。第四章先介紹了與分?jǐn)?shù)階微分方程有關(guān)的一些概念,利用分?jǐn)?shù)階微積分的特點(diǎn)和性質(zhì),研究了一類分?jǐn)?shù)階常微分方程解振動(dòng)性質(zhì)及一類分?jǐn)?shù)階偏微分方程解的強(qiáng)迫振動(dòng)性質(zhì),得到了方程的解振動(dòng)及強(qiáng)迫振動(dòng)的充分條件,這些結(jié)論可以看做是分?jǐn)?shù)階微分方程振動(dòng)性研究新的補(bǔ)充。第五章對(duì)本文的研究?jī)?nèi)容和主要結(jié)果進(jìn)行了歸納和總結(jié),并對(duì)今后的研究工作進(jìn)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.
【學(xué)位授予單位】：中國(guó)科學(xué)院研究生院(武漢物理與數(shù)學(xué)研究所)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175

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本文編號(hào)：1638642