本文選題：振動(dòng)性　切入點(diǎn)：二階　出處：《曲阜師范大學(xué)》2016年碩士論文

【摘要】：近幾年來,隨著微分方程的發(fā)展,越來越多的人對(duì)微分方程的振動(dòng)性和非振動(dòng)性感興趣,從而出現(xiàn)了新的關(guān)于振動(dòng)準(zhǔn)則的理論.微分方程的解的振動(dòng)性理論是微分方程定性理論的一個(gè)重要的分支.在許多實(shí)際應(yīng)用中都出現(xiàn)了關(guān)于方程解的振動(dòng)性的問題,尤其是對(duì)二階微分方程的研究最多.常微分方程的振動(dòng)性是方程解的性態(tài)之一,對(duì)自然科學(xué)和生產(chǎn)技術(shù)中的應(yīng)用問題有重要意義,具有物理背景和數(shù)學(xué)模型的作用.本篇文章是對(duì)特殊函數(shù)和含阻尼項(xiàng)的一般方程的解的振動(dòng)性的研究.在第二章和第三章的證明過程中都采用了Riccati變換.根據(jù)內(nèi)容本文分為以下三章:第一章緒論.是關(guān)于微分方程解的振動(dòng)性的發(fā)展歷程及其意義.從整體上對(duì)微分方程的解的振動(dòng)性有一定的了解.第二章關(guān)于一類特殊的二階非線性微分方程振動(dòng)準(zhǔn)則.給定方程通過Riccati變換證明其振動(dòng)性.其證明過程的特殊性在于要充分利用給定函數(shù)的特殊性:對(duì)于所有的.這些振動(dòng)結(jié)果是在文獻(xiàn)[1]和[2]的基礎(chǔ)上得到的.新的定理推廣了文獻(xiàn)中的相關(guān)結(jié)果.第三章含有阻尼項(xiàng)的二階非線性微分方程振動(dòng)準(zhǔn)則.第三章則是在文獻(xiàn)[2]的方程基礎(chǔ)上加入了阻尼項(xiàng),使方程變?yōu)槠渲?使微分方程解的振動(dòng)更一般化,而在證明方法上同樣是利用了Riccati變換,并且參考了文獻(xiàn)[3],并且滿足條件,且是兩個(gè)正奇數(shù)的商.假設(shè),并且當(dāng)足夠大時(shí),不恒為零;并且,對(duì)所有的和某些成立,并且且.
[Abstract]:In recent years, with the development of differential equations, more and more people are interested in the oscillation and non-vibration sexy of differential equations.The oscillatory theory of solutions of differential equations is an important branch of qualitative theory of differential equations.In many practical applications, there are many problems about the oscillation of the solution of the equation, especially the second order differential equation.The oscillation of ordinary differential equation is one of the properties of the solution of the equation, which is of great significance to the application of natural science and production technology, and has the function of physical background and mathematical model.This paper studies the oscillation of solutions of special functions and general equations with damping term.In the second and third chapters, the Riccati transform is used in the proof process.According to the content of this paper is divided into the following three chapters: the first chapter is an introduction.It is about the development and significance of the oscillation of the solution of differential equation.On the whole, we have a certain understanding of the oscillation of the solutions of differential equations.The second chapter deals with the Oscillation criteria for a class of special second order nonlinear differential equations.The oscillation of given equation is proved by Riccati transform.The particularity of the proof process is to make full use of the particularity of the given function: for all.These vibration results are obtained on the basis of references [1] and [2].The new theorem generalizes the related results in the literature.Chapter 3 Oscillation criteria for second order nonlinear differential equations with damping term.In the third chapter, the damping term is added to the equation in reference [2].The oscillation of the solution of differential equation is generalized, and the Riccati transformation is also used in the proof method, and the condition is satisfied and the quotient of two positive odd numbers is satisfied with reference [3].Suppose, and when large enough, not always zero; and, for all and some, and...
【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175

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