二阶非线性微分方程振动准则

发布时间：2018-03-31 10:27

本文选题：振动性　切入点：二阶　出处：《曲阜师范大学》2016年硕士论文

【摘要】：近几年来,随着微分方程的发展,越来越多的人对微分方程的振动性和非振动性感兴趣,从而出现了新的关于振动准则的理论.微分方程的解的振动性理论是微分方程定性理论的一个重要的分支.在许多实际应用中都出现了关于方程解的振动性的问题,尤其是对二阶微分方程的研究最多.常微分方程的振动性是方程解的性态之一,对自然科学和生产技术中的应用问题有重要意义,具有物理背景和数学模型的作用.本篇文章是对特殊函数和含阻尼项的一般方程的解的振动性的研究.在第二章和第三章的证明过程中都采用了Riccati变换.根据内容本文分为以下三章:第一章绪论.是关于微分方程解的振动性的发展历程及其意义.从整体上对微分方程的解的振动性有一定的了解.第二章关于一类特殊的二阶非线性微分方程振动准则.给定方程通过Riccati变换证明其振动性.其证明过程的特殊性在于要充分利用给定函数的特殊性:对于所有的.这些振动结果是在文献[1]和[2]的基础上得到的.新的定理推广了文献中的相关结果.第三章含有阻尼项的二阶非线性微分方程振动准则.第三章则是在文献[2]的方程基础上加入了阻尼项,使方程变为其中.使微分方程解的振动更一般化,而在证明方法上同样是利用了Riccati变换,并且参考了文献[3],并且满足条件,且是两个正奇数的商.假设,并且当足够大时,不恒为零;并且,对所有的和某些成立,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...
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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