分数阶动力方程振动性研究及其应用

发布时间：2018-01-02 03:04

本文关键词：分数阶动力方程振动性研究及其应用　出处：《济南大学》2015年硕士论文　论文类型：学位论文

【摘要】：微分方程振动性理论是微分方程定性理论的一个重要分支,它刻画了方程的解关于x轴上下扰动的情况,并且在实际的生产生活中都具有重要的价值。例如,在研究水体漂浮的船只模型时,其晃动的频率与程度可以由带阻尼项的微分方程的解的振动性过程来刻画;又如,在经济学领域,生产消费之间的时滞现象,商品价格的波动,都涉及到了相应的泛函微分方程的解的振动性理论;工业上的机械振动,电磁感应现象等也都与微分方程的振动性理论有关。因此,微分方程振动性理论在控制学、生态学、经济学、生物学、生命科学、工程领域等方面具有广泛的应用,对它的研究也成为了倍受人们关注的热点内容和重要的研究课题之一。随着振动性理论研究的深入,所研究的方程对象不仅仅局限于传统的线性常微分方程上,人们的目光开始放到了差分方程、偏微分方程、泛函微分方程以及时间尺度上的动力方程上。并且所研究的微分方程的导数也由最开始的一阶方程、二阶方程推广到了高阶微分方程上,并取得了大量的理论成果,这使得微分方程的振动性理论已经得到了长足的完善与发展。目前,关于分数阶微分方程的振动性研究还处在开始阶段,并且这一新的领域开始受到越来越多学者的关注。分数阶微分方程,即指具有特定分数阶导数的微分方程,在某些情况下,具有比整数阶方程更好的模拟性,其导数的特殊性质使得分数阶微分方程在物理学、生物学、通讯工程等多个领域都有应用。目前,分数阶微分方程的多项理论都得到了深入的研究,但关于分数阶微分方程的振动性理论所研究的还甚少,这一新的领域亟待人们的研究与关注。因此,本文正是抱着这一目的对分数阶微分方程的振动性问题,从多方面进行了试探性的研究,克服了分数阶导数较整数阶导数难以计算的问题,探索了判断方程振动性的振动准则。此外,本文还从经典的振动性理论出发,对目前的热门问题,时间尺度上的动力方程的振动性进行了研究,并得到了新颖的结果。本文主要研究了几类分数阶非线性微分方程、分数阶中立型时滞微分方程和具常系数的分数阶线性微分方程的振动性。此外,还包括时间尺度上的二阶超线性动力方程、三阶Emden-Fowler型动力方程以及高阶动力方程的振动性问题,并得到了多项较好的研究结果。第一章主要介绍了分数阶微分方程振动性理论目前已取得的研究成果,并给出了分数阶导数的基本定义,以及分数阶微积分的历史背景。第二章通过采用Riccati变换法以及不等式技巧研究一类具Riemann-Liouville型分数阶导数的非线性分数阶微分方程解的振动性问题,并给出满足方程振动的几个充分条件。第三章通过比较定理研究一类具有修正型Riemann-Liouville分数阶导数的中立型分数阶微分方程解的振动性,给出方程振动所需的充分条件。第四章利用Laplace变换以及特征方程的一些理论,研究了具有常系数的时滞分数阶微分方程的振动性,给出了方程振动的充要条件。第五章研究两类时间尺度上动力方程解的振动性。利用Hille-Nehari型振动准则,给出了两类动力方程的若干振动条件。第六章通过Kwong-Wong等人所给出的积分不等式定理,建立了一类关于二阶动力方程的振动准则。第七章总结与展望。归纳总结本文研究的主要工作和创新点,并对未来的研究工作进行展望。
[Abstract]:The differential equation of vibration theory is an important branch of the qualitative theory of differential equation, which describes a X axis under the perturbation equation, and it has important value in practical life. For example, in the research of water floating vessel model, the frequency and degree of the sloshing can be made with differential equation of damped oscillation of the solutions of the process to describe; again, in the field of economics, time lag between production and consumption, commodity price fluctuations are related to the functional differential equations corresponding to the oscillation of the solutions of the mechanical vibration theory; industry, electromagnetic induction phenomenon also oscillation theory with the differential equations. Therefore, differential equation of vibration theory in control science, ecology, economics, biology, life science, has a wide application engineering field, the research on it has become more people One of the hot topics of concern and an important research topic. With the in-depth study of the vibration theory, the linear equation of the object is not limited to the traditional differential equation, people begin to differential equations, partial differential equations, dynamic equations and functional differential equations on time scales of the differential equations. The research and the derivative by a first order equation, two order equation is extended to the high order differential equation, and obtained a lot of theoretical results, which makes the oscillation theory of differential equations has been considerable improvement and development. At present, the research on the vibration of the fractional differential equation is at the beginning, and this new area began to receive more and more attention of scholars. The fractional differential equations, i.e. differential equations with specific fractional derivatives, in some cases, compared with Simulation of integer order equation better, the special character of the derivative of fractional order differential equations in physics, biology, many fields of communication engineering etc. are applied. At present, many theories of fractional differential equations have been studied, but the research on vibration theory of fractional differential equations is little this, a new field to study and attention. Therefore, it is with the purpose of vibration problems of fractional differential equations, the tentative research from many aspects, overcome the fractional derivative with integer derivative to calculate, the vibration equation of exploration vibration criteria. In addition, this article also from the classical theory of vibration based on the popular questions, the vibration of dynamic equations on time scales are studied, and obtained new results. This paper mainly studies Several kinds of fractional order nonlinear differential equation, fractional order neutral delay differential equation and fractional order linear differential equation with constant coefficient oscillation. In addition, also includes two order super linear dynamic equations on time scales, the three order Emden-Fowler dynamic equations and high order equations of vibration problems, and obtained the a number of good results. The first chapter mainly introduces the research results of the fractional differential equation of vibration theory has been made, and some basic definitions of fractional derivative and fractional calculus, the historical background of the second chapter. By using Riccati transform method and inequality technique is studied for a class of Riemann-Liouville type with nonlinear fractional derivative the fractional order differential equations of vibration problems, meet some sufficient conditions for the oscillation of the equation is given. The third chapter through the comparison theorem of a correction Oscillation of neutral differential equations of fractional order Riemann-Liouville type fractional derivative solution, sufficient conditions for the vibration equation. The fourth chapter uses Laplace transform and some theories of the characteristic equation of oscillation of delay fractional differential equations with constant coefficients, and give the necessary and sufficient conditions for the oscillation of the equation. The fifth chapter vibration study on two kinds of solutions of dynamic equations on time scales. By using Hille-Nehari type oscillation criteria, some vibration conditions for two types of dynamic equation is given. The sixth chapter through the integral inequality given by Kwong-Wong et al theory, established the oscillation criteria for a class of two order dynamic equations. The seventh chapter is summary and prospect. The main work this paper summarizes research and innovation, and the future research work is prospected.

【学位授予单位】：济南大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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