發(fā)布時(shí)間：2018-01-02 03:04

  本文關(guān)鍵詞：分?jǐn)?shù)階動(dòng)力方程振動(dòng)性研究及其應(yīng)用　出處：《濟(jì)南大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 分?jǐn)?shù)階微分方程 振動(dòng)性 時(shí)間尺度 動(dòng)力方程

【摘要】：微分方程振動(dòng)性理論是微分方程定性理論的一個(gè)重要分支,它刻畫了方程的解關(guān)于x軸上下擾動(dòng)的情況,并且在實(shí)際的生產(chǎn)生活中都具有重要的價(jià)值。例如,在研究水體漂浮的船只模型時(shí),其晃動(dòng)的頻率與程度可以由帶阻尼項(xiàng)的微分方程的解的振動(dòng)性過(guò)程來(lái)刻畫;又如,在經(jīng)濟(jì)學(xué)領(lǐng)域,生產(chǎn)消費(fèi)之間的時(shí)滯現(xiàn)象,商品價(jià)格的波動(dòng),都涉及到了相應(yīng)的泛函微分方程的解的振動(dòng)性理論;工業(yè)上的機(jī)械振動(dòng),電磁感應(yīng)現(xiàn)象等也都與微分方程的振動(dòng)性理論有關(guān)。因此,微分方程振動(dòng)性理論在控制學(xué)、生態(tài)學(xué)、經(jīng)濟(jì)學(xué)、生物學(xué)、生命科學(xué)、工程領(lǐng)域等方面具有廣泛的應(yīng)用,對(duì)它的研究也成為了倍受人們關(guān)注的熱點(diǎn)內(nèi)容和重要的研究課題之一。隨著振動(dòng)性理論研究的深入,所研究的方程對(duì)象不僅僅局限于傳統(tǒng)的線性常微分方程上,人們的目光開(kāi)始放到了差分方程、偏微分方程、泛函微分方程以及時(shí)間尺度上的動(dòng)力方程上。并且所研究的微分方程的導(dǎo)數(shù)也由最開(kāi)始的一階方程、二階方程推廣到了高階微分方程上,并取得了大量的理論成果,這使得微分方程的振動(dòng)性理論已經(jīng)得到了長(zhǎng)足的完善與發(fā)展。目前,關(guān)于分?jǐn)?shù)階微分方程的振動(dòng)性研究還處在開(kāi)始階段,并且這一新的領(lǐng)域開(kāi)始受到越來(lái)越多學(xué)者的關(guān)注。分?jǐn)?shù)階微分方程,即指具有特定分?jǐn)?shù)階導(dǎo)數(shù)的微分方程,在某些情況下,具有比整數(shù)階方程更好的模擬性,其導(dǎo)數(shù)的特殊性質(zhì)使得分?jǐn)?shù)階微分方程在物理學(xué)、生物學(xué)、通訊工程等多個(gè)領(lǐng)域都有應(yīng)用。目前,分?jǐn)?shù)階微分方程的多項(xiàng)理論都得到了深入的研究,但關(guān)于分?jǐn)?shù)階微分方程的振動(dòng)性理論所研究的還甚少,這一新的領(lǐng)域亟待人們的研究與關(guān)注。因此,本文正是抱著這一目的對(duì)分?jǐn)?shù)階微分方程的振動(dòng)性問(wèn)題,從多方面進(jìn)行了試探性的研究,克服了分?jǐn)?shù)階導(dǎo)數(shù)較整數(shù)階導(dǎo)數(shù)難以計(jì)算的問(wèn)題,探索了判斷方程振動(dòng)性的振動(dòng)準(zhǔn)則。此外,本文還從經(jīng)典的振動(dòng)性理論出發(fā),對(duì)目前的熱門問(wèn)題,時(shí)間尺度上的動(dòng)力方程的振動(dòng)性進(jìn)行了研究,并得到了新穎的結(jié)果。本文主要研究了幾類分?jǐn)?shù)階非線性微分方程、分?jǐn)?shù)階中立型時(shí)滯微分方程和具常系數(shù)的分?jǐn)?shù)階線性微分方程的振動(dòng)性。此外,還包括時(shí)間尺度上的二階超線性動(dòng)力方程、三階Emden-Fowler型動(dòng)力方程以及高階動(dòng)力方程的振動(dòng)性問(wèn)題,并得到了多項(xiàng)較好的研究結(jié)果。第一章主要介紹了分?jǐn)?shù)階微分方程振動(dòng)性理論目前已取得的研究成果,并給出了分?jǐn)?shù)階導(dǎo)數(shù)的基本定義,以及分?jǐn)?shù)階微積分的歷史背景。第二章通過(guò)采用Riccati變換法以及不等式技巧研究一類具Riemann-Liouville型分?jǐn)?shù)階導(dǎo)數(shù)的非線性分?jǐn)?shù)階微分方程解的振動(dòng)性問(wèn)題,并給出滿足方程振動(dòng)的幾個(gè)充分條件。第三章通過(guò)比較定理研究一類具有修正型Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù)的中立型分?jǐn)?shù)階微分方程解的振動(dòng)性,給出方程振動(dòng)所需的充分條件。第四章利用Laplace變換以及特征方程的一些理論,研究了具有常系數(shù)的時(shí)滯分?jǐn)?shù)階微分方程的振動(dòng)性,給出了方程振動(dòng)的充要條件。第五章研究?jī)深悤r(shí)間尺度上動(dòng)力方程解的振動(dòng)性。利用Hille-Nehari型振動(dòng)準(zhǔn)則,給出了兩類動(dòng)力方程的若干振動(dòng)條件。第六章通過(guò)Kwong-Wong等人所給出的積分不等式定理,建立了一類關(guān)于二階動(dòng)力方程的振動(dòng)準(zhǔn)則。第七章總結(jié)與展望。歸納總結(jié)本文研究的主要工作和創(chuàng)新點(diǎn),并對(duì)未來(lái)的研究工作進(jìn)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.

【學(xué)位授予單位】：濟(jì)南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175

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