本文關(guān)鍵詞： 積分不等式 Hermite-Hadamard型不等式 Hadamard分?jǐn)?shù)階積分 修正的Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù) 時(shí)滯分?jǐn)?shù)階微分方程　出處：《曲阜師范大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：近幾十年來,隨著分?jǐn)?shù)階微分計(jì)算的興起,分?jǐn)?shù)階微積分理論已經(jīng)在數(shù)學(xué)、信號(hào)處理系統(tǒng)、熱學(xué)和光學(xué)系統(tǒng)及其它應(yīng)用領(lǐng)域里取得了許多重要的成果,分?jǐn)?shù)階微分方程的研究也越來越受到國(guó)內(nèi)外廣大學(xué)者的關(guān)注.結(jié)合常微分方程的經(jīng)典理論,對(duì)于很多實(shí)際問題,都可以從中抽象出分?jǐn)?shù)階微分方程的模型,并且相關(guān)的研究已經(jīng)出現(xiàn)了一系列有價(jià)值的結(jié)果.在研究分?jǐn)?shù)階微分方程解的性質(zhì)中作為重要工具的分?jǐn)?shù)階積分不等式,也成為數(shù)學(xué)工作者的研究熱點(diǎn).各類積分不等式及其推廣形式在研究分?jǐn)?shù)階微分方程解的有界性、唯一性及對(duì)初值的連續(xù)依賴性等方面繼續(xù)發(fā)揮重要作用.本文在參考文獻(xiàn)[2,3,11,17,30,31]的基礎(chǔ)上,將相關(guān)積分不等式推廣到分?jǐn)?shù)階積分不等式,并得到一些新的結(jié)果.根據(jù)內(nèi)容本文分為以下四章：第一章 緒論,介紹本文研究的主要問題及其背景.第二章 結(jié)合參考文獻(xiàn)[2]中一些已知的積分不等式,推導(dǎo)出如下的結(jié)果：第三章 研究在修正的Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù)及積分定義下的一些新的Gronwall-Bellman不等式,推廣到如下的積分不等式：并應(yīng)用其研究分?jǐn)?shù)階微分方程解的有界性、唯一性以乃對(duì)初值的連續(xù)依賴性第四章應(yīng)用修正的iemann-Liouville數(shù)階導(dǎo)數(shù)及積分的性質(zhì),研究如下的為未知函數(shù)u(t)提供了明確的邊界,并應(yīng)用這些結(jié)論來研究分?jǐn)?shù)階微分方程解的有界性,唯一性,以及對(duì)初值的連續(xù)依賴性.
[Abstract]:In recent decades, with the rise of fractional differential computing, fractional calculus theory has made many important achievements in mathematics, signal processing systems, thermal and optical systems and other applications. The research of fractional differential equation has been paid more and more attention by many scholars at home and abroad. Combined with the classical theory of ordinary differential equation, the model of fractional differential equation can be abstracted from it for many practical problems. And a series of valuable results have been found in related studies. In the study of the properties of solutions of fractional differential equations, fractional integral inequalities are used as important tools. All kinds of integral inequalities and their generalized forms are used to study the boundedness of solutions of fractional differential equations. Uniqueness and continuous dependence on initial values continue to play an important role. On the basis of reference [2 / 3 / 11 / 1730 / 31], this paper generalizes the relevant integral inequalities to fractional integral inequalities. Some new results are obtained. According to the content of this paper, there are four chapters as follows: the first chapter introduces the main problems and the background of this paper. Chapter two combines with some known integral inequalities in [2]. The following results are derived: in Chapter 3, some new Gronwall-Bellman inequalities under the modified Riemann-Liouville fractional derivative and integral definitions are studied, which are generalized to the following integral inequalities: the boundedness of the solutions of fractional differential equations is also studied. Uniqueness is the properties of modified iemann-Liouville order derivatives and integrals for continuous dependence on initial values in Chapter 4th. The following studies provide a definite boundary for the unknown function ut), and apply these conclusions to study the boundedness of the solutions of fractional differential equations. Uniqueness and continuous dependence on initial values.
【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】：2015
【分類號(hào)】：O172

【共引文獻(xiàn)】

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1 王虎;時(shí)滯分?jǐn)?shù)階Hopfield神經(jīng)網(wǎng)絡(luò)的動(dòng)力學(xué)分析[D];北京交通大學(xué);2015年

相關(guān)碩士學(xué)位論文 前1條

1 田晶磊;分?jǐn)?shù)階捕食者—食餌系統(tǒng)的動(dòng)力學(xué)研究[D];北京交通大學(xué);2015年

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本文編號(hào)：1508033