本文關鍵詞： 積分不等式 Hermite-Hadamard型不等式 Hadamard分數(shù)階積分 修正的Riemann-Liouville分數(shù)階導數(shù) 時滯分數(shù)階微分方程　出處：《曲阜師范大學》2015年碩士論文　論文類型：學位論文

【摘要】：近幾十年來,隨著分數(shù)階微分計算的興起,分數(shù)階微積分理論已經(jīng)在數(shù)學、信號處理系統(tǒng)、熱學和光學系統(tǒng)及其它應用領域里取得了許多重要的成果,分數(shù)階微分方程的研究也越來越受到國內外廣大學者的關注.結合常微分方程的經(jīng)典理論,對于很多實際問題,都可以從中抽象出分數(shù)階微分方程的模型,并且相關的研究已經(jīng)出現(xiàn)了一系列有價值的結果.在研究分數(shù)階微分方程解的性質中作為重要工具的分數(shù)階積分不等式,也成為數(shù)學工作者的研究熱點.各類積分不等式及其推廣形式在研究分數(shù)階微分方程解的有界性、唯一性及對初值的連續(xù)依賴性等方面繼續(xù)發(fā)揮重要作用.本文在參考文獻[2,3,11,17,30,31]的基礎上,將相關積分不等式推廣到分數(shù)階積分不等式,并得到一些新的結果.根據(jù)內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 結合參考文獻[2]中一些已知的積分不等式,推導出如下的結果：第三章 研究在修正的Riemann-Liouville分數(shù)階導數(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：并應用其研究分數(shù)階微分方程解的有界性、唯一性以乃對初值的連續(xù)依賴性第四章應用修正的iemann-Liouville數(shù)階導數(shù)及積分的性質,研究如下的為未知函數(shù)u(t)提供了明確的邊界,并應用這些結論來研究分數(shù)階微分方程解的有界性,唯一性,以及對初值的連續(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.
【學位授予單位】：曲阜師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O172

【共引文獻】

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1 田晶磊;分數(shù)階捕食者—食餌系統(tǒng)的動力學研究[D];北京交通大學;2015年

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本文編號：1508033