几类新的分数阶积分不等式及其应用

发布时间：2018-02-13 11:18

本文关键词： 积分不等式 Hermite-Hadamard型不等式 Hadamard分数阶积分修正的Riemann-Liouville分数阶导数时滞分数阶微分方程　出处：《曲阜师范大学》2015年硕士论文　论文类型：学位论文

【摘要】：近几十年来,随着分数阶微分计算的兴起,分数阶微积分理论已经在数学、信号处理系统、热学和光学系统及其它应用领域里取得了许多重要的成果,分数阶微分方程的研究也越来越受到国内外广大学者的关注.结合常微分方程的经典理论,对于很多实际问题,都可以从中抽象出分数阶微分方程的模型,并且相关的研究已经出现了一系列有价值的结果.在研究分数阶微分方程解的性质中作为重要工具的分数阶积分不等式,也成为数学工作者的研究热点.各类积分不等式及其推广形式在研究分数阶微分方程解的有界性、唯一性及对初值的连续依赖性等方面继续发挥重要作用.本文在参考文献[2,3,11,17,30,31]的基础上,将相关积分不等式推广到分数阶积分不等式,并得到一些新的结果.根据内容本文分为以下四章：第一章绪论,介绍本文研究的主要问题及其背景.第二章结合参考文献[2]中一些已知的积分不等式,推导出如下的结果：第三章研究在修正的Riemann-Liouville分数阶导数及积分定义下的一些新的Gronwall-Bellman不等式,推广到如下的积分不等式：并应用其研究分数阶微分方程解的有界性、唯一性以乃对初值的连续依赖性第四章应用修正的iemann-Liouville数阶导数及积分的性质,研究如下的为未知函数u(t)提供了明确的边界,并应用这些结论来研究分数阶微分方程解的有界性,唯一性,以及对初值的连续依赖性.
[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.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O172

【共引文献】