几类分数阶模糊微分方程初边值问题及其应用
发布时间：2018-08-01 18:32
【摘要】：随着科学技术的不断发展,分数阶微分方程已成为微分方程理论研究中的一个重要分支,在工程、力学、天文学、经济学、控制论及生物学等领域中的许多问题都会涉及到分数阶微分方程.但是,由于观测、实验和维护引起的误差,我们得到的变量和参数通常是模糊的、信息不完全的,而非精确的.将这些不确定性引入分数阶微分方程,称之为分数阶模糊微分方程.分数阶模糊微分方程初值问题是分数阶模糊微分方程定性理论的基本研究对象之一.近年来,由于分数阶方程和模糊方程在各个领域中的广泛应用,对分数阶模糊微分方程初值问题以及相关理论的研究逐渐成为研究热点.同时,分数阶模糊微分方程边值问题则是一个相对更新颖的领域,一直以来鲜有人问津.随着实际领域中需求的不断出现,分数阶模糊微分方程边值问题也逐渐引起人们的关注.但是,由于模糊数空间中的分析学和代数学理论远没有经典理论那样完善,各类导数特别是高阶导数定义复杂繁琐而且条件苛刻,从微分方程到积分方程的等价转化也不像经典问题那样简单易行.所以,研究分数阶模糊微分方程需要可靠的分析方法和高效的数值技术.也正是这些问题和对这些问题的不懈研究,极大地推动了分数阶模糊微积分和模糊微分方程的发展.纵观经典分数阶微分方程的发展,我们不难发现,过去很多数学上束手无策的问题,往往由于分数阶微积分的使用迎刃而解,显示出分数阶微分方程的非凡魅力.所以,对分数阶模糊微分方程基本理论和基本性质进行更加深入和系统地研究,不仅可以为分数阶模糊微分方程理论的进一步发展奠定坚实的基础,也可以为其他科学领域提供强有力的理论支撑.鉴于此,本文系统地研究了分数阶模糊微分方程初边值问题,涉及解的存在性、唯一性和稳定性,并将本文的研究方法应用于各种科学和工程中的实际问题模型求解.全文共分为七章.第一章详细阐述分数阶模糊微分方程的研究背景、发展进程和研究现状以及分数阶模糊微分方程初边值问题在理论与实际应用中的研究意义,并列出相关基本定义、引理和本文的主要研究方法,最后简明扼要地介绍本文的主要研究内容和结构框架.第二章研究两类分数阶模糊微分方程初值问题解的存在性和唯一性.利用不动点定理和逐次逼近法得到解的存在性和唯一性.第三章研究一类分数阶模糊微分方程的稳定性.借助分数阶双曲函数、Banach压缩映像原理和不等式技术,得到解的存在性、唯一性和Eq-Ulam型稳定性.第四章研究高阶分数阶模糊微分方程初值问题解的存在性和唯一性.利用逐次逼近法和Banach压缩映像原理,得到解的存在性和唯一性以及解对初值的连续依赖性.第五章研究分数阶模糊微分方程(系统)周期边值问题的可解性.利用切换点概念、Schauder不动点定理、Leray-Schauder非线性抉择定理和Banach压缩映像原理等技术,得到几类非线性方程(系统)解存在唯一的若干充分条件.第六章研究一类高阶分数阶模糊微分方程边值问题的可解性.利用Schauder不动点定理、广义Gronwall不等式得到该类边值问题解的存在性和唯一性.第七章为全文的总结与展望.概括总结本文的主要工作和创新点,并对该领域相关研究工作进行展望.
[Abstract]:With the continuous development of science and technology, fractional differential equations have become an important branch of the theoretical research of differential equations. Many problems in the fields of engineering, mechanics, astronomy, economics, control and biology will involve fractional differential equations. However, we get the error caused by observation, experiment and maintenance. The variables and parameters are usually fuzzy, information incomplete and not accurate. Introducing these uncertainties into fractional differential equations is called fractional order fuzzy differential equations. The initial value problem of fractional order fuzzy differential equations is one of the basic research objects of the qualitative theory of fractional order fuzzy differential equations. In recent years, the fractional order equation is used. As well as the widespread application of fuzzy equations in various fields, the research on the initial value problem of fractional order fuzzy differential equations and the related theories gradually become a hot spot. At the same time, the boundary value problem of fractional order fuzzy differential equations is a relatively new field. The boundary value problem of fractional order fuzzy differential equations has also gradually aroused people's attention. However, because the analysis and algebra theory in the fuzzy number space are far from the classical theory, the definitions of various derivatives, especially the high order derivatives are complicated and harsh, and the equivalent transformation from the differential equation to the integral equation is not like the classical question. Therefore, the study of fractional order fuzzy differential equations requires reliable analytical methods and efficient numerical techniques. It is these problems and the unremitting studies of these problems that greatly promote the development of fractional fuzzy calculus and fuzzy differential equations. It is found that many of the problems in Mathematics in the past are easily solved by the use of fractional calculus, which shows the extraordinary charm of fractional differential equations. Therefore, the basic theory and basic properties of fractional order fuzzy differential equations are more deeply and systematically studied, not only for the theory of fractional order fuzzy differential equations. It lays a solid foundation for further development and provides strong theoretical support for other scientific fields. In view of this, this paper systematically studies the initial boundary value problem of fractional order fuzzy differential equations, involving the existence, uniqueness and stability of the solution, and applies the research method in this paper to the practical problems in various science and engineering. The full text is divided into seven chapters. The first chapter describes the background of the fractional fuzzy differential equation, the development process and the research status as well as the significance of the theoretical and practical application of the initial boundary value problems of fractional order fuzzy differential equations, and lists the relevant basic definitions, introduction and the main research methods of this paper. In the second chapter, the existence and uniqueness of the solution for the initial value problem of the two class fractional fuzzy differential equations are studied. The existence and uniqueness of the solution are obtained by the fixed point theorem and the successive approximation method. In the third chapter, the stability of a class of fractional fuzzy differential equations is studied. With the help of fractional hyperbolic Function, Banach compression mapping principle and inequality technology, obtain the existence, uniqueness and Eq-Ulam type stability of the solution. The fourth chapter studies the existence and uniqueness of the solution of the initial value problem of the higher order fractional order fuzzy differential equation. By using the successive approximation method and the Banach compression mapping principle, the existence and uniqueness of the solution and the connection of the solution to the initial value are obtained. The fifth chapter studies the solvability of the periodic boundary value problems of fractional order fuzzy differential equations (Systems). By using the concept of the switching point, the Schauder fixed point theorem, the Leray-Schauder nonlinear choice theorem and the Banach compression mapping principle, some sufficient conditions for the existence of several nonlinear equations (system) solutions are obtained. The sixth chapter studies The solvability of a class of boundary value problems of a class of higher order fractional differential equations. The existence and uniqueness of the solution of this class of boundary value problems are obtained by using the Schauder fixed point theorem and the generalized Gronwall inequality. The seventh chapter is the summary and Prospect of the full text. The main work and innovation of this paper are summarized and the related research work in this field is prospected.
【学位授予单位】：济南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
本文编号：2158467
[Abstract]:With the continuous development of science and technology, fractional differential equations have become an important branch of the theoretical research of differential equations. Many problems in the fields of engineering, mechanics, astronomy, economics, control and biology will involve fractional differential equations. However, we get the error caused by observation, experiment and maintenance. The variables and parameters are usually fuzzy, information incomplete and not accurate. Introducing these uncertainties into fractional differential equations is called fractional order fuzzy differential equations. The initial value problem of fractional order fuzzy differential equations is one of the basic research objects of the qualitative theory of fractional order fuzzy differential equations. In recent years, the fractional order equation is used. As well as the widespread application of fuzzy equations in various fields, the research on the initial value problem of fractional order fuzzy differential equations and the related theories gradually become a hot spot. At the same time, the boundary value problem of fractional order fuzzy differential equations is a relatively new field. The boundary value problem of fractional order fuzzy differential equations has also gradually aroused people's attention. However, because the analysis and algebra theory in the fuzzy number space are far from the classical theory, the definitions of various derivatives, especially the high order derivatives are complicated and harsh, and the equivalent transformation from the differential equation to the integral equation is not like the classical question. Therefore, the study of fractional order fuzzy differential equations requires reliable analytical methods and efficient numerical techniques. It is these problems and the unremitting studies of these problems that greatly promote the development of fractional fuzzy calculus and fuzzy differential equations. It is found that many of the problems in Mathematics in the past are easily solved by the use of fractional calculus, which shows the extraordinary charm of fractional differential equations. Therefore, the basic theory and basic properties of fractional order fuzzy differential equations are more deeply and systematically studied, not only for the theory of fractional order fuzzy differential equations. It lays a solid foundation for further development and provides strong theoretical support for other scientific fields. In view of this, this paper systematically studies the initial boundary value problem of fractional order fuzzy differential equations, involving the existence, uniqueness and stability of the solution, and applies the research method in this paper to the practical problems in various science and engineering. The full text is divided into seven chapters. The first chapter describes the background of the fractional fuzzy differential equation, the development process and the research status as well as the significance of the theoretical and practical application of the initial boundary value problems of fractional order fuzzy differential equations, and lists the relevant basic definitions, introduction and the main research methods of this paper. In the second chapter, the existence and uniqueness of the solution for the initial value problem of the two class fractional fuzzy differential equations are studied. The existence and uniqueness of the solution are obtained by the fixed point theorem and the successive approximation method. In the third chapter, the stability of a class of fractional fuzzy differential equations is studied. With the help of fractional hyperbolic Function, Banach compression mapping principle and inequality technology, obtain the existence, uniqueness and Eq-Ulam type stability of the solution. The fourth chapter studies the existence and uniqueness of the solution of the initial value problem of the higher order fractional order fuzzy differential equation. By using the successive approximation method and the Banach compression mapping principle, the existence and uniqueness of the solution and the connection of the solution to the initial value are obtained. The fifth chapter studies the solvability of the periodic boundary value problems of fractional order fuzzy differential equations (Systems). By using the concept of the switching point, the Schauder fixed point theorem, the Leray-Schauder nonlinear choice theorem and the Banach compression mapping principle, some sufficient conditions for the existence of several nonlinear equations (system) solutions are obtained. The sixth chapter studies The solvability of a class of boundary value problems of a class of higher order fractional differential equations. The existence and uniqueness of the solution of this class of boundary value problems are obtained by using the Schauder fixed point theorem and the generalized Gronwall inequality. The seventh chapter is the summary and Prospect of the full text. The main work and innovation of this paper are summarized and the related research work in this field is prospected.
【学位授予单位】：济南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
【参考文献】
相关期刊论文 前1条
1 王立社;模糊数值正弦函数与余弦函数[J];聊城师院学报(自然科学版);1997年02期
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