【摘要】：隨著科學(xué)技術(shù)的不斷發(fā)展,分?jǐn)?shù)階微分方程已成為微分方程理論研究中的一個(gè)重要分支,在工程、力學(xué)、天文學(xué)、經(jīng)濟(jì)學(xué)、控制論及生物學(xué)等領(lǐng)域中的許多問題都會(huì)涉及到分?jǐn)?shù)階微分方程.但是,由于觀測(cè)、實(shí)驗(yàn)和維護(hù)引起的誤差,我們得到的變量和參數(shù)通常是模糊的、信息不完全的,而非精確的.將這些不確定性引入分?jǐn)?shù)階微分方程,稱之為分?jǐn)?shù)階模糊微分方程.分?jǐn)?shù)階模糊微分方程初值問題是分?jǐn)?shù)階模糊微分方程定性理論的基本研究對(duì)象之一.近年來(lái),由于分?jǐn)?shù)階方程和模糊方程在各個(gè)領(lǐng)域中的廣泛應(yīng)用,對(duì)分?jǐn)?shù)階模糊微分方程初值問題以及相關(guān)理論的研究逐漸成為研究熱點(diǎn).同時(shí),分?jǐn)?shù)階模糊微分方程邊值問題則是一個(gè)相對(duì)更新穎的領(lǐng)域,一直以來(lái)鮮有人問津.隨著實(shí)際領(lǐng)域中需求的不斷出現(xiàn),分?jǐn)?shù)階模糊微分方程邊值問題也逐漸引起人們的關(guān)注.但是,由于模糊數(shù)空間中的分析學(xué)和代數(shù)學(xué)理論遠(yuǎn)沒有經(jīng)典理論那樣完善,各類導(dǎo)數(shù)特別是高階導(dǎo)數(shù)定義復(fù)雜繁瑣而且條件苛刻,從微分方程到積分方程的等價(jià)轉(zhuǎn)化也不像經(jīng)典問題那樣簡(jiǎn)單易行.所以,研究分?jǐn)?shù)階模糊微分方程需要可靠的分析方法和高效的數(shù)值技術(shù).也正是這些問題和對(duì)這些問題的不懈研究,極大地推動(dòng)了分?jǐn)?shù)階模糊微積分和模糊微分方程的發(fā)展.縱觀經(jīng)典分?jǐn)?shù)階微分方程的發(fā)展,我們不難發(fā)現(xiàn),過去很多數(shù)學(xué)上束手無(wú)策的問題,往往由于分?jǐn)?shù)階微積分的使用迎刃而解,顯示出分?jǐn)?shù)階微分方程的非凡魅力.所以,對(duì)分?jǐn)?shù)階模糊微分方程基本理論和基本性質(zhì)進(jìn)行更加深入和系統(tǒng)地研究,不僅可以為分?jǐn)?shù)階模糊微分方程理論的進(jìn)一步發(fā)展奠定堅(jiān)實(shí)的基礎(chǔ),也可以為其他科學(xué)領(lǐng)域提供強(qiáng)有力的理論支撐.鑒于此,本文系統(tǒng)地研究了分?jǐn)?shù)階模糊微分方程初邊值問題,涉及解的存在性、唯一性和穩(wěn)定性,并將本文的研究方法應(yīng)用于各種科學(xué)和工程中的實(shí)際問題模型求解.全文共分為七章.第一章詳細(xì)闡述分?jǐn)?shù)階模糊微分方程的研究背景、發(fā)展進(jìn)程和研究現(xiàn)狀以及分?jǐn)?shù)階模糊微分方程初邊值問題在理論與實(shí)際應(yīng)用中的研究意義,并列出相關(guān)基本定義、引理和本文的主要研究方法,最后簡(jiǎn)明扼要地介紹本文的主要研究?jī)?nèi)容和結(jié)構(gòu)框架.第二章研究?jī)深惙謹(jǐn)?shù)階模糊微分方程初值問題解的存在性和唯一性.利用不動(dòng)點(diǎn)定理和逐次逼近法得到解的存在性和唯一性.第三章研究一類分?jǐn)?shù)階模糊微分方程的穩(wěn)定性.借助分?jǐn)?shù)階雙曲函數(shù)、Banach壓縮映像原理和不等式技術(shù),得到解的存在性、唯一性和Eq-Ulam型穩(wěn)定性.第四章研究高階分?jǐn)?shù)階模糊微分方程初值問題解的存在性和唯一性.利用逐次逼近法和Banach壓縮映像原理,得到解的存在性和唯一性以及解對(duì)初值的連續(xù)依賴性.第五章研究分?jǐn)?shù)階模糊微分方程(系統(tǒng))周期邊值問題的可解性.利用切換點(diǎn)概念、Schauder不動(dòng)點(diǎn)定理、Leray-Schauder非線性抉擇定理和Banach壓縮映像原理等技術(shù),得到幾類非線性方程(系統(tǒng))解存在唯一的若干充分條件.第六章研究一類高階分?jǐn)?shù)階模糊微分方程邊值問題的可解性.利用Schauder不動(dòng)點(diǎn)定理、廣義Gronwall不等式得到該類邊值問題解的存在性和唯一性.第七章為全文的總結(jié)與展望.概括總結(jié)本文的主要工作和創(chuàng)新點(diǎn),并對(duì)該領(lǐng)域相關(guān)研究工作進(jìn)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.
【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】：2017
【分類號(hào)】：O175.8

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相關(guān)期刊論文 前1條

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本文編號(hào)：2158467