本文選題：分數(shù)階 + Mittag-Leffler函數(shù)��； 參考：《昆明理工大學》2017年碩士論文

【摘要】：本文主要應(yīng)用Mittag-Leffler函數(shù)和分數(shù)階微分方程的比較定理研究了兩類分數(shù)階數(shù)學模型的持久性、漸近穩(wěn)定性與漸近周期性,推廣并改進了已知文獻中的結(jié)果.全文共分為五章.第一章主要闡述了選題的背景和意義,介紹了分數(shù)階微分方程的發(fā)展、目前在各個領(lǐng)域的應(yīng)用以及前人所做的相關(guān)工作.簡要介紹了兩類數(shù)學模型:食餌模型、Mackey-Glass呼吸道模型,而且對本文的主要工作做了簡單介紹.第二章主要給出了分數(shù)階微分方程的一些基本定義、引理和Mittag-Leffler函數(shù)的相關(guān)性質(zhì),并在這些性質(zhì)的基礎(chǔ)上,得到了一些重要的引理.為了得到模型的持久性,證明了分數(shù)階微分方程的比較定理.為了研究模型的漸近穩(wěn)定性及漸近周期性,引入了分數(shù)階微分方程的特征方程.第三、四章通過運用Mittag-Leffler函數(shù)的性質(zhì)、分數(shù)階微分方程的比較定理以及拉普拉斯變換的特征方程與穩(wěn)定性的關(guān)系,得到了分數(shù)階食餌模型以及分數(shù)階Mackey-Glass呼吸道模型的持久性、漸近穩(wěn)定性與漸近周期性.最后,舉例說明了我們研究結(jié)果的可行性.第五章主要對全文作一個簡短的總結(jié)和回顧,并且探討了未來的一些研究方向.
[Abstract]:In this paper, by using the comparison theorem of Mittag-Leffler function and fractional differential equation, we study the persistence, asymptotic stability and asymptotic periodicity of two kinds of fractional order mathematical models, and generalize and improve the known results. The full text is divided into five chapters. The first chapter describes the background and significance of the topic, introduces the development of fractional differential equations, the current applications in various fields and related work done by predecessors. This paper briefly introduces two kinds of mathematical models: the prey model and the Mackey-Glass respiratory tract model, and gives a brief introduction to the main work of this paper. In the second chapter, we give some basic definitions of fractional differential equations, the properties of Lemma and Mittag-Leffler function, and get some important Lemma on the basis of these properties. In order to obtain the permanence of the model, the comparison theorem of fractional differential equations is proved. In order to study the asymptotic stability and asymptotic periodicity of the model, the characteristic equations of fractional differential equations are introduced. In the third and fourth chapter, by using the properties of Mittag-Leffler function, the comparison theorem of fractional differential equation and the relation between the characteristic equation of Laplace transform and the stability, the persistence of fractional prey model and fractional Mackey-Glass respiratory model is obtained. Asymptotic stability and asymptotic periodicity. Finally, examples are given to illustrate the feasibility of our research results. Chapter five gives a brief summary and review of the full text, and discusses some research directions in the future.
【學位授予單位】：昆明理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175

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