本文選題：分?jǐn)?shù)階導(dǎo)數(shù) + 偏微分方程��； 參考：《曲阜師范大學(xué)》2017年碩士論文

【摘要】：微分方程理論研究和應(yīng)用幾乎滲透所有學(xué)科和領(lǐng)域,因此微分方程的定性理論研究受到很多專(zhuān)家學(xué)者的重視.振動(dòng)性作為微分方程定性性質(zhì)的一部分也成為研究的熱點(diǎn).不僅在整數(shù)階常微分方程方面,同時(shí)分?jǐn)?shù)階常微分方程、分?jǐn)?shù)階偏微分方程振動(dòng)性的研究在近年來(lái)都引起了眾多專(zhuān)家學(xué)者的興趣和關(guān)注,并取得了一系列研究成果.本文在借鑒前人研究方法的基礎(chǔ)上,利用廣義的Riccati變換、積分平均方法以及算子法研究了三類(lèi)分?jǐn)?shù)階偏微分方程的振動(dòng)性準(zhǔn)則.根據(jù)內(nèi)容本文分為以下四章:第一章緒論里主要介紹本文用到的關(guān)于分?jǐn)?shù)階微積分的基本定義、性質(zhì)以及引理.第二章在前人研究方法的啟發(fā)下,利用新的方法探討了方程:(?)的振動(dòng)性.第三章在文獻(xiàn)[1]的形式下,加入強(qiáng)迫項(xiàng),研究方程:(?)的振動(dòng)性,得到方程振動(dòng)的充分條件.第四章研究如下類(lèi)型的非線性分?jǐn)?shù)階偏微分方程的振動(dòng)性準(zhǔn)則:三類(lèi)方程具有以下邊界條件:或者其中,?是Rn中具有分段光滑邊界??的一個(gè)有界區(qū)域,α是一個(gè)常數(shù),且α∈(0,1),R+=(0,∞).Dα+,tu(x,t)是u相對(duì)于t的α階Riemann-Liouville導(dǎo)數(shù),?是Rn中的拉普拉斯算子.N是邊界??的單位外法向量,g(x,t)是??×R_+上的非負(fù)連續(xù)函數(shù)。
[Abstract]:The research and application of differential equation theory permeate almost all disciplines and fields, so the qualitative theory of differential equation is paid much attention by many experts and scholars. As a part of qualitative properties of differential equations, oscillation has also become a hot topic. Not only in integer order ordinary differential equation, but also in fractional order ordinary differential equation, fractional partial differential equation oscillation has attracted the interest and attention of many experts and scholars in recent years, and a series of research results have been obtained. In this paper, the oscillatory criteria of three kinds of fractional partial differential equations are studied by using generalized Riccati transform, integral averaging method and operator method on the basis of previous research methods. This paper is divided into four chapters according to the content: in the first chapter, the basic definition, properties and Lemma of fractional calculus used in this paper are introduced. In the second chapter, inspired by the previous research methods, we use a new method to discuss the equation: (1) The vibrancy of In the third chapter, in the form of [1], the forced term is added to study the equation. A sufficient condition for the oscillation of the equation is obtained. In chapter 4, we study the oscillatory criteria of nonlinear fractional partial differential equations: three kinds of equations have the following boundary conditions: or? It is that rn has piecewise smooth boundary? A bounded region of a, 偽 is a constant, and 偽 鈭，

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