分数阶偏微分方程的振动性研究

发布时间：2018-05-31 14:02

  本文选题：分数阶导数 + 偏微分方程　； 参考：《曲阜师范大学》2017年硕士论文

【摘要】：微分方程理论研究和应用几乎渗透所有学科和领域,因此微分方程的定性理论研究受到很多专家学者的重视.振动性作为微分方程定性性质的一部分也成为研究的热点.不仅在整数阶常微分方程方面,同时分数阶常微分方程、分数阶偏微分方程振动性的研究在近年来都引起了众多专家学者的兴趣和关注,并取得了一系列研究成果.本文在借鉴前人研究方法的基础上,利用广义的Riccati变换、积分平均方法以及算子法研究了三类分数阶偏微分方程的振动性准则.根据内容本文分为以下四章:第一章绪论里主要介绍本文用到的关于分数阶微积分的基本定义、性质以及引理.第二章在前人研究方法的启发下,利用新的方法探讨了方程:(?)的振动性.第三章在文献[1]的形式下,加入强迫项,研究方程:(?)的振动性,得到方程振动的充分条件.第四章研究如下类型的非线性分数阶偏微分方程的振动性准则:三类方程具有以下边界条件:或者其中,?是Rn中具有分段光滑边界??的一个有界区域,α是一个常数,且α∈(0,1),R+=(0,∞).Dα+,tu(x,t)是u相对于t的α阶Riemann-Liouville导数,?是Rn中的拉普拉斯算子.N是边界??的单位外法向量,g(x,t)是??×R_+上的非负连续函数。
[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, 伪 is a constant, and 伪 鈭，

本文编号：1959979