分数阶时滞微分方程与偏微分方程的振动性-应用数学专业论文.docxVIP

I I 摘 要 在过去的几十年里, 整数阶微分方程与偏微分方程解的振动性理论研究发 展迅速, 研究成果层出不穷. 虽然分数阶微分方程也得到了很大的发展, 但与整 数微分方程相比, 理论研究还有很多不足, 有些甚至还没有起步. 据我们所知, 现 在还没有关于分数阶泛函微分方程与偏微分方程解的振动性理论研究. 本文主要讨论了分数阶时滞微分方程与偏微分方程的解的振动性理论两部 分内容. 在第 2 章, 我们通过 Laplace 变换来研究 Riemann-Liouville 分数阶时滞 微分方程解的振动性, 并给出振动解的充分条件及若干推论. 在第 3 章, 我们通 过格林公式与 Laplace 变换来探究 Riemann-Liouville 分数偏微分方程在三类边 界条件下解的振动性, 并结合第 2 章的内容给出振动解的充分条件及若干推论. 关键词: 分数微分方程; 分数偏微分方程; 时滞; 振动性; Laplace 变换; 格林 公式; Riemann-Liouville 导数 I II Abstract Oscillation theory of functional di?erential equations and partial di?erential equations with integer derivative has been developed rapidly in the past decade, and a lot of theories has been obtained. Though fractional di?erential equations have gained more development, there is not enough theories comparison with the integer order, even not be started. To the best of our knowledge, nothing has been done with oscillations of fractional functional di?erential equations and partial di?erential equations. In this paper, we discuss the theorem of oscillation for fractional delay di?er- ential equations and partial di?erential equations. In Chapter 2, we present the oscillation for the fractional delay di?erential equations with Riemann-Liouville fractional derivative, su?cient conditions and some corollaries for all solutions of fractional delay di?erential equations to be oscillatory are obtained by using the Laplace transforms. In chapter 3, we survey the oscillation for the frac- tional partial di?erential equations with Riemann-Liouville fractional derivative, in addition with the chapter 2, su?cient conditions and some corollaries for all solutions of fractional partial di?erential equations to be oscillatory are obtained by using the Laplace transforms and Green’s formula. Keywords: Fractional di?erential equations; fractional partial di?erential equations; delay; Oscillation; Laplace transform; Green’s formula; Riemann– Liouville derivative 目 录 第 1 章 绪 论 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .