分数阶微积分及分数阶方程初步研究.docx

分数阶微积分及分数阶方程初步研究

[摘要]分数阶微积分及分数阶方程是当今国内外研究的最热的研究课题，理论及相关问题的研究还处在初级阶段。本文旨在通过引入分数阶导数及其相关问题，初步介绍和研究了分数阶微积分的若干性质。本文分别给出分数阶导数常见的四种定义：Grünwald-Letnikov分数阶导数定义、Riemann-Liouville分数阶导数定义、Caputo分数阶导数定义、Weyl分数阶导数定义，讨论了其联系与区别。在整数阶微积分的基础上进一步延伸了Riemann-Liouville分数阶导数定义下分数阶的运算法则、基本性质。最后简要介绍了线性分数阶微分方程初值问题解的唯一存在性。

[关键词]分数阶导数；分数阶方程；Grünwald-Letniko分数阶导数；Riemann-Liouville分数阶导数；Caputo分数阶导数.

Preliminarystudiesoffractionalcalculusandfractionalequation

[Abstract]FractionalCalculusandFractionalequationsarethehottestresearchtopicintoday'sdomesticandinternationalresearch,theoreticalandrelatedissuesisstillinitsinfancy.Thispaperaimstointroducefractionalderivativesandrelatedissues,initialpresentationandstudysomepropertiesoffractionalcalculus.Thisarticlegivesfourcommondefinitionofthefractionalderivatives:GrunwaldLetnikovfractionalderivative,RiemannLiouvillefractionalderivative,Caputofractionalderivative,Weylfractionalderivativeandtherelationanddistinctionbetweenthem.Onthebasisoftheinteger-ordercalculus,furtherextendthefractionalalgorithmsandbasicnatureunderthedefinitionoftheRiemann-Liouvillefractionalderivative.Finally,brieflyintroducedtheexistenceanduniquenessofthesolutionsoflinearfractionaldifferentialequations.

[Keywords]Fractionalderivative;Fractionalequations;Grünwald-Letnikovfractional
derivative;Riemann-Liouvillefractionalderivative;Caputofractionalderivative.

目录

TOC\o"1-2"\h\z\uHYPERLINK\l"_Toc357790957"1引言 1
HYPERLINK\l"_Toc357790958"1.1分数阶导数的研究背景、意义 2
HYPERLINK\l"_Toc357790959"1.2分数阶微积分理论的研究现状 2
HYPERLINK\l"_Toc357790961"1.3本文的组织结构 3
HYPERLINK\l"_Toc357790962"2分数阶微积分的基本概念 3
HYPERLINK\l"_Toc357790963"2.1Grünwald-Letnikov分数阶导数定义 3
HYPERLINK\l"_Toc357790964"2.2Riemann-Liouville分数阶导数定义 4
HYPERLINK\l"_Toc357790965"2.3Caputo分数阶导数定义 6
HYPERLINK\l"_Toc357790965"2.4Weyl分数阶导数定义 6
HYPERLINK\l"_Toc357790966"2.5三种分数阶导数的关系及其与整数阶导数的区别 8
HYPERLINK\l"_Toc357790966"2.5.1Riemann-Liouville定义与Grünwald-Letnikov定义的比较 8
HYPERLINK\l"_Toc357790966"2.5.2Grünwa