二维、三维空间Riesz 分数阶扩散方程的基本解

J4 ›› 2011, Vol. 46 ›› Issue (8): 23-30.

二维、三维空间Riesz 分数阶扩散方程的基本解

王学彬

武夷学院数学与计算机系, 福建武夷山 354300

收稿日期:2011-03-12 出版日期:2011-08-20 发布日期:2011-09-08

Fundamental solutions of fractional-in-space diffusion equation with Riesz fractional derivative in two and three dimensions

WANG Xue-bin

Department of Mathematics and Computer, Wuyi University, Wuyishan 354300, Fujian, China

Received:2011-03-12 Online:2011-08-20 Published:2011-09-08
About author:WANG Xuebin(1976- ), Male, Lecturer, his research mainly focuses on fractional calculus.Email:wxbnp@163.com
Supported by:
Supported by the Natural Science Foundation of Fujian Province (2008J0204)； Fujian Provincial Department of Education Category the Projects(JA09242); Wuyi University Special Research Fund for Young Teachers(xq201022)

摘要/Abstract

摘要：

讨论二维、三维空间Riesz 分数阶扩散方程的解,用特征函数幂级数形式定义二维、三维分数阶拉普拉斯算子,并给出分数阶拉普拉斯算子与Riesz 分数阶导数的关系。最后用谱表示法导出二维、三维空间Riesz 分数阶扩散方程在齐次和非齐次情况下的在有界区间上满足一定初边值条件的基本解。

关键词: Riesz 分数阶导数;空间分数阶扩散方程;RimannLiouville分数阶导数

Abstract:

The fundamental solutions of fractional-in-space diffusion equation is considered with Riesz fractional derivative (RFDE) in two and three dimensions. The existing definitions of the fractional Laplacian (two dimensions and three dimensions) are investigated and discussed by using eigenfunction expansion, and the relations between fractional Laplacian and Riesz fractional derivative are given. Finally, the fundamental solutions of homogeneous and non-homogeneous RFDE with an initial and boundary condition is derived on a finite domain using a spectral representation.

Key words: Riesz fractional derivative; fractional-in-space diffusion equation; Rimann-Liouville fractional derivative

王学彬. 二维、三维空间Riesz 分数阶扩散方程的基本解[J]. J4, 2011, 46(8): 23-30.

WANG Xue-bin. Fundamental solutions of fractional-in-space diffusion equation with Riesz fractional derivative in two and three dimensions[J]. J4, 2011, 46(8): 23-30.

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