Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation

doi:10.4208/cicp.020709.221209a

科研成果详情

题名	Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation
作者	Li，Xianjuan 1,2; Xu，Chuanju 2
发表日期	2010
发表期刊	Communications in Computational Physics
ISSN/eISSN	1815-2406
卷号	8 期号:5 页码:1016-1051
摘要	In this paper, we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions. Two definitions, i.e., Riemann-Liouville definition and Caputo one, of the fractional derivative are considered in parallel. In both cases, we establish the well-posedness of the weak solution. Moveover, based on the proposed weak formulation, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: First, a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed. We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem, and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems. Secondly, we show that in the case of Riemann-Liouville definition, the well-posedness of the space-time fractional diffusion equation does not require any initial conditions. This contrasts with the case of Caputo definition, in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness. Finally, thanks to the weak formulation, we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem. © 2010 Global-Science Press.
关键词	Error estimates Existence and uniqueness Space-time fractional diffusion equation Spectral methods
DOI	10.4208/cicp.020709.221209a
URL	查看来源
语种	英语English
Scopus入藏号	2-s2.0-77954143774
引用统计	被引频次：282[WOS] [WOS记录] [WOS相关记录]
文献类型	期刊论文
条目标识符	https://repository.uic.edu.cn/handle/39GCC9TT/6638
专题	北师香港浸会大学
通讯作者	Xu，Chuanju
作者单位	1.1 BNU-HKBU United International College,Zhuhai 519085,China 2.School of Mathematical Sciences,Xiamen University,361005 Xiamen,China
第一作者单位	北师香港浸会大学
推荐引用方式 GB/T 7714	Li，Xianjuan,Xu，Chuanju. Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation[J]. Communications in Computational Physics, 2010, 8(5): 1016-1051.
APA	Li，Xianjuan, & Xu，Chuanju. (2010). Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Communications in Computational Physics, 8(5), 1016-1051.
MLA	Li，Xianjuan,et al."Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation". Communications in Computational Physics 8.5(2010): 1016-1051.