A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation

Song, J., Yu, Q., Liu, F. and Turner, I. (2014) A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation. Numerical Algorithms, 66 4: 911-932. doi:10.1007/s11075-013-9768-x

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Author Song, J.
Yu, Q.
Liu, F.
Turner, I.
Title A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation
Journal name Numerical Algorithms   Check publisher's open access policy
ISSN 1017-1398
1572-9265
Publication date 2014-08-01
Year available 2013
Sub-type Article (original research)
DOI 10.1007/s11075-013-9768-x
Open Access Status DOI
Volume 66
Issue 4
Start page 911
End page 932
Total pages 22
Place of publication New York, United States
Publisher Springer
Language eng
Formatted abstract
In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from Fractional Calculus. Recently, a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (Magin et al., J. Magn. Reson. 190(2), 255–270, 2008), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE with a nonlinear source term on a finite domain in three-dimensions. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a spatially second-order accurate implicit numerical method (INM) for the ST-FBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is uniquely solvable, unconditionally stable and convergent, and the order of convergence of the implicit numerical method is O(τ2−α+τ+h2x+h2y+h2z) . Finally, some numerical results are presented to support our theoretical analysis.
Keyword Fractional Bloch-Torrey equation
Fractional calculus
Implicit numerical method
Fractional centered difference
Q-Index Code C1
Q-Index Status Provisional Code
Grant ID DP120103770
Institutional Status Non-UQ
Additional Notes Published online 17 September 2013

Document type: Journal Article
Sub-type: Article (original research)
Collections: Non HERDC
Centre for Advanced Imaging Publications
 
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Citation counts: TR Web of Science Citation Count  Cited 14 times in Thomson Reuters Web of Science Article | Citations
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Created: Thu, 02 Oct 2014, 23:44:55 EST by Shona Osborne on behalf of Centre for Advanced Imaging