Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation

Yu, Q., Liu, F., Turner, I. and Burrage, K. (2013) Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371 1990: . doi:10.1098/rsta.2012.0150

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Author Yu, Q.
Liu, F.
Turner, I.
Burrage, K.
Title Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation
Journal name Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences   Check publisher's open access policy
ISSN 1364-503X
1471-2962
Publication date 2013-05-13
Year available 2013
Sub-type Article (original research)
DOI 10.1098/rsta.2012.0150
Open Access Status File (Author Post-print)
Volume 371
Issue 1990
Total pages 19
Place of publication London, United Kingdom
Publisher The Royal Society Publishing
Language eng
Subject 2600 Mathematics
3100 Physics and Astronomy
2200 Engineering
Formatted abstract
Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here, the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments, where processes are often anisotropic. Anomalous diffusion in the human brain using fractional-order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional-order calculus with respect to time and space. However, effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) in both fractional Laplacian and Riesz derivative form is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated.We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.
Keyword Convergence
Implicit numerical method
Space and time fractional Bloch-Torrey equation
Stability
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: Non HERDC
Centre for Advanced Imaging Publications
 
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