Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Yu, Qiang, Liu, Fawang, Turner, Ian, & Burrage, Kevin (2011) Analytical and numerical solutions of the space and time fractional bloch-torrey equation. In Jalili, Nader (Ed.) Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Elsevier, Chicago, Illinois, pp. 1-10.

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Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are 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 (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). 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) 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 with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

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ID Code: 52821
Item Type: Conference Paper
Refereed: Yes
Keywords: Brownian motion, Gaussian probability function
DOI: 10.1115/DETC2011-47613
ISBN: 9780791854808
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000)
Divisions: Current > QUT Faculties and Divisions > Science & Engineering Faculty
Copyright Owner: Copyright 2012 ASME
Deposited On: 07 Aug 2012 23:54
Last Modified: 16 Jul 2017 12:02

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