Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Yu, Qiang, Liu, Fawang, Turner, Ian, & Burrage, Kevin (2012) Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D. In Cheng, Wen, Sun, HongGuang, & Baleanu, Dumitru (Eds.) Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, Hohai University, Hohai University, Nanjing, China, pp. 1-8.

Abstract

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

Impact and interest:

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ID Code: 51437
Item Type: Conference Paper
Refereed: Yes
Keywords: Fractional Bloch-Torrey Equation, fractional calculus, implicit numerical method, alternating direction method, stability, convergence
Divisions: Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Deposited On: 04 Jul 2012 22:18
Last Modified: 12 Jun 2013 14:54

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