Alternating Direction Implicit Numerical Method for The Space and Time Fractional Bloch-Torrey Equation In 3-D
Recently, some authors have considered a new diffusion modelflipace and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., α = 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 e.ective 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 con.rm 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 e.cient. In this paper, we consider the numerical solution of a ST-FBTE on a .nite 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.
Fractional Bloch-Torrey Equation fractional calculus implicit numerical method alternating direction method stability convergence
Q. Yu F. Liu I. Turner K. Burrage
Mathematical Sciences, Queensland University of Technology,Brisbane, Australia Mathematical Sciences, Queensland University of Technology, Brisbane, Australia; COMLAB and OCISB, O
国际会议
The Fifth Symposium on Fractional Differentiation and Its Applications(第五届国际自动控制联合会分数阶导数及其应用会议)
南京
英文
1-8
2012-05-14（万方平台首次上网日期，不代表论文的发表时间）