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

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 ? 2 ? ? + ? + h x 2 + h y 2 + h z 2 $O\left (\tau ^{2-\alpha }+\tau +h_{x}^{2}+h_{y}^{2}+h_{z}^{2}\right )$ . Finally, some numerical results are presented to support our theoretical analysis.