A fractional alternating-direction implicit method for a multi-term time–space fractional Bloch–Torrey equations in three dimensions

Abstract

The time–space fractional Bloch–Torrey equation (TS-FBTE) has been proposed to simulate anomalous diffusion in the human brain. However, the development of numerical methods and theoretical analysis for multi-term time–space fractional Bloch–Torrey equations in three dimensions is still limited. In this paper, we consider a class of 3-D multi-term time–space fractional Bloch–Torrey equations (3D-MTTS-FBTEs). Firstly, we adopt a fractional centred difference scheme to discretize the Riesz fractional derivative and propose a fractional alternating-direction implicit method (FADIM) for the model. Secondly, the solvability, stability and convergence of the method are investigated. Finally, numerical results are presented to support our theoretical analysis. In addition, to demonstrate the applicability of our method, we consider a coupled 3-D FBTE as an example to solve and exhibit the effects on the behaviour of the transverse magnetization.