Computational Multi-term Time-space Fractional Bloch-Torrey Models in Three-dimensions

Abstract

In this paper, we will develop new mathematical tools for mapping tissue microstructural properties via the use of space-time fractional calculus methods. We propose the following two new computational multi-term time-space fractional diffusion models in three dimensions, which can be used to simulate multi-term time-space fractional Bloch-Torrey models in three dimensions: Model I: Finite element method for the multi-term time-space fractional diffusion model with Riesz fractional operator; Model II: Finite difference method for the multi-term time-space fractional diffusion equation with fractional Laplacian operator. Firstly, the three-dimensional multi-term time-space fractional Bloch-Torrey models are decoupled; the problem is then equivalent to solving the three-dimensional time-space fractional diffusion equations (Model I and Model II). Secondly, we propose the finite element method for Model I and finite difference method for Model II, respectively. These methods can be directly used to simulate three-dimensional multi-term time-space fractional Bloch-Torrey models. Finally, some numerical examples are given to demonstrate the versatility and application of the models.