A finite volume method for the two-dimensional time and space variable-order fractional Bloch-Torrey equation with variable coefficients on irregular domains

Abstract

A new generalised two-dimensional time and space variable-order fractional Bloch-Torrey equation is developed in this study. The variable-order Riesz fractional derivative and variable diffusion coefficient are introduced to simulate diffusion phenomena in heterogeneous, irregularly shaped biological tissues. The fractional Bloch-Torrey equation is discretised by the weighted and shifted Grünwald-Letnikov formula with respect to time and by finite volume method with respect to space. Additionally, to improve the accuracy of the numerical method for dealing with non-smooth solutions, some appropriate correction terms are introduced in the time approximation. Numerical examples on different irregular domains with various non-smooth solutions are explored to verify the effectiveness of the presented numerical scheme. Furthermore, we also solve the coupled variable-order fractional Bloch-Torrey equation on a human brain-like domain which is composed of white matter and grey matter. The solution behaviour of this model is compared with that of the constant-order fractional model, and the transverse magnetisation in magnetic resonance imaging on different biological micro-environments are graphically analysed. Results suggest that incorporation of the non-local property and spatial heterogeneity in the model by use of fractional operators can lead to a better capability for capturing the complexities of diffusion phenomena in biological tissues. This research may provide a basis for further research on the application of fractional calculus to clinical research and medical imaging.