A finite volume method for the two-dimensional time and space variable-order fractional Bloch-Torrey equation with variable coefficients on irregular domains. (15th September 2021)

• Record Type: Journal Article
• Authors: Zhang, Mengchen; Liu, Fawang; Turner, Ian W.; Anh, Vo V.; Feng, Libo
• Is Part Of: Computers & mathematics with applications. Volume 98(2021)
• Page Start: 81
• Page End: 98
• Publication Date: 2021-09-15
• Subjects: Variable-order fractional derivative -- Bloch-Torrey equation -- Finite volume method -- Variable coefficients -- Non-smooth solutions; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2021.06.013
• Languages: English
• ISSNs: 0898-1221

Abstract

Highlights: A new time-space variable-order fractional Bloch-Torrey equation is developed. Some appropriate correction terms are introduced to deal with non-smooth solutions. A finite volume method is employed for solving the fractional dynamical system. This fractional model can better capture the diffusion in heterogeneous media. 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.