Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Abstract

Abstract:Recently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain.In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator. Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions.PACS (2008)

Tissue microstructure features derived from anomalous diffusion measurements in magnetic resonance imaging

“…These methods have not been evaluated and validated in detail, which is likely due to a lack of in‐depth problem formulation and associated numerical methods of solution. Recently, fractional order calculus was used to investigate the link between fractional order dynamics and diffusion by solving the space and time fractional Bloch‐Torrey equation (ST‐FBTE) [Magin et al, ; Yu et al, ; Yu et al, ; Yu et al, ; Zhou et al, ] where , i is the complex identity, , is the gyromagnetic ratio, G(t) and D are the magnetic field gradient and the diffusion coefficient, respectively. is the Caputo time fractional derivative of order , is a sequential Riesz fractional order operator in space [Kilbas et al, ].…”

Finite difference scheme for time‐space fractional diffusion equation with fractional boundary conditions

“…Recently, fractional differential equations are widely used in physics, geology, finance, and so on, as they can provide an adequate and accurate description of the models, which cannot be modeled properly by integer‐order differential equations. Time‐space fractional diffusion equation has been successfully applied to analyze diffusion images of human brain tissues and provide new insights into further investigations of tissue structures and the microenvironment . And a considerable number of numerical methods have been developed for time‐space fractional diffusion equations.…”

Finite element approximation of fractional order elliptic boundary value problems

“…Nowadays, higher order approximations [13], [14], ADI methods [15] are available for the corresponding numerical simulations. Moreover, recently, several kinds of linear [16], [17] and non-linear problems are studied containing fractional order Laplacian operators [18], [19].…”

“…This is called the matrix transformation or matrix transfer method and was originally proposed for finite difference approximations of fractional order diffusion problems in [29] and [30]. Its usefulness has been verified experimentally in [31], [32], [17]. Additionally, in these works efficient methods are developed to compute (or approximate) the corresponding matrix powers.…”