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

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Yu Q., Liu F., Turner I., Burrage K.

Open Physics

2013

vol. 11

issue 6

646-665

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.

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1 Open Physics

1 2013

1 Burrage K.

1 Liu F.

1 Turner I.

1 Yu Q.

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