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Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

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Liu F., Chen S., Turner I., Burrage K., Anh V.

Open Physics

2013

vol. 11

issue 10

1221-1232

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

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1 Open Physics

1 2013

1 Anh V.

1 Burrage K.

1 Chen S.

1 Liu F.

1 Turner I.

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Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term