Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

2015 ◽  
Vol 66 (3) ◽  
pp. 1281-1312 ◽  
Author(s):  
Guang-hua Gao ◽  
Zhi-zhong Sun
Keyword(s):  
Fractional Diffusion ◽  
Difference Schemes ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fractional Diffusion Equations ◽  
Alternating Direction ◽  
Implicit Difference Schemes ◽  
Distributed Order

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

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Fawang Liu ◽  
Shiping Chen ◽  
Ian Turner ◽  
Kevin Burrage ◽  
Vo Anh
Keyword(s):  
Fractional Diffusion ◽  
Riesz Space ◽  
Two Dimensional ◽  
Alternating Direction Implicit Method ◽  
Alternating Direction Implicit ◽  
Fractional Diffusion Equations ◽  
Reaction Term ◽  
Nonlinear Reaction ◽  
Alternating Direction ◽  
Space Fractional Diffusion Equation

AbstractFractional 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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