scholarly journals Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space‐fractional diffusion equations in two space dimensions

2020 ◽  
Vol 37 (1) ◽  
pp. 818-835
Author(s):  
Huan Liu ◽  
Xiangcheng Zheng ◽  
Hongfei Fu ◽  
Hong Wang
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Diffusion Equations ◽  
Alternating Direction Implicit ◽  
Fractional Diffusion Equations ◽  
Alternating Direction ◽  
Two Space Dimensions ◽  
Volume Method

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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