A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation

2020 ◽  
Vol 158 ◽  
pp. 271-291 ◽  
Author(s):  
Mostafa Abbaszadeh ◽  
Mehdi Dehghan
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Reduced Order ◽  
Alternating Direction ◽  
Space Fractional Diffusion Equation ◽  
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.

scholarly journals Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation

2020 ◽  
Vol 4 (3) ◽  
pp. 39
Author(s):  
Rafał Brociek ◽  
Agata Chmielowska ◽  
Damian Słota
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Parameter Identification ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Fractional Diffusion Equation ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Space Fractional Diffusion Equation ◽  
Swarm Intelligence Algorithm

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm’s accuracy.

scholarly journals Compact ADI method for two-dimensional Riesz space fractional diffusion equation

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1543-1554
Author(s):  
Sohrab Valizadeh ◽  
Alaeddin Malek ◽  
Abdollah Borhanifar
Keyword(s):  
Diffusion Equation ◽  
Maximum Error ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Fractional Diffusion Equation ◽  
Matrix Analysis ◽  
Two Dimensional ◽  
Adi Method ◽  
Space Fractional Diffusion Equation ◽  
The Matrix

In this paper, a compact alternating direction implicit (ADI) method has been developed for solving two-dimensional Riesz space fractional diffusion equation. The precision of the discretization method used in spatial directions is twice the order of the corresponding fractional derivatives. It is proved that the proposed method is unconditionally stable via the matrix analysis method and the maximum error in achieving convergence is discussed. Numerical example is considered aiming to demonstrate the validity and applicability of the proposed technique.

scholarly journals A new compact alternating direction implicit method for solving two dimensional time fractional diffusion equation with Caputo-Fabrizio derivative

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3609-3626
Author(s):  
Mehran Taghipour ◽  
Hossein Aminikhah
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
Theoretical Considerations

In this paper, a new compact alternating direction implicit (ADI) difference scheme is proposed for the solution of two dimensional time fractional diffusion equation. Theoretical considerations are discussed. We show that the proposed method is fourth order accurate in space and two order accurate in time. The stability and convergence of the compact ADI method are presented by the Fourier analysis method. Numerical examples confirm the theoretical results and high accuracy of the proposed scheme.

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