An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations
A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. Thi...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/4/4/57 |
| _version_ | 1797544757510012928 |
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| author | Moreno Concezzi Renato Spigler |
| author_facet | Moreno Concezzi Renato Spigler |
| author_sort | Moreno Concezzi |
| collection | DOAJ |
| description | A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the <i>PageRank</i> algorithm. Some numerical examples are given. |
| first_indexed | 2024-03-10T14:05:56Z |
| format | Article |
| id | doaj.art-7c0702192f45446bb385f83b511f69eb |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T14:05:56Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-7c0702192f45446bb385f83b511f69eb2023-11-21T00:42:45ZengMDPI AGFractal and Fractional2504-31102020-12-01445710.3390/fractalfract4040057An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion EquationsMoreno Concezzi0Renato Spigler1Department of Mathematics and Physics, Roma Tre University, 1, Largo S. Leonardo Murialdo, 00146 Rome, ItalyDepartment of Mathematics and Physics, Roma Tre University, 1, Largo S. Leonardo Murialdo, 00146 Rome, ItalyA numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the <i>PageRank</i> algorithm. Some numerical examples are given.https://www.mdpi.com/2504-3110/4/4/573D anomalous diffusion3D fractional diffusion3D fractional reaction-diffusionCaputo derivativeAlternating Direction Implicit (ADI) methodsextrapolation techniques |
| spellingShingle | Moreno Concezzi Renato Spigler An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations Fractal and Fractional 3D anomalous diffusion 3D fractional diffusion 3D fractional reaction-diffusion Caputo derivative Alternating Direction Implicit (ADI) methods extrapolation techniques |
| title | An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations |
| title_full | An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations |
| title_fullStr | An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations |
| title_full_unstemmed | An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations |
| title_short | An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations |
| title_sort | adi method for the numerical solution of 3d fractional reaction diffusion equations |
| topic | 3D anomalous diffusion 3D fractional diffusion 3D fractional reaction-diffusion Caputo derivative Alternating Direction Implicit (ADI) methods extrapolation techniques |
| url | https://www.mdpi.com/2504-3110/4/4/57 |
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