Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients
This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potentia...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1260 |
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| author | Elsayed I. Mahmoud Viktor N. Orlov |
| author_facet | Elsayed I. Mahmoud Viktor N. Orlov |
| author_sort | Elsayed I. Mahmoud |
| collection | DOAJ |
| description | This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative and the Riemann–Liouville derivative are considered in the temporal and spatial directions, respectively. The Riemann–Liouville derivative is approximated by the standard Grünwald approximation and the shifted Grünwald approximation. The stability and convergence of the numerical scheme are discussed. Finally, we provide a numerical example to test the theoretical analysis. |
| first_indexed | 2024-03-10T10:51:24Z |
| format | Article |
| id | doaj.art-183056472b58475b91faa68c5dd0058b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:51:24Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-183056472b58475b91faa68c5dd0058b2023-11-21T22:12:47ZengMDPI AGMathematics2227-73902021-05-01911126010.3390/math9111260Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable CoefficientsElsayed I. Mahmoud0Viktor N. Orlov1Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, EgyptDepartment of Applied Math, Moscow State University of Civil Engineering, Yaroslavskoe Shosse, 26, 129337 Moscow, RussiaThis paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative and the Riemann–Liouville derivative are considered in the temporal and spatial directions, respectively. The Riemann–Liouville derivative is approximated by the standard Grünwald approximation and the shifted Grünwald approximation. The stability and convergence of the numerical scheme are discussed. Finally, we provide a numerical example to test the theoretical analysis.https://www.mdpi.com/2227-7390/9/11/1260two-dimensional time–space fractional Fokker–Planck equationstandard and shifted Grünwald approximationRiemann–Liouville fractional derivativeCaputo fractional derivativeimplicit finite difference schemestability and convergence |
| spellingShingle | Elsayed I. Mahmoud Viktor N. Orlov Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients Mathematics two-dimensional time–space fractional Fokker–Planck equation standard and shifted Grünwald approximation Riemann–Liouville fractional derivative Caputo fractional derivative implicit finite difference scheme stability and convergence |
| title | Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients |
| title_full | Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients |
| title_fullStr | Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients |
| title_full_unstemmed | Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients |
| title_short | Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients |
| title_sort | numerical solution of two dimensional time space fractional fokker planck equation with variable coefficients |
| topic | two-dimensional time–space fractional Fokker–Planck equation standard and shifted Grünwald approximation Riemann–Liouville fractional derivative Caputo fractional derivative implicit finite difference scheme stability and convergence |
| url | https://www.mdpi.com/2227-7390/9/11/1260 |
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