Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative
In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated b...
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MDPI AG
2018-04-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/20/5/321 |
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| author | Sadia Arshad Dumitru Baleanu Jianfei Huang Maysaa Mohamed Al Qurashi Yifa Tang Yue Zhao |
| author_facet | Sadia Arshad Dumitru Baleanu Jianfei Huang Maysaa Mohamed Al Qurashi Yifa Tang Yue Zhao |
| author_sort | Sadia Arshad |
| collection | DOAJ |
| description | In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis. |
| first_indexed | 2024-12-10T07:18:54Z |
| format | Article |
| id | doaj.art-2ba3dfa001124fc4a808c8bcc5cdb548 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-12-10T07:18:54Z |
| publishDate | 2018-04-01 |
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| series | Entropy |
| spelling | doaj.art-2ba3dfa001124fc4a808c8bcc5cdb5482022-12-22T01:57:52ZengMDPI AGEntropy1099-43002018-04-0120532110.3390/e20050321e20050321Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz DerivativeSadia Arshad0Dumitru Baleanu1Jianfei Huang2Maysaa Mohamed Al Qurashi3Yifa Tang4Yue Zhao5The State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaDepartment of Mathematics, Cankaya University, Ankara 06530, TurkeyCollege of Mathematical Sciences, Yangzhou University, Yangzhou 225002, ChinaDepartment of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi ArabiaThe State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaThe State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaIn this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis.http://www.mdpi.com/1099-4300/20/5/321fractional advection dispersion equationriesz derivativecaputo derivativetrapezoidal formula |
| spellingShingle | Sadia Arshad Dumitru Baleanu Jianfei Huang Maysaa Mohamed Al Qurashi Yifa Tang Yue Zhao Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative Entropy fractional advection dispersion equation riesz derivative caputo derivative trapezoidal formula |
| title | Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative |
| title_full | Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative |
| title_fullStr | Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative |
| title_full_unstemmed | Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative |
| title_short | Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative |
| title_sort | finite difference method for time space fractional advection diffusion equations with riesz derivative |
| topic | fractional advection dispersion equation riesz derivative caputo derivative trapezoidal formula |
| url | http://www.mdpi.com/1099-4300/20/5/321 |
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