High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates
A coupled system of fractional order Gross-Pitaevskii equations is under consideration in which the time-fractional derivative is given in Caputo sense and the spatial fractional order derivative is of Riesz type. This kind of model may shed light on some time-evolution properties of the rotating tw...
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AIMS Press
2023-07-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231160?viewType=HTML |
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| author | A.S. Hendy R.H. De Staelen A.A. Aldraiweesh M.A. Zaky |
| author_facet | A.S. Hendy R.H. De Staelen A.A. Aldraiweesh M.A. Zaky |
| author_sort | A.S. Hendy |
| collection | DOAJ |
| description | A coupled system of fractional order Gross-Pitaevskii equations is under consideration in which the time-fractional derivative is given in Caputo sense and the spatial fractional order derivative is of Riesz type. This kind of model may shed light on some time-evolution properties of the rotating two-component Bose¢ Einstein condensates. An unconditional convergent high-order scheme is proposed based on L2-$ 1_{\sigma} $ finite difference approximation in the time direction and Galerkin Legendre spectral approximation in the space direction. This combined scheme is designed in an easy algorithmic style. Based on ideas of discrete fractional Grönwall inequalities, we can prove the convergence theory of the scheme. Accordingly, a second order of convergence and a spectral convergence order in time and space, respectively, without any constraints on temporal meshes and the specified degree of Legendre polynomials $ N $. Some numerical experiments are proposed to support the theoretical results. |
| first_indexed | 2024-03-12T20:52:37Z |
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| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-12T20:52:37Z |
| publishDate | 2023-07-01 |
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| series | AIMS Mathematics |
| spelling | doaj.art-dab17cd4ad304b8fb4e2ef5c0b9f7e2b2023-08-01T01:32:14ZengAIMS PressAIMS Mathematics2473-69882023-07-01810227662278810.3934/math.20231160High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensatesA.S. Hendy 0R.H. De Staelen 1https://orcid.org/0000-0001-9193-2011A.A. Aldraiweesh 2M.A. Zaky31. Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia 2. Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt3. Beheer en Algemene Directie, Ghent University Hospital, Corneel Heymanslaan 10, B-9000 Ghent, Belgium 4. Research Department, Ghent University, Sint-Pietersnieuwstraat 25, B-9000 Ghent, Belgium5. Educational Technology Department, College of Education, King Saud University, Riyadh, Saudi Arabia5. Educational Technology Department, College of Education, King Saud University, Riyadh, Saudi Arabia 6. Department of Applied Mathematics, National Research Centre, Dokki, Cairo 12622, EgyptA coupled system of fractional order Gross-Pitaevskii equations is under consideration in which the time-fractional derivative is given in Caputo sense and the spatial fractional order derivative is of Riesz type. This kind of model may shed light on some time-evolution properties of the rotating two-component Bose¢ Einstein condensates. An unconditional convergent high-order scheme is proposed based on L2-$ 1_{\sigma} $ finite difference approximation in the time direction and Galerkin Legendre spectral approximation in the space direction. This combined scheme is designed in an easy algorithmic style. Based on ideas of discrete fractional Grönwall inequalities, we can prove the convergence theory of the scheme. Accordingly, a second order of convergence and a spectral convergence order in time and space, respectively, without any constraints on temporal meshes and the specified degree of Legendre polynomials $ N $. Some numerical experiments are proposed to support the theoretical results. https://www.aimspress.com/article/doi/10.3934/math.20231160?viewType=HTMLgalerkin-legendre spectral methodl2-$ 1_{\sigma} $ schemetime-space fractional coupled gross¢pitaevskii equationconvergence analysis |
| spellingShingle | A.S. Hendy R.H. De Staelen A.A. Aldraiweesh M.A. Zaky High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates AIMS Mathematics galerkin-legendre spectral method l2-$ 1_{\sigma} $ scheme time-space fractional coupled gross¢pitaevskii equation convergence analysis |
| title | High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates |
| title_full | High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates |
| title_fullStr | High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates |
| title_full_unstemmed | High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates |
| title_short | High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates |
| title_sort | high order approximation scheme for a fractional order coupled system describing the dynamics of rotating two component bose einstein condensates |
| topic | galerkin-legendre spectral method l2-$ 1_{\sigma} $ scheme time-space fractional coupled gross¢pitaevskii equation convergence analysis |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231160?viewType=HTML |
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