High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates

A.S. Hendy; R.H. De Staelen; A.A. Aldraiweesh; M.A. Zaky; A.S. Hendy; R.H. De Staelen; A.A. Aldraiweesh; M.A. Zaky

doi:10.3934/math.20231160

AIMS Mathematics

2023, Volume 8, Issue 10: 22766-22788. doi: 10.3934/math.20231160

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Research article

High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates

1.
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, Egypt
3.
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, Belgium
5.
Educational Technology Department, College of Education, King Saud University, Riyadh, Saudi Arabia
6.
Department of Applied Mathematics, National Research Centre, Dokki, Cairo 12622, Egypt

Received: 12 May 2023 Revised: 19 June 2023 Accepted: 25 June 2023 Published: 18 July 2023
MSC : 78M22, 65M06, 34K37

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.
Citation: 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[J]. AIMS Mathematics, 2023, 8(10): 22766-22788. doi: 10.3934/math.20231160

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Abstract

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.

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