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Open AccessArticle

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

1
Department of Applied Mathematics, National Research Centre, Dokki, Cairo 12622, Egypt
2
Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., 620002 Yekaterinburg, Russia
3
Department of Mathematics, Faculty of Science Benha University, Benha 13511, Egypt
4
Dean’s Office of the Faculty of Medicine and Health Sciences, Ghent University, C. Heymanslaan 10, 9000 Gent, Belgium
5
Beheer en Algemene Directie, Ghent University Hospital, C. Heymanslaan 10, 9000 Gent, Belgium
*
Authors to whom correspondence should be addressed.
Mathematics 2021, 9(2), 183; https://doi.org/10.3390/math9020183
Received: 23 December 2020 / Revised: 13 January 2021 / Accepted: 15 January 2021 / Published: 18 January 2021
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims. View Full-Text
Keywords: generalized fractional coupled Ginzburg–Landau system; Alikhanov difference formula; Galerkin spectral scheme; discrete fractional Grönwall inequality generalized fractional coupled Ginzburg–Landau system; Alikhanov difference formula; Galerkin spectral scheme; discrete fractional Grönwall inequality
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MDPI and ACS Style

Zaky, M.A.; Hendy, A.S.; De Staelen, R.H. Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System. Mathematics 2021, 9, 183. https://doi.org/10.3390/math9020183

AMA Style

Zaky MA, Hendy AS, De Staelen RH. Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System. Mathematics. 2021; 9(2):183. https://doi.org/10.3390/math9020183

Chicago/Turabian Style

Zaky, Mahmoud A.; Hendy, Ahmed S.; De Staelen, Rob H. 2021. "Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System" Mathematics 9, no. 2: 183. https://doi.org/10.3390/math9020183

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