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

Author

Listed:
  • Mahmoud A. Zaky

    (Department of Applied Mathematics, National Research Centre, Dokki, Cairo 12622, Egypt)

  • Ahmed S. Hendy

    (Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., 620002 Yekaterinburg, Russia
    Department of Mathematics, Faculty of Science Benha University, Benha 13511, Egypt)

  • Rob H. De Staelen

    (Dean’s Office of the Faculty of Medicine and Health Sciences, Ghent University, C. Heymanslaan 10, 9000 Gent, Belgium
    Beheer en Algemene Directie, Ghent University Hospital, C. Heymanslaan 10, 9000 Gent, Belgium)

Abstract

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L 2 - 1 σ 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.

Suggested Citation

  • Mahmoud A. Zaky & Ahmed S. Hendy & Rob H. De Staelen, 2021. "Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System," Mathematics, MDPI, vol. 9(2), pages 1-22, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:183-:d:482346
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    References listed on IDEAS

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    1. Abbaszadeh, Mostafa & Dehghan, Mehdi, 2021. "Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    2. Tarasov, Vasily E. & Zaslavsky, George M., 2005. "Fractional Ginzburg–Landau equation for fractal media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 354(C), pages 249-261.
    3. Liu, Haiyu & Lü, Shujuan, 2019. "Galerkin spectral method for nonlinear time fractional Cable equation with smooth and nonsmooth solutions," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 32-47.
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    Cited by:

    1. Omran, A.K. & Zaky, M.A. & Hendy, A.S. & Pimenov, V.G., 2022. "An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 218-239.
    2. Abbaszadeh, Mostafa & Zaky, Mahmoud A. & Hendy, Ahmed S. & Dehghan, Mehdi, 2024. "Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 77-103.

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