IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i2p183-d482346.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/2/183/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/9/2/183/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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).
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Saffarian, Marziyeh & Mohebbi, Akbar, 2021. "Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    2. Sun, Fengxin & Wang, Jufeng & Xu, Ying, 2024. "An improved stabilized element-free Galerkin method for solving steady Stokes flow problems," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    3. Hong Lu & Linlin Wang & Mingji Zhang, 2022. "Dynamics of Fractional Stochastic Ginzburg–Landau Equation Driven by Nonlinear Noise," Mathematics, MDPI, vol. 10(23), pages 1-36, November.
    4. El-Ajou, Ahmad & Abu Arqub, Omar & Al-Smadi, Mohammed, 2015. "A general form of the generalized Taylor’s formula with some applications," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 851-859.
    5. Pavlos, G.P. & Karakatsanis, L.P. & Iliopoulos, A.C. & Pavlos, E.G. & Xenakis, M.N. & Clark, Peter & Duke, Jamie & Monos, D.S., 2015. "Measuring complexity, nonextensivity and chaos in the DNA sequence of the Major Histocompatibility Complex," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 188-209.
    6. Ivars, Salim B. & Botey, Muriel & Herrero, Ramon & Staliunas, Kestutis, 2023. "Stabilisation of spatially periodic states by non-Hermitian potentials," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    7. Qiu, Yunli & Malomed, Boris A. & Mihalache, Dumitru & Zhu, Xing & Zhang, Li & He, Yingji, 2020. "Soliton dynamics in a fractional complex Ginzburg-Landau model," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    8. Korabel, Nickolay & Zaslavsky, George M., 2007. "Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(2), pages 223-237.
    9. Heydari, M.H. & Razzaghi, M., 2023. "Piecewise fractional Chebyshev cardinal functions: Application for time fractional Ginzburg–Landau equation with a non-smooth solution," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    10. Tarasov, Vasily E. & Zaslavsky, George M., 2007. "Fractional dynamics of systems with long-range space interaction and temporal memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 383(2), pages 291-308.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:183-:d:482346. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.