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Stability analysis and error estimates of local discontinuous Galerkin method for nonlinear fractional Ginzburg–Landau equation with the fractional Laplacian

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Abstract

In this paper, we present and analyze local discontinuous Galerkin (LDG) method to solve nonlinear fractional Ginzburg–Landau equation (FGLE) with the fractional Laplacian. This method transforms the nonlinear FGLE with fractional Laplacian of order \(\sigma \) into a system of first-order equations and approximates the solution of the equation by selecting the appropriate basis functions. The stability analysis for FGLE has been investigated, and the optimal convergence rates \(O(h^{N+1})\) of the semi-discrete scheme have been established. Finally, numerical examples are displayed to verify the theoretical results.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors would like to express special thanks to anonymous referees for their valuable comments and suggestions, which significantly improved the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Qassim, Saudi Arabia

    Tarek Aboelenen

  2. Department of Mathematics, College of Science and Arts, Qassim University, Ar Rass, Saudi Arabia

    Mohammed Alqawba

  3. Department of Mathematics, Assiut University, Assiut, 71516, Egypt

    Tarek Aboelenen

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  1. Tarek Aboelenen
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Correspondence to Tarek Aboelenen.

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Aboelenen, T., Alqawba, M. Stability analysis and error estimates of local discontinuous Galerkin method for nonlinear fractional Ginzburg–Landau equation with the fractional Laplacian. Eur. Phys. J. Spec. Top. 232, 2607–2617 (2023). https://doi.org/10.1140/epjs/s11734-023-00921-6

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