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Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method

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  • Saffarian, Marziyeh
  • Mohebbi, Akbar

Abstract

At the present work, we propose a new numerical scheme for the solution of two and three dimensional time fractional nonlinear damped Klein–Gordon equation (DKGE). To this end, we use the Legendre spectral element method to discretize the equation in the spatial directions and for the time stepping, an alternating direction implicit (ADI) method based on a scheme of order O(τ2) is considered. We prove that the presented method is unconditionally stable and obtain an error estimate for the fully discrete scheme. The presented numerical experiments demonstrate the good accuracy and efficiency of the proposed ADI spectral element method.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:405:y:2021:i:c:s0096300321002721
    DOI: 10.1016/j.amc.2021.126182
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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. Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    3. Zahra, W.K. & Nasr, M.A. & Van Daele, M., 2019. "Exponentially fitted methods for solving time fractional nonlinear reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 468-490.
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    Cited by:

    1. Qiao, Leijie & Qiu, Wenlin & Xu, Da, 2023. "Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 205-231.
    2. Behnood, Maryam & Shokri, Ali, 2022. "A Legendre spectral element method for the family of regularized long wave equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 239-253.

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