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Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel

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  • Hosseininia, M.
  • Heydari, M.H.

Abstract

This paper investigates a novel version for the nonlinear 2D telegraph equation involving variable-order (V-O) time fractional derivatives in the Atangana–Baleanu–Caputo sense with Mittag–Leffler non-singular kernel. A meshfree method based on the moving least squares (MLS) shape functions is proposed for the numerical solution of this class of problems. More precisely, the V-O fractional derivatives in this model are approximated by the finite difference scheme at first. Then, the θ-weighted method is utilized to derive a recursive algorithm. Next, the solution of the problem is expanded in terms of the MLS shape functions with undetermined coefficients. Eventually, by substituting this expansion and its partial derivatives into the original equation, solution of the problem in each time step is reduced to the solution of a linear system of algebraic equations. Several numerical examples are investigated to show the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the established method is high accurate in solving such V-O fractional models.

Suggested Citation

  • Hosseininia, M. & Heydari, M.H., 2019. "Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 389-399.
    • Handle: RePEc:eee:chsofr:v:127:y:2019:i:c:p:389-399
    DOI: 10.1016/j.chaos.2019.07.015
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    Cited by:

    1. Heydari, M.H., 2020. "Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana–Baleanu–Caputo variable-order fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    2. Ahmed, Hoda F. & Hashem, W.A., 2023. "A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 388-408.
    3. 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.
    4. Heydari, M.H. & Atangana, A., 2019. "A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 339-348.
    5. Abdelkawy, M.A. & Lopes, António M. & Babatin, Mohammed M., 2020. "Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    6. Fouladi, Somayeh & Dahaghin, Mohammad Shafi, 2022. "Numerical investigation of the variable-order fractional Sobolev equation with non-singular Mittag–Leffler kernel by finite difference and local discontinuous Galerkin methods," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    7. Pho, Kim-Hung & Heydari, M.H. & Tuan, Bui Anh & Mahmoudi, Mohammad Reza, 2020. "Numerical study of nonlinear 2D optimal control problems with multi-term variable-order fractional derivatives in the Atangana-Baleanu-Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    8. Heydari, M. H. & Atangana, A., 2020. "An optimization method based on the generalized Lucas polynomials for variable-order space-time fractional mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).

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