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A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative

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  • Heydari, M.H.
  • Atangana, A.

Abstract

This paper is concerned with an operational matrix method based on the shifted Legendre cardinal functions for solving the nonlinear variable-order time fractional Schrödinger equation. The variable-order fractional derivative operator is defined in the Atangana–Baleanu–Caputo sense. Through the way, a new operational matrix of variable-order fractional derivative is derived for the shifted Legendre cardinal functions and used in the established method. More precisely, the unknown solution is separated into the real and imaginary parts, and then these parts are expanded in terms of the shifted Legendre cardinal functions with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain an approximate solution for the problem. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm that the suggested approach is high accurate in providing satisfactory results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:128:y:2019:i:c:p:339-348
    DOI: 10.1016/j.chaos.2019.08.009
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    References listed on IDEAS

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    1. Heydari, Mohammad Hossein & Avazzadeh, Zakieh & Yang, Yin, 2019. "A computational method for solving variable-order fractional nonlinear diffusion-wave equation," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 235-248.
    2. Sweilam, N.H. & Abou Hasan, M.M., 2017. "Numerical solutions for 2-D fractional Schrödinger equation with the Riesz–Feller derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 140(C), pages 53-68.
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    5. 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.
    6. Heydari, Mohammad Hossein & Avazzadeh, Zakieh & Haromi, Malih Farzi, 2019. "A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 215-228.
    7. Hosseininia, M. & Heydari, M.H., 2019. "Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 400-407.
    8. Heydari, Mohammad Hossein & Avazzadeh, Zakieh, 2018. "Legendre wavelets optimization method for variable-order fractional Poisson equation," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 180-190.
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