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Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel

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  • Hosseininia, M.
  • Heydari, M.H.

Abstract

In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of nonlinear variable-order (V-O) time fractional 2D reaction-diffusion equations. The V-O time fractional derivative is defined in the Caputo sense with Mittag-Leffler non-singular kernel of order α(x,t)∈(0,1) (known as the Atangana–Baleanu–Caputo fractional derivative). First, the V-O fractional derivative is approximated via the finite difference scheme and the theta-weighted method is utilized to derive a recursive algorithm. Then, the unknown solution of the intended problem is expanded via the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the solution of the problem is reduced to solution of a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the applicability, validity and accuracy of the presented wavelet method, some numerical test examples are solved. The achieved numerical results reveal that the established method is highly accurate in solving the introduced new V-O fractional model.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:127:y:2019:i:c:p:400-407
    DOI: 10.1016/j.chaos.2019.07.017
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    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. 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.
    3. 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).
    4. Mostafa M. A. Khater & Aliaa Mahfooz Alabdali, 2021. "Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation," Mathematics, MDPI, vol. 9(12), pages 1-13, June.
    5. Ullah, Malik Zaka & Mallawi, Fouad & Baleanu, Dumitru & Alshomrani, Ali Saleh, 2020. "A new fractional study on the chaotic vibration and state-feedback control of a nonlinear suspension system," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    6. 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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