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A meshless method to solve nonlinear variable-order time fractional 2D reaction–diffusion equation involving Mittag-Leffler kernel

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Abstract

In this paper, an efficient and accurate meshless method based on the moving least squares (MLS) shape functions is developed to solve the generalized variable-order (V-O) time fractional nonlinear 2D reaction–diffusion equation. The V-O fractional derivative is considered in the Atangana–Baleanu–Caputo sense with Mittag-Leffler non-singular kernel. The numerical method is based on the following steps: First, the V-O fractional derivative is approximated by finite differences, and the \(\theta \)-weighted method has been used to derive a recursive algorithm. Then, the solution of the problem is expanded by the MLS shape functions. Finally, by a substitution of this series expansion and corresponding its partial derivatives into the main equation, the problem is reduced to a linear system of algebraic equations to be solved at each time step. Several numerical examples are also given to illustrate the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the proposed method is highly accurate in solving the introduced V-O fractional model.

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Authors and Affiliations

  1. Faculty of Mathematics, Yazd University, Yazd, Iran

    M. Hosseininia

  2. Department of Mathematics, Shiraz University of Technology, Shiraz, Iran

    M. H. Heydari

  3. Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran

    J. Rouzegar

  4. Engineering School (DEIM), University of Tuscia, Viterbo, Italy

    C. Cattani

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  1. M. Hosseininia
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  2. M. H. Heydari
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Correspondence to M. H. Heydari.

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Hosseininia, M., Heydari, M.H., Rouzegar, J. et al. A meshless method to solve nonlinear variable-order time fractional 2D reaction–diffusion equation involving Mittag-Leffler kernel. Engineering with Computers 37, 731–743 (2021). https://doi.org/10.1007/s00366-019-00852-8

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