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Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative

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  • Sene, Ndolane
  • Abdelmalek, Karima

Abstract

In this paper, we analyze two types of diffusion processes obtained with the fractional diffusion equations described by the Atangana-Baleanu-Caputo (ABC) fractional derivative. The mean square displacement (MSD) concept has been used to discuss the types of diffusion processes obtained when the order of the fractional derivative take certain values. Many types of diffusion processes exist and depend to the value of the order of the used fractional derivatives: the fractional diffusion equation with the subdiffusive process, the fractional diffusion equation with the superdiffusive process, the fractional diffusion equation with the ballistic diffusive process and the fractional diffusion equation with the hyper diffusive process. Here we use the Atangana-Baleanu fractional derivative and analyze the subdiffusion process obtained when the order of ABC α is into (0,1) and the normal diffusion obtained in the limiting case α=1. The Laplace transform of the Atangana-Baleanu-Caputo fractional derivative has been used for getting the mean square displacement of the fractional diffusion equation. The central limit theorem has been discussed too, and the main results illustrated graphically.

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  • Sene, Ndolane & Abdelmalek, Karima, 2019. "Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 158-164.
    • Handle: RePEc:eee:chsofr:v:127:y:2019:i:c:p:158-164
    DOI: 10.1016/j.chaos.2019.06.036
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    References listed on IDEAS

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    Cited by:

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    2. dos Santos, Maike A.F., 2020. "Mittag-Leffler functions in superstatistics," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    3. Uroosa Arshad & Mariam Sultana & Ali Hasan Ali & Omar Bazighifan & Areej A. Al-moneef & Kamsing Nonlaopon, 2022. "Numerical Solutions of Fractional-Order Electrical RLC Circuit Equations via Three Numerical Techniques," Mathematics, MDPI, vol. 10(17), pages 1-16, August.
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    6. Maike A. F. dos Santos, 2019. "Mittag–Leffler Memory Kernel in Lévy Flights," Mathematics, MDPI, vol. 7(9), pages 1-13, August.

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