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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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    1. Owolabi, Kolade M. & Atangana, Abdon, 2018. "Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations," Chaos, Solitons & Fractals, Elsevier, vol. 111(C), pages 119-127.
    2. Abdeljawad, Thabet, 2018. "Different type kernel h−fractional differences and their fractional h−sums," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 146-156.
    3. Fall, Aliou Niang & Ndiaye, Seydou Nourou & Sene, Ndolane, 2019. "Black–Scholes option pricing equations described by the Caputo generalized fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 108-118.
    4. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    5. Sene, Ndolane, 2018. "Stokes’ first problem for heated flat plate with Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 68-75.
    6. O. Tasbozan & A. Esen & N. M. Yagmurlu & Y. Ucar, 2013. "A Numerical Solution to Fractional Diffusion Equation for Force-Free Case," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-6, May.
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    Cited by:

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