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Modeling diffusive transport with a fractional derivative without singular kernel

Author

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  • Gómez-Aguilar, J.F.
  • López-López, M.G.
  • Alvarado-Martínez, V.M.
  • Reyes-Reyes, J.
  • Adam-Medina, M.

Abstract

In this paper we present an alternative representation of the diffusion equation and the diffusion–advection equation using the fractional calculus approach, the spatial-time derivatives are approximated using the fractional definition recently introduced by Caputo and Fabrizio in the range β,γ∈(0;2] for the space and time domain respectively. In this representation two auxiliary parameters σx and σt are introduced, these parameters related to equation results in a fractal space–time geometry provide an entire new family of solutions for the diffusion processes. The numerical results showed different behaviors when compared with classical model solutions. In the range β,γ∈(0;1), the concentration exhibits the non-Markovian Lévy flights and the subdiffusion phenomena; when β=γ=1 the classical case is recovered; when β,γ∈(1;2] the concentration exhibits the Markovian Lévy flights and the superdiffusion phenomena; finally when β=γ=2 the concentration is anomalous dispersive and we found ballistic diffusion.

Suggested Citation

  • Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Reyes-Reyes, J. & Adam-Medina, M., 2016. "Modeling diffusive transport with a fractional derivative without singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 467-481.
    • Handle: RePEc:eee:phsmap:v:447:y:2016:i:c:p:467-481
    DOI: 10.1016/j.physa.2015.12.066
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    Citations

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    Cited by:

    1. Saad, Khaled M. & Gómez-Aguilar, J.F., 2018. "Analysis of reaction–diffusion system via a new fractional derivative with non-singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 703-716.
    2. Owolabi, Kolade M., 2017. "Mathematical modelling and analysis of two-component system with Caputo fractional derivative order," Chaos, Solitons & Fractals, Elsevier, vol. 103(C), pages 544-554.
    3. Owolabi, Kolade M. & Atangana, Abdon, 2017. "Numerical approximation of nonlinear fractional parabolic differential equations with Caputo–Fabrizio derivative in Riemann–Liouville sense," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 171-179.
    4. Abdulhameed, M. & Vieru, D. & Roslan, R., 2017. "Modeling electro-magneto-hydrodynamic thermo-fluidic transport of biofluids with new trend of fractional derivative without singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 233-252.
    5. Yu, Xiangnan & Zhang, Yong & Sun, HongGuang & Zheng, Chunmiao, 2018. "Time fractional derivative model with Mittag-Leffler function kernel for describing anomalous diffusion: Analytical solution in bounded-domain and model comparison," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 306-312.
    6. Al-khedhairi, A. & Elsadany, A.A. & Elsonbaty, A., 2019. "Modelling immune systems based on Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 25-39.
    7. Sun, HongGuang & Hao, Xiaoxiao & Zhang, Yong & Baleanu, Dumitru, 2017. "Relaxation and diffusion models with non-singular kernels," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 590-596.
    8. Gómez-Aguilar, J.F., 2017. "Irving–Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 179-186.
    9. Owolabi, Kolade M. & Atangana, Abdon, 2018. "Chaotic behaviour in system of noninteger-order ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 362-370.
    10. Costa, F.S. & Oliveira, D.S. & Rodrigues, F.G. & de Oliveira, E.C., 2019. "The fractional space–time radial diffusion equation in terms of the Fox’s H-function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 403-418.
    11. Shuai Yang & Qing Wei & Lu An, 2022. "Fractional Advection Diffusion Models for Radionuclide Migration in Multiple Barriers System of Deep Geological Repository," Mathematics, MDPI, vol. 10(14), pages 1-7, July.
    12. Owolabi, Kolade M. & Atangana, Abdon, 2017. "Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 111-119.
    13. Liu, Zhengguang & Cheng, Aijie & Li, Xiaoli, 2017. "A second order Crank–Nicolson scheme for fractional Cattaneo equation based on new fractional derivative," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 361-374.
    14. Habibirad, Ali & Azin, Hadis & Hesameddini, Esmail, 2023. "A capable numerical meshless scheme for solving distributed order time-fractional reaction–diffusion equation," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    15. Owolabi, Kolade M., 2018. "Analysis and numerical simulation of multicomponent system with Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 127-134.
    16. Al-Refai, Mohammed & Jarrah, Abdulla M., 2019. "Fundamental results on weighted Caputo–Fabrizio fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 7-11.

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