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Time fractional derivative model with Mittag-Leffler function kernel for describing anomalous diffusion: Analytical solution in bounded-domain and model comparison

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  • Yu, Xiangnan
  • Zhang, Yong
  • Sun, HongGuang
  • Zheng, Chunmiao

Abstract

Non-Fickian or anomalous diffusion had been well documented in material transport through heterogeneous systems at all scales, whose dynamics can be quantified by the time fractional derivative equations (fDEs). While analytical or numerical solutions have been developed for the standard time fDE in bounded domains, the standard time fDE suffers from the singularity issue due to its power-law function kernel. This study aimed at deriving the analytical solutions for the time fDE models with a modified kernel in bounded domains. The Mittag-Leffler function was selected as the alternate kernel to improve the standard power-law function in defining the time fractional derivative, which was known to be able to overcome the singularity issue of the standard fractional derivative. Results showed that the method of variable separation can be applied to derive the analytical solution for various time fDEs with absorbing and/or reflecting boundary conditions. Finally, numerical examples with detailed comparison for fDEs with different kernels showed that the models and solutions obtained by this study can capture anomalous diffusion in bounded domains.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:115:y:2018:i:c:p:306-312
    DOI: 10.1016/j.chaos.2018.08.026
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    References listed on IDEAS

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    1. 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.
    2. Chang, Ailian & Sun, HongGuang & Zheng, Chunmiao & Lu, Bingqing & Lu, Chengpeng & Ma, Rui & Zhang, Yong, 2018. "A time fractional convection–diffusion equation to model gas transport through heterogeneous soil and gas reservoirs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 356-369.
    3. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
    4. Wu, Guo-Cheng & Baleanu, Dumitru & Luo, Wei-Hua, 2017. "Lyapunov functions for Riemann–Liouville-like fractional difference equations," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 228-236.
    5. 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.
    6. Metzler, Ralf & Klafter, Joseph, 2000. "Boundary value problems for fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(1), pages 107-125.
    7. 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.
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    Cited by:

    1. Guo, Liujie & Gao, Fei & Zhan, Hui, 2022. "Existence, uniqueness and L∞-bound for weak solutions of a time fractional Keller-Segel system," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    2. Wei, Q. & Yang, S. & Zhou, H.W. & Zhang, S.Q. & Li, X.N. & Hou, W., 2021. "Fractional diffusion models for radionuclide anomalous transport in geological repository systems," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    3. Qiu, Lin & Chen, Wen & Wang, Fajie & Lin, Ji, 2019. "A non-local structural derivative model for memristor," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 169-177.
    4. dos Santos, Maike A.F., 2019. "Analytic approaches of the anomalous diffusion: A review," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 86-96.
    5. Zhang, Yong & Yu, Xiangnan & Sun, HongGuang & Tick, Geoffrey R. & Wei, Wei & Jin, Bin, 2020. "Applicability of time fractional derivative models for simulating the dynamics and mitigation scenarios of COVID-19," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    6. Zheng, Bibo & Wang, Zhanshan, 2022. "Mittag-Leffler synchronization of fractional-order coupled neural networks with mixed delays," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    7. 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.

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