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Fractional differential models for anomalous diffusion

Author

Listed:
  • Sun, HongGuang
  • Chen, Wen
  • Li, Changpin
  • Chen, YangQuan

Abstract

In this study, we investigate three kinds of fractional differential models (distributed-order model, variable-order model and random-order model) for characterization of anomalous diffusion. The characteristics, physical advantages and potential applications of each model are highlighted. The numerical simulations also validate our analytical and comparison results. Furthermore, a generalized distributed–variable-order model and a more generalized distributed–variable–random-order model are proposed to combine the advantages of distributed-order model, variable-order model and random-order model.

Suggested Citation

  • Sun, HongGuang & Chen, Wen & Li, Changpin & Chen, YangQuan, 2010. "Fractional differential models for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2719-2724.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:14:p:2719-2724
    DOI: 10.1016/j.physa.2010.02.030
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    Cited by:

    1. Kostrobij, P.P. & Markovych, B.M. & Viznovych, O.V. & Tokarchuk, M.V., 2019. "Generalized transport equation with nonlocality of space–time. Zubarev’s NSO method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 63-70.
    2. Ahmed, Hoda F. & Hashem, W.A., 2023. "A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 388-408.
    3. Solís-Pérez, J.E. & Gómez-Aguilar, J.F. & Atangana, A., 2018. "Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 175-185.
    4. Sun, HongGuang & Li, Zhipeng & Zhang, Yong & Chen, Wen, 2017. "Fractional and fractal derivative models for transient anomalous diffusion: Model comparison," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 346-353.
    5. Meng, Zhijun & Yi, Mingxu & Huang, Jun & Song, Lei, 2018. "Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 454-464.
    6. Saad, Khaled M. & Srivastava, H.M. & Gómez-Aguilar, J.F., 2020. "A Fractional Quadratic autocatalysis associated with chemical clock reactions involving linear inhibition," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    7. Nagy, A.M., 2017. "Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc–Chebyshev collocation method," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 139-148.
    8. Gómez-Aguilar, J.F., 2018. "Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 52-75.
    9. Ávalos-Ruiz, L.F. & Zúñiga-Aguilar, C.J. & Gómez-Aguilar, J.F. & Escobar-Jiménez, R.F. & Romero-Ugalde, H.M., 2018. "FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 177-189.
    10. Kumar, Kamlesh & Pandey, Rajesh K. & Yadav, Swati, 2019. "Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    11. Hashemi, M.S. & Inc, Mustafa & Yusuf, Abdullahi, 2020. "On three-dimensional variable order time fractional chaotic system with nonsingular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    12. Hayman Thabet & Subhash Kendre & Dimplekumar Chalishajar, 2017. "New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations," Mathematics, MDPI, vol. 5(4), pages 1-15, September.
    13. Tawfik, Ashraf M. & Fichtner, Horst & Elhanbaly, A. & Schlickeiser, Reinhard, 2018. "Analytical solution of the space–time fractional hyperdiffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 510(C), pages 178-187.

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