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Fractional vector calculus for fractional advection–dispersion

Author

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  • Meerschaert, Mark M.
  • Mortensen, Jeff
  • Wheatcraft, Stephen W.

Abstract

We develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient, divergence, and curl, and a fractional divergence theorem and Stokes theorem. These basic tools are then applied to provide a physical explanation for the fractional advection–dispersion equation for flow in heterogeneous porous media.

Suggested Citation

  • Meerschaert, Mark M. & Mortensen, Jeff & Wheatcraft, Stephen W., 2006. "Fractional vector calculus for fractional advection–dispersion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 181-190.
    • Handle: RePEc:eee:phsmap:v:367:y:2006:i:c:p:181-190
    DOI: 10.1016/j.physa.2005.11.015
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    References listed on IDEAS

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    1. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    2. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    3. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    4. Piryatinska, A. & Saichev, A.I. & Woyczynski, W.A., 2005. "Models of anomalous diffusion: the subdiffusive case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(3), pages 375-420.
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    Cited by:

    1. Vasily E. Tarasov, 2021. "General Fractional Vector Calculus," Mathematics, MDPI, vol. 9(21), pages 1-87, November.
    2. Cai, Rui-Yang & Cheng, Lan & Zhou, Hua-Cheng, 2024. "Sliding mode control for the stabilization of fractional heat equations subject to boundary uncertainty," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    3. Prodanov, Dimiter, 2016. "Characterization of strongly non-linear and singular functions by scale space analysis," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 14-19.
    4. Owolabi, Kolade M., 2016. "Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 89-98.
    5. Ochoa-Tapia, J. Alberto & Valdes-Parada, Francisco J. & Alvarez-Ramirez, Jose, 2007. "A fractional-order Darcy's law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(1), pages 1-14.
    6. Wang, Yiduan & Zheng, Shenzhou & Zhang, Wei & Wang, Guochao & Wang, Jun, 2018. "Fuzzy entropy complexity and multifractal behavior of statistical physics financial dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 486-498.
    7. Owolabi, Kolade M., 2020. "High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    8. Che, Han & Wang, Yu-Lan & Li, Zhi-Yuan, 2022. "Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 149-163.
    9. Adán J. Serna-Reyes & Jorge E. Macías-Díaz & Nuria Reguera, 2021. "A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate," Mathematics, MDPI, vol. 9(12), pages 1-22, June.
    10. Troparevsky, M.I. & Muszkats, J.P. & Seminara, S.A. & Zitto, M.E. & Piotrkowski, R., 2022. "Modeling particulate pollutants dispersed in the atmosphere using fractional turbulent diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 599(C).
    11. Hejazi, S. Reza & Saberi, Elaheh & Mohammadizadeh, Fatemeh, 2021. "Anisotropic non-linear time-fractional diffusion equation with a source term: Classification via Lie point symmetries, analytic solutions and numerical simulation," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    12. Bolster, Diogo & Benson, David A. & Meerschaert, Mark M. & Baeumer, Boris, 2013. "Mixing-driven equilibrium reactions in multidimensional fractional advection–dispersion systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(10), pages 2513-2525.

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