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Advection and dispersion in time and space

Author

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  • Baeumer, B.
  • Benson, D.A.
  • Meerschaert, M.M.

Abstract

Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly.

Suggested Citation

  • Baeumer, B. & Benson, D.A. & Meerschaert, M.M., 2005. "Advection and dispersion in time and space," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 350(2), pages 245-262.
    • Handle: RePEc:eee:phsmap:v:350:y:2005:i:2:p:245-262
    DOI: 10.1016/j.physa.2004.11.008
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    Cited by:

    1. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2008. "Triangular array limits for continuous time random walks," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1606-1633, September.
    2. Wei, Song & Chen, Wen & Hon, Y.C., 2016. "Characterizing time dependent anomalous diffusion process: A survey on fractional derivative and nonlinear models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 1244-1251.
    3. Fernandez-Anaya, G. & Valdes-Parada, F.J. & Alvarez-Ramirez, J., 2011. "On generalized fractional Cattaneo’s equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4198-4202.
    4. Liwei Shi, 2018. "A Petrov-Galerkin Finite Element Method for Solving the Time-fractional Diffusion Equation with Interface," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 10(4), pages 136-150, August.
    5. Baeumer, B. & Meerschaert, M.M., 2007. "Fractional diffusion with two time scales," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 237-251.
    6. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2006. "Stochastic model for ultraslow diffusion," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1215-1235, September.

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