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Beyond monofractional kinetics

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  • Sandev, Trifce
  • Sokolov, Igor M.
  • Metzler, Ralf
  • Chechkin, Aleksei

Abstract

We discuss generalized integro-differential diffusion equations whose integral kernels are not of a simple power law form, and thus these equations themselves do not belong to the family of fractional diffusion equations exhibiting a monoscaling behavior. They instead generate a broad class of anomalous nonscaling patterns, which correspond either to crossovers between different power laws, or to a non-power-law behavior as exemplified by the logarithmic growth of the width of the distribution. We consider normal and modified forms of these generalized diffusion equations and provide a brief discussion of three generic types of integral kernels for each form, namely, distributed order, truncated power law and truncated distributed order kernels. For each of the cases considered we prove the non-negativity of the solution of the corresponding generalized diffusion equation and calculate the mean squared displacement.

Suggested Citation

  • Sandev, Trifce & Sokolov, Igor M. & Metzler, Ralf & Chechkin, Aleksei, 2017. "Beyond monofractional kinetics," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 210-217.
    • Handle: RePEc:eee:chsofr:v:102:y:2017:i:c:p:210-217
    DOI: 10.1016/j.chaos.2017.05.001
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    References listed on IDEAS

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    1. Sokolov, I.M & Chechkin, A.V & Klafter, J, 2004. "Fractional diffusion equation for a power-law-truncated Lévy process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 245-251.
    2. 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.
    3. Liemert, André & Sandev, Trifce & Kantz, Holger, 2017. "Generalized Langevin equation with tempered memory kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 356-369.
    4. Saxena, Ram K. & Pagnini, Gianni, 2011. "Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(4), pages 602-613.
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    Cited by:

    1. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    2. Wang, Zhaoyang & Lin, Ping & Wang, Erhui, 2021. "Modeling multiple anomalous diffusion behaviors on comb-like structures," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    3. Trifce Sandev, 2017. "Generalized Langevin Equation and the Prabhakar Derivative," Mathematics, MDPI, vol. 5(4), pages 1-11, November.

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