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From continuous time random walks to the generalized diffusion equation

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Abstract

We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.

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Authors and Affiliations

  1. Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020, Skopje, Macedonia

    Trifce Sandev

  2. Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University in Skopje, P.O. Box 162, 1001, Skopje, Macedonia

    Trifce Sandev

  3. Research Center for Computer Science and Information Technologies Macedonian Academy of Sciences and Arts Bul, Krste Misirkov 2, 1000, Skopje, Macedonia

    Trifce Sandev

  4. Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-St. 24/25, D-14776, Potsdam-Golm, Germany

    Ralf Metzler & Aleksei Chechkin

  5. Akhiezer Institute for Theoretical Physics, Akademicheskaya St. 1, Kharkov, 61108, Ukraine

    Aleksei Chechkin

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  1. Trifce Sandev
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  2. Ralf Metzler
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  3. Aleksei Chechkin
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Correspondence to Trifce Sandev.

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Sandev, T., Metzler, R. & Chechkin, A. From continuous time random walks to the generalized diffusion equation. FCAA 21, 10–28 (2018). https://doi.org/10.1515/fca-2018-0002

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