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The Stretched Exponential Behavior and Its Underlying Dynamics. The Phenomenological Approach

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Abstract

We show that the anomalous diffusion equations with a fractional spatial derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution is conceived as integral transform whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and for different initial conditions.

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Authors and Affiliations

  1. H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eljasza-Radzikowskiego 152, PL, 31342, Kraków, Poland

    Katarzyna Górska & Andrzej Horzela

  2. Sorbonne Universités, Université Pierre et Marie Curie (Paris VI) CNRS UMR 7600 Laboratoire de Physique Théorique de la Matiére Condensée (LPTMC), Tour 13-5ième ét., B.C. 121, 4 pl. Jussieu, F 75252, Paris Cedex 05, France

    Karol A. Penson

  3. ENEA–Centro Ricerche Frascati, via E. Fermi, 45, IT 00044, Frascati (Roma), Italy

    Giuseppe Dattoli

  4. Institut Galilée CNRS UMR 7030, Université Paris XIII, LIPN, 99 Av. J.-B. Clement, F 93430, Villetaneuse, France

    Gerard H. E. Duchamp

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  1. Katarzyna Górska
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  2. Andrzej Horzela
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  3. Karol A. Penson
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  4. Giuseppe Dattoli
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  5. Gerard H. E. Duchamp
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Correspondence to Katarzyna Górska.

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Górska, K., Horzela, A., Penson, K.A. et al. The Stretched Exponential Behavior and Its Underlying Dynamics. The Phenomenological Approach. FCAA 20, 260–283 (2017). https://doi.org/10.1515/fca-2017-0014

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