An Analytical Study of Fractional Klein–Kramers Approximations for Describing Anomalous Diffusion of Energetic Particles

  • Ashraf M. Tawfik
  • Horst Fichtner
  • A. Elhanbaly
  • Reinhard Schlickeiser


In this article, anomalous transport models of energetic particles in space plasmas are developed by deriving the fractional force-less Fokker–Planck equation and the fractional diffusion-advection equation from the Klein–Kramers equation. Analytical solutions of the space-time fractional equations are obtained by use of the Caputo and the Riesz fractional derivatives with the Laplace–Fourier technique. The solutions are given in terms of Fox’s H-function and the profiles of the particles densities are discussed in each case for different values of fractional orders.


Fractional calculus Anomalous diffusion Energetic particles 


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Authors and Affiliations

  • Ashraf M. Tawfik
    • 1
    • 2
  • Horst Fichtner
    • 1
  • A. Elhanbaly
    • 2
  • Reinhard Schlickeiser
    • 1
  1. 1.Institut für Theoretische Physik IV, Ruhr-Universität BochumBochumGermany
  2. 2.Theoretical Physics Research GroupMansoura UniversityMansouraEgypt

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