Analytical study of fractional equations describing anomalous diffusion of energetic particles

Frontiers in Theoretical and Applied Physics/UAE 2017

Tawfik, A.M.^1,2 ; Fichtner, H.^1 ; Schlickeiser, R.^1 ; Elhanbaly, A.^2

Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, Bochum

D-44780, Germany^1

Theoretical Physics Research Group, Mansoura University, Mansoura

35516, Egypt^2

关键词: Anomalous diffusion;    Caputo fractional derivatives;    Energetic particles;    Fractional derivative model;    Fractional equation;    Geometric functions;    Mittag-Leffler functions;    Travelling pulse solutions;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/869/1/012050/pdf
DOI  :  10.1088/1742-6596/869/1/012050

来源: IOP

【 摘 要 】

To present the main influence of anomalous diffusion on the energetic particle propagation, the fractional derivative model of transport is developed by deriving the fractional modified Telegraph and Rayleigh equations. Analytical solutions of the fractional modified Telegraph and the fractional Rayleigh equations, which are defined in terms of Caputo fractional derivatives, are obtained by using the Laplace transform and the Mittag-Leffler function method. The solutions of these fractional equations are given in terms of special functions like Fox's H, Mittag-Leffler, Hermite and Hyper-geometric functions. The predicted travelling pulse solutions are discussed in each case for different values of fractional order.