Open AccessFundamental solutions of the fractional diffusion and the fractional Fokker–Planck equations
Author(s) -
E. A. Abdel-Rehim
Publication year - 2016
Publication title -
journal of the egyptian mathematical society
Language(s) - English
Resource type - Journals
eISSN - 2090-9128
pISSN - 1110-256X
DOI - 10.1016/j.joems.2015.08.006
Subject(s) - mathematics , fokker–planck equation , fractional calculus , skewness , mathematical analysis , anomalous diffusion , diffusion equation , space (punctuation) , differential equation , invariant (physics) , diffusion , mathematical physics , physics , innovation diffusion , statistics , knowledge management , linguistics , philosophy , economy , service (business) , computer science , economics , thermodynamics
The solutions of the space–time fractional diffusion equations and that of the space–time fractional Fokker–Planck equation are probabilities evolving in time and stable in the sense of Lévy. The fundamental solution, Green function, of the space–time fractional diffusion equation, is early obtained by using the scale invariant method. In this paper, I use this reduced Green functions and the scale invariant method to obtain the fundamental solution, Green function, of the fractional diffusion equation and henceforth I obtain the solution of the space–time fractional Fokker–Planck equation, by applying the Billerś transformation between the independent spatial coordinates of these fractional differential equations. Henceforth, I simulate these solutions in the 3D for all the possible values of the space and time fractional orders and also for different values of the skewness