Fundamental solutions of the fractional diffusion and the fractional Fokker–Planck equations

• Main Author: E.A. Abdel-Rehim
• Format: Article
• Language: English
• Published: SpringerOpen 2016-07-01
• Series: Journal of the Egyptian Mathematical Society
• Subjects: α-Stable distribution; Green function; Similarity variable; Feller operator; Scale invariant method; Fractional diffusion equations
• Online Access: http://www.sciencedirect.com/science/article/pii/S1110256X15000711

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.