Fractional representation of Fokker-Planck equation

From the definition of the characteristic function and Kramers-Moyal forward expansion, one can obtain the fractional Fokker-Planck equation (FFPE) in the domain of fractal time evolution with a critical exponent beta (0 < beta less than or equal to 1) [El-Wakil SA, Zahran MA. Chaos, Solitons g( Fractals 11 (2000) 791-98]. The solutions of Fokker-Planck equation will establish in three different cases of mean-square displacement as follows: (i) ((x(t + tau) - x(t)(2)) similar to tau, (ii) ((x(t + tau) - x(t))(2)) tau (beta), 0 < beta less than or equal to 1, (iii) ((x(t + tau) - x(t))2) similar to x-0 tau beta, theta = d(w) -2 The distribution function of each case can be obtained in a closed form of Fox-function. (C) 2001 Elsevier Science Ltd. All rights reserved.