Fractional integral representation of master equation

Using the definition of Liouville-Riemann (L-R) fractional integral operator, master equation can be represented in the domain of fractal time evolution with a critical exponent a(0 < a less than or equal to 1). The relation between the continuous time random walks (CTRW) and fractional master equation (FME) has been achieved by obtaining the corresponding waiting time density (WTD) psi(t). The latter is obtained in a closed form in terms of the generalized Mittag-Leffler (M-L) function. The asymptotic expansion of the (M-L) function show the same behavior considered in the theory of random walk. Applying the Fourier and Laplace-Mellin transforms to (FME), one obtains the solution, in closed form, in terms of the Fox function. (C) 1999 Elsevier Science Ltd. All rights reserved.