Fractional radial diffusion in a cylinder

The standard integral equation describing the inward radial diffusion from the surface of a cylinder of radius R and unit length has been generalized to a fractional integral equation describing fractional radial diffusion. The solution for the case when the concentration at the boundary is held constant, can be expressed as the inverse finite Hankel transform of Mittag-Leffler functions, resulting in an infinite series in terms of Bessel functions. The solution giving the variation of concentration in reduced variables has been evaluated numerically. In addition, numerical results for the total amount that has diffused and second moment of the concentration distribution as functions of time are presented. All the quantities exhibit a universal behavior in terms of the dimensionless parameter beta = Dt(alpha)/R-2. As a result, the usual distinction between sub- (alpha < 1) and super-diffusion (alpha > 1) normally discussed in the context of one-dimensional diffusion does not exist in this two-dimensional system. (C) 2004 Elsevier B.V. All rights reserved.