Applications of integral transforms in fractional diffusion processes
by
Francesco Mainardi
Department of Physics, University of Bologna, Italy

In this lecture we deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order \alpha in (0, 2] and skewness \theta (|\theta| <= min {\alpha, 2-\alpha}), and the first-order time derivative with a Caputo derivative of order \beta in (0, 2]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional diffusion {0 < \alpha <= 2 ,  \beta = 1}, time-fractional diffusion {\alpha = 2 ,  0 < \beta <= 2 }, and neutral-fractional diffusion {0 < \alpha = \beta <= 2 }, for which the fundamental solution can be interpreted as a spatial probability density function evolving in time. Then, by using the Mellin transform, we provide a general representation of the Green functions in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation to the ranges {0 < \alpha <= 2} \cap {0 < \beta <= 1 } and {1 < \beta <= \alpha <= 2}. Furthermore, from this representation we derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the spatial probability densities for different values of the relevant parameters \alpha, \theta, \beta. This lecture is based on Author's recent works carried out with R. Gorenflo, Yu. Luchko and G. Pagnini.

Date received: May 15, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahk-45.