The partial differential equation of Gaussian diffusion is generalized by
using the time-fractional derivative of distributed order between 0 and 1, in
both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general
distribution of time orders we provide the fundamental solution, that is still
a probability density, in terms of an integral of Laplace type. The kernel
depends on the type of the assumed fractional derivative except for the single
order case where the two approaches turn to be equivalent. We consider with
some detail two cases of order distribution: the double-order and the uniformly
distributed order. For these cases we exhibit plots of the corresponding
fundamental solutions and their variance, pointing out the remarkable
difference between the two approaches for small and large times.Comment: 30 pages, 4 figures. International Workshop on Fractional
Differentiation and its Applications (FDA06), 19-21 July 2006, Porto,
Portugal. Journal of Vibration and Control, in press (2007
