Plenary Lecture

Fractional Relaxation and Diffusion of Distributed Order

Professor Francesco Mainardi
Department of Physics
University of Bologna & INFN

Abstract: The standard differential equations of exponential relaxation and Gaussian diffusion are generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville and the Caputo sense. For a general distribution of time orders we provide the fundamental solutions 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 out to be equivalent .We consider in some detail two cases of order distribution: double-order, and uniformly distributed order.
Plots of the corresponding fundamental solutions are provided for these cases, pointing out the remarkable difference between the two approaches for small and large times.
The considered processes are relevant in the framework of more general phenomena of anomalous kinetics in several fields of applied science and engineering.

Brief Biography of the Speaker: Presently Francesco MAINARDI is retired professor of Mathematical Physics from the University of Bologna (since November 2013) where he has taught this course since 40 years. Even if retired, he continues to carry out teaching and research activity. His fields of research concern several topics of applied mathematics, including diffusion and wave problems, asymptotic methods, integral transforms, special functions, fractional calculus and non-Gaussian stochastic processes.
At present his H-index is > 50.
For a full biography, list of references on author's papers and books see:
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