Ricerc@Sapienza

Our aim is to model phenomena where the associated dynamics is well described by fractional equations (FE). The FE we are dealing with arise from applications of a large class of phenomena, namely population dynamics, cell growth, bacterial motion, smart materials, bird flight, pedestrian motion. A problem we approach is concerned with the fractional growth of populations and therefore, with the number of interacting particles which grows according to a non-linear fractional dynamic given by the logistic model. In this scenario, we are interested in studying the behaviour of a particle and the behaviour of a large number of particles where the underlying dynamic for each particle is given by a fractional diffusion rather than a standard diffusion. Fractional diffusions are driven by governing equations with fractional operators in time as the well-known Caputo derivative. Recently, new fractional operators in time have been considered: such operators are obtained by convolution and characterized by a general class of kernels associated to Bernstein functions. The pseudo-differential symbol of the fractional operator has a Bernstein representation given in terms of Lévy measure. Thus, from a probabilistic view-point, the theory of fractional calculus we consider here, well accords with the theory of the time-changes for Markov processes. The study of innovative materials such as polymers (it is well-known that textile, petroleum and pharmaceutical industries have strong interests in the investigation of smart materials) brings about to challenging mathematical questions. These are related with solutions to integro-differential equations and Mittag-Leffler functions. The mathematical studies of this class of equations need advanced notions of fractional calculus. With respect to the dynamic population motion, we consider a large number of people each associated with a time-changed process where the underlying dynamic of the individual is given by a fractional equation