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
The project is characterized by a multidisciplinary point of view and profits of the synergy of the complementary expertise involved, namely: Probability and Statistic, for the definition of the important quantities to be considered; Mathematical Analysis and Applied Mathematics, to construct and investigate models; Numerical Analysis, to implement the required simulations. These synergies represent a strong point which we are confident will lead to fruitful interactions between the researchers and also to productive exchanges between the different subjects the team is going to investigate.
To compare the purpose of this research project let us consider thee main arguments that are inside.
Logistic equation -> fractional logistic equation.
This is a well-known Verhulst model a prototype for many other studies and also the basic equation for coupling and the study of more complicated processes. Its fractional logistic equation has been discussed in several context and it was not completely solved from two point of view. What does it means fractional setting of this equation since the equation is non-linear and Laplace transform method is not applicable. Then, it is not so direct to give fractional logistic equation with respect to this problem we give a study for short and large time. Recent papers of people of the research group give a partial answer to this question. Just to describe briefly we found the connection with Euler¿s numbers. This is an intriguing argument and leads to further directions of research. Our feeling is that the analysis of the fractional logistic equation associated with the class of Bernstein functions (and therefore to a class of Volterra-like equations) leads to new fractional Euler's numbers with new mathematical properties. The potential field of applications are so many as evidenced by the numerous articles in the literature concerning the standard case.
Mean field games ->fractional mean field games.
In the standard Mean Field Games model, the dynamics of the single agent is governed by a Gaussian diffusion process. Hence the underlying environment has no role in the problem or, in other words, is isotropic, an assumption not satisfied in several applications. On the contrary, subdiffusive processes display local motion occasionally interrupted by long sojourns, a trapping effects due to the anisotropy of the medium. Aim of this part of the project is to introduce a class of MFG model in which the dynamics of the agent is subdiffusive rather diffusive as in the Lasry-Lions model. A subdiffusive regime is considered to be a better model not only for several transport phenomena in physics, but also, for example, in the study of volatility of financial markets, bacterial motion, bird flight, etc.
According to the optimal control interpretation of the problem, we get a system involving fractional time derivatives for the Hamilton-Jacobi-Bellman and the Fokker-Planck equations. A first important point
is to understand the correct formulation of the MFG system in this framework. Time-fractional FP
equation governing the evolution of the PDF of a subdiffusive process were first derived for the case
of a space-dependent drift and constant diffusion coefficient, hence the theory has been
progressively generalized to include the case of space-time dependent coefficients which is relevant for
our study.
Memory kernels for viscoelastic materials -> Memory kernels for smart materials
In literature, classes of integro-differential equation representing evolution process of viscoelastic materials as oscillations are studied. They are models of integro-differential equations in which the integral term is of convolution type with a nonnegative kernel satisfying conditions that yield a nonincreasing energy of the solution to the problem. From a mathematical point of view the study of innovative materials as polymers leads to the Burger model where the memory term is the sum of decreasing exponentials functions. This is a generalization of the well-known models as Kelvin-Voigt solid models (spring in parallel with a dashpot) and Maxwell fluid models (spring in series with a dashpot). An analysis may be done when we consider Mittag-Leffler functions. As well-known, roughly speaking, the Mittag-Leffler functions generalize the exponential functions and, in the linear case, the analysis will be done although it may contain unexpected difficulties. In particular, the study of oscillations leads to a wave-integro-differential equations. A spectral study is needed to understand the behaviour of the solution and it is preliminary to reachability problem in control theory. We have described the closer new results to our research interest. These are few examples of promising research lines. Our contributions are published in international Journals. Moreover, senior participants have been invited in national and international conferences to give lectures on related results.