A wide variety of processes in biomathematics, engineering and applied science exhibit aspects in analysis and probability that can be model by a fractional setting. Fractional calculus (integrals and derivatives of non-integer order) is considered in models to take into account macroscopic effect. The use of fractional derivatives in the equations means that as global effect we get a slowdown in the process. In this project we consider effect of fractional time derivatives in evolution equations of large use in mathematics and engineering. We consider linear equations such as wave equations, Petrovsky systems, and nonlinear equations such as viscous Hamilton Jacobi equations.
In all this class we mainly study existence and regularity of solutions once that operators in time are replaced by fractional time derivatives in the Caputo sense.
In the context of analysis, modelling and simulation of biological systems, we
consider infective disease which does not confer immunity and which is transmitted through contact between people, the so called SIS models. SIS means that people are splitted into two disjoint classes which evolve in time: the susceptibles and the infectives. The SIS model is a disease model without immunity, where the individuals recovered from the infection come back into the class of susceptibles. The use of fractional derivatives in the model means that we consider global effects as slowdown in the process. We study the question to combine different scale of fractional derivatives to model a different slowdown between susceptible and infective and to combine the action of the Caputo fractional derivatives and the Caputo-Fabrizio operator.
In this project we consider effect of fractional time derivatives in evolution equations of large use in mathematics and engineering.
The term fractional is referred to a new definition of derivates that makes sense for non integer order. Numerous contributions have been given to definitions for fractional derivatives and integrals. Here we use the Caputo fractional derivative. Due the presence of a singular kernel that works as a memory and it gives a non local character to the models, the analysis is technically difficult and requires a deep study. However these non local aspects of models are realistic in applications, and we are interested to give a mathematical setting of these models in terms of evolution equations.
1 Hidden regularity for fractional time wave and Petrovsky systems
In the non-fractional setting wave equations and Petrovsky systems model respectively vibrations of strings and beams. According to the boundary conditions, the models describe physical models of vibrations respectively of hinged strings and hinged beams. Both problems have been studied in control theory.
In the project we study the fractional-time derivative effects and the results behind the fractional setting. Main difficulties are the meaning of solutions and how to choose the appropriate spaces where to set the problem. Also, due to the singularity of the integral kernels, we need to avoid integration by parts in time. To establish the estimates we have to find alternative methods.
Recent references papers to the subject of research are:
[GLY] R. Gorenflo, Y. Luchko, M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces Fract. Calc. Appl. Anal. 18, no. 3, 799-820,(2015).
[LS1] P. Loreti, D. Sforza, Hidden regularity for wave equations with memory, Riv. Math. Univ. Parma (N.S.) 7, 391-405,(2016).
[LS2] P. Loreti, D. Sforza, A semilinear integro-differential equation: global existence and hidden regularity. Trends in control theory and partial differential equations, 157-180, Springer IN- dAM Ser., 32, Springer, Cham, (2019).
[SY] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion- wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (1) 426-447,(2011).
2. Subdiffusive Mean Field Games and time-fractional Hamilton-Jacobi equations
We consider a class of MFG problems in which the dynamics of the agents are subdiffusive. The main point is to understand the correct formulation of the MFG theory in this framework and to show the well-posedness of the corresponding MFG system ( [CDM] for some preliminary results).
In addition to the theoretical study of the subdiffusive MFG problem, we also propose to reconsider in this context some classical models studied through the theory of MFGs in order to highlight how this class of models is able to more accurately grasp some physical characteristics. In particular, applications to the study of financial markets and on-line trading seem particularly interesting precisely due to the ability of subdiffusive models to capture memory effects that have been highlighted in various works in the literature.
[CDM] F. Camilli F, R. De Maio A time-fractional mean field game. Adv. Diff. Equ. 24(9-10):531-554 (2019).
3. Fractional Models in epidemiology
In the project we consider infective disease which does not confer immunity and which is transmitted through contact between people, the so called SIS models. SIS means that people are splitted into two disjoint classes which evolve in time: the susceptibles and the infectives. It is then described by a system of ordinary differential equations for each class of individuals . The model is largely used, and it is possible to find many adaptations according to the applications.
For the fractional logistic equations see [DL]
The use of fractional derivatives in the model means that we consider global effect that may produce slowdown in the process [BDL] . We plan to study the question to combine different scale of fractional derivatives to model a different slowdown between susceptible and infected and to combine the action of Caputo fractional derivatives and the Caputo-Fabrizio operator.
For non-linear equations, we still observe the classical delaying effect introduced by the fractional derivative for linear equations. Some probabilistic connections in terms of time changes can be also given (see for example [D]).
Recent references papers to the subject of research are
[BDL] C. Balzotti, M. D'Ovidio, P. Loreti, Fractional SIS Epidemic Models Fractal Fract. 4, no. 3: 44. (2020)
[D] M. D'Ovidio, Non-local logistic equations from the probability viewpoint. Theory of Probability and Mathematical Statistics (to appear)
[DL] M. D'Ovidio, P. Loreti, Solutions of fractional logistic equations by Euler's numbers, Physica A: Statistical Mechanics and its Applications, 506 (2018)