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.
