Abstract
We present a general series representation formula for the local solution of the Bernoulli equation with Caputo fractional derivatives. We then focus on a generalization of the fractional logistic equation and present some related numerical simulations.
Highlights
Models arising with aspects related to non-local behavior need to be studied in the fractional setting; see [1,2] for an overview and [2,3] for fractional growth models for social and biological dynamics
The fractional differential calculus is as old as the ordinary one, the complexity of the computations involved still prevents a full exploitation of its applicative potential
The presence of a singular kernel introduces a memory, and it gives a non-local character to the dynamics
Summary
Moving to the fractional setting, we have similar approaches that cannot be followed As it is well known, the solution of the fractional logistic equation—corresponding to p = 1 and a0 = a1 = −1 in (1)—was an open problem, and in [6], the first and the third author were able to solve the fractional logistic equation by series representation, giving a detailed formula involving Euler numbers for u0 = 1/2. In the particular case of social dynamics, the need to model memory effects and non-local behavior lead to a decline of the above models in a fractional setting. At this early stage, the problem of addressing non-linear dynamics and series expansions proved, at least in a local setting, to be a viable approach.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
