Abstract
We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the state.
Highlights
The exact birthday of fractional calculus, and the idea of non-integer differentiation, goes back to the 17th century, precisely to 30 September 1695, when L’Hôpital wrote a question to ∂nLeibniz about the meaning of n in the case n =([1], pp. 301–302)
∂t mathematicians have investigated around this question
Heymans and Podlubny have shown that it is possible to attribute a physical meaning to initial conditions expressed in terms of Riemann–Liouville fractional derivatives on the field of viscoelasticity, which is more appropriate than standard initial conditions [4]
Summary
The exact birthday of fractional calculus, and the idea of non-integer differentiation, goes back to the 17th century, precisely to 30 September 1695, when L’Hôpital wrote a question to. In the last decades, fractional calculus has been recognized as one of the best tools to describe long-memory processes and materials, anomalous diffusion, long-range interactions, long-term behaviours, power laws, and allometric scaling [5] Such models are those described by differential equations containing fractional order derivatives. Several works deal with the problem of regional observability, which we study here in the fractional context, investigating the possibility to reconstruct the initial state or gradient only on a subregion ω of the evolution domain Ω [31,32,33,34,35].
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