While for the integer-order derivatives, the commutative and semigroup (or associative) properties hold, the same is not true, in general, for the fractional derivatives. We show that, when the function is analytic, the Caputo derivative enjoys the above mentioned properties, at least when the fractional indices are smaller than one. This result is proved under assumptions on the derivatives (evaluated in zero) that are much less restrictive than the usual requirement of vanishing in the origin. Finally, we study the same prop- erties for a generalization of the above fractional derivative, i.e. the Caputo-type convolu- tion operator defined in [12] and [25]. In this more general setting (which includes the fractional, as particular case), we prove that the commutative property holds, while the associative property must be formulated accordingly.

Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator / Beghin, Luisa; Caputo, Michele. - In: COMMUNICATIONS IN NONLINEAR SCIENCE & NUMERICAL SIMULATION. - ISSN 1007-5704. - 89:(2020), pp. 1-7. [10.1016/j.cnsns.2020.105338]

### Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator

#### Abstract

While for the integer-order derivatives, the commutative and semigroup (or associative) properties hold, the same is not true, in general, for the fractional derivatives. We show that, when the function is analytic, the Caputo derivative enjoys the above mentioned properties, at least when the fractional indices are smaller than one. This result is proved under assumptions on the derivatives (evaluated in zero) that are much less restrictive than the usual requirement of vanishing in the origin. Finally, we study the same prop- erties for a generalization of the above fractional derivative, i.e. the Caputo-type convolu- tion operator defined in [12] and [25]. In this more general setting (which includes the fractional, as particular case), we prove that the commutative property holds, while the associative property must be formulated accordingly.
##### Scheda breve Scheda completa
2020
Caputo fractional derivative; semigroup property; commutative property; Caputo-type convolution operator
01 Pubblicazione su rivista::01a Articolo in rivista
Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator / Beghin, Luisa; Caputo, Michele. - In: COMMUNICATIONS IN NONLINEAR SCIENCE & NUMERICAL SIMULATION. - ISSN 1007-5704. - 89:(2020), pp. 1-7. [10.1016/j.cnsns.2020.105338]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/1402916`