Ricerc@Sapienza

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.

Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties.