Ricerc@Sapienza

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.

We use a collocation method in refinable spline spaces to solve a linear dynamical system having fractional derivative in time. The method takes advantage of an explicit differentiation rule for the B-spline basis that allows us to efficiently evaluate the collocation matrices appearing in the method. We prove the convergence of the method and show some numerical results.