Venue
Aula Magna – Dipartimento di Matematica.
Abstract
Fractional derivatives are a mathematical tool that receivedmuch attention in the last decades because of their non-local behaviorwhich has been demonstrated to be useful when modeling anomalousdiffusion phenomena appearing, e.g., in imaging or electrophysiology.Two of the most famous definitions of fractional derivatives are theRiemann-Liouville and the Caputo ones. The two formulations are relatedby a well-known formula that expresses a Riemann-Liouville derivative asa Caputo one plus a term that depends on the function and itsderivatives at the boundary. As a consequence of this relation,Riemann-Liouville and Caputo derivatives coincide only for sufficientlysmooth functions that satisfy homogeneous conditions at the boundary.Aiming at uncovering how much this discrepancy reflects on theirdiscretized counterparts, we focus on the relation betweenRiemann-Liouville and Caputo derivatives once approximated by acollocation method based on B-splines, i.e., high order finite elementswith maximum regularity. We show that when the fractional orderα ranges in (1,2) their difference in terms of matrices corresponds to arank-1 correction whose spectral norm increases with the mesh-size n and is o(√n). On one hand, this implies that the spectraldistribution for the B-spline collocation matrices corresponding to theRiemann-Liouville and Caputo derivatives coincide; on the other hand,the presence of the rank-1 correction makes the Caputo matrices worseconditioned for α tending to 1 due to a larger maximum singularvalue. Some linear algebra consequences of all this knowledge arediscussed, and a selection of numerical experiments that validate ourfindings is provided together with a numerical study of theapproximation behavior of B-spline collocation.