B-spline collocation discretizations of Caputo and Riemann-Liouville derivatives: a matrix comparison

Two of the most famous definitions of fractional derivatives are the Riemann-Liouville and the Caputo ones. In principle, these formulations are not equivalent and ask for different levels of regularity of the considered function. By focusing on a B-spline collocation discretization of both kind of derivatives, we show that when the fractional order a ranges in (1, 2) their difference in terms of matrices corresponds to a rank-1 correction whose spectral norm increases with the mesh-size n and is o(root n). On one hand, this implies that the spectral distribution for the B-spline collocation matrices corresponding to the Riemann-Liouville and Caputo derivatives coincide; on the other hand, the presence of the rank-1 correction makes the Caputo matrices worse conditioned for a tending to 1 due to a larger maximum singular value. Some linear algebra consequences of all this knowledge are discussed, and a selection of numerical experiments that validate our findings is provided.