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.

Mazza, M. (2021). B-spline collocation discretizations of Caputo and Riemann-Liouville derivatives: a matrix comparison. FRACTIONAL CALCULUS & APPLIED ANALYSIS, 24(6), 1670-1698 [10.1515/fca-2021-0072].

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

Mazza M.
2021-01-01

Abstract

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.
2021
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/08
English
fractional calculus operators
B-spline collocation
Toeplitz matrices
eigenvalues
conditioning
Mazza, M. (2021). B-spline collocation discretizations of Caputo and Riemann-Liouville derivatives: a matrix comparison. FRACTIONAL CALCULUS & APPLIED ANALYSIS, 24(6), 1670-1698 [10.1515/fca-2021-0072].
Mazza, M
Articolo su rivista
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/344050
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
social impact