B-spline collocation discretizations of caputo and Riemann-Liouville derivatives: A matrix comparison

References

[1] F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali, G. Sangalli, Isogeometric collocation methods. Math. Models Methods Appl. Sci. 20 (2010), 2075–2107.10.1142/S0218202510004878Search in Google Scholar

[2] O. Axelsson, G. Lindskog, On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math. 48 (1986), 499–523.10.1007/BF01389448Search in Google Scholar

[3] G. Barbarino, S. Serra-Capizzano, Non-Hermitian perturbations of Hermitian matrix-sequences and applications to the spectral analysis of the numerical approximation of partial differential equations. Numer. Linear Algebra Appl. 27 (2020), # e2286.10.1002/nla.2286Search in Google Scholar

[4] D. Bertaccini, F. Durastante, Solving mixed classical and fractional partial differential equations using short–memory principle and approximate inverses. Numer. Algor. 74 (2017), 1061–1082.10.1007/s11075-016-0186-8Search in Google Scholar

[5] A. Böttcher, S.M. Grudsky, On the condition numbers of large semidefinite Toeplitz matrices. Linear Algebra Appl. 279 (1998), 285–301.10.1016/S0024-3795(98)00015-9Search in Google Scholar

[6] J. Bai, X. Feng, Fractional-order anisotropic diffusion for image denoising. IEEE Trans. on Image Process. 16 (2007), 2492–2502.10.1109/TIP.2007.904971Search in Google Scholar PubMed

[7] A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez, K. Burrage, Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization. J. Royal Soc. Interface 11 (2014), # 20140352.10.1098/rsif.2014.0352Search in Google Scholar PubMed PubMed Central

[8] M. Ciesielsky, J. Leszhynski, Numerical simulation of anomalous diffusion. CMM-2003, Poland (Reprinted in arXiv:math-ph/0309007).Search in Google Scholar

[9] C. de Boor, A Practical Guide to Splines. Springer-Verlag, New York (2001).Search in Google Scholar

[10] K. Diethelm, R. Garrappa, A. Giusti, M. Stynes, Why fractional derivatives with nonsingular kernels should not be used. Fract. Calc. Appl. Anal. 23, No 3 (2020), 610–634; 10.1515/fca-2020-0032; https://www.degruyter.com/journal/key/fca/23/3/html.Search in Google Scholar

[11] M. Donatelli, M. Mazza, S. Serra-Capizzano, Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307 (2016), 262–279.10.1016/j.jcp.2015.11.061Search in Google Scholar

[12] M. Donatelli, M. Mazza, S. Serra-Capizzano, Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations. SIAM J. Sci. Comput. 40 (2018), A4007–A4039.10.1137/17M115164XSearch in Google Scholar

[13] A. Esen, O. Tasbozan, Y. Ucar, N. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations. Tbil. Math. J. 8 (2015), 181–193.10.1515/tmj-2015-0020Search in Google Scholar

[14] Z.W. Fang, M.K. Ng, H.W. Sun, Circulant preconditioners for a kind of spatial fractional diffusion equations. Numer. Algor. 82 (2019), 729–747.10.1007/s11075-018-0623-ySearch in Google Scholar

[15] L.L. Ferrás, N. Ford, M.L. Morgado, M. Rebelo, High-order methods for systems of fractional ordinary differential equations and their application to time-fractional diffusion equations. Math. Comput. Sci. 15 (2021), 535–551.10.1007/s11786-019-00448-xSearch in Google Scholar

[16] N.J. Ford, M.L. Morgado, M. Rebelo, Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract. Calc. Appl. Anal. 16, No 4 (2013), 874–891; 10.2478/s13540-013-0054-3; https://www.degruyter.com/journal/key/fca/16/4/html.Search in Google Scholar

[17] C. Garoni, C. Manni, F. Pelosi, S. Serra-Capizzano, H. Speleers, On the spectrum of stiffness matrices arising from isogeometric analysis. Numer. Math 127 (2014), 751–799.10.1007/s00211-013-0600-2Search in Google Scholar

[18] C. Garoni, S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I. Springer, Cham (2017).10.1007/978-3-319-53679-8Search in Google Scholar

[19] G.H. Golub, C.F. Van Loan, Matrix Computations. Vol. 52. Baltimore: Johns Hopkins University Press (1983).Search in Google Scholar

[20] U. Grenander, G. Szegö, Toeplitz Forms and Their Applications. Second Edition, Chelsea, New York (1984).Search in Google Scholar

[21] N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45 (2006), C765–C771.10.1007/s00397-005-0043-5Search in Google Scholar

[22] R.R. Hiemstra, T.J.R. Hughes, A. Reali, and D. Schillinger, Removal of spurious outlier frequencies and modes from isogeometric discretizations of second- and fourth-order problems in one, two, and three dimensions. Oden Institute Report 21-07 (2021).10.1016/j.cma.2021.114115Search in Google Scholar

[23] Y. Jiang, J. Ma, High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235 (2011), 3285–3290.10.1016/j.cam.2011.01.011Search in Google Scholar

[24] X.-L. Lin, M.K. Ng, H.-W. Sun, Efficient preconditioner of one-sided space fractional diffusion equation. BIT Numer. Math. 8 (2018), 729–748.10.1007/s10543-018-0699-8Search in Google Scholar

[25] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007), 1533–1552.10.1016/j.jcp.2007.02.001Search in Google Scholar

[26] M. Mazza, M. Donatelli, C. Manni, H. Speleers, On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties. ArXiv:2106.14834.Search in Google Scholar

[27] Z. Mao, G.E. Karniakadis, A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56 (2018), 24–49.10.1137/16M1103622Search in Google Scholar

[28] G. Pan, W. Chen, K.Y. Sze, Gauss-Jacobi-type quadrature rules for fractional directional integrals. Comput. Math. Appl. 66 (2013), 597–607.10.1016/j.camwa.2013.04.020Search in Google Scholar

[29] F. Pitolli, Optimal B-spline bases for numerical solution of fractional differential problems. Axioms 7 (2018), # 46.10.3390/axioms7030046Search in Google Scholar

[30] Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56 (2006), 193–209.10.1016/j.apnum.2005.03.003Search in Google Scholar

[31] F. Zhou, X. Xu, Numerical solution of time-fractional diffusion-wave equations via Chebyshev wavelets collocation method. Adv. Math. Phys. (2017), 2610804.10.1155/2017/2610804Search in Google Scholar