Abstract
A two-point boundary value problem whose highest-order derivative is a Riemann–Liouville–Caputo derivative of order \(\alpha \in (1,2)\) is considered. A similar problem was considered in Gracia et al. (BIT 60:411–439, 2020) but under a simplifying assumption that excluded singular solutions. In the present paper, this assumption is not imposed; furthermore, the finite difference method of the BIT paper, which was proved to attain 1st-order convergence under a sign restriction on the convective term, is replaced by a piecewise polynomial collocation method which can give any desired integer order of convergence on a suitably graded mesh. An error analysis of the collocation method is given which removes the above sign restriction and numerical results are presented to support our theoretical conclusions. The tools devised for this analysis include new comparison principles for Caputo initial-value problems and weakly singular Volterra integral equations that are of independent interest. Numerical experiments demonstrate the sharpness of our theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Baeumer B, Kovács M, Meerschaert MM, Sankaranarayanan H (2018) Boundary conditions for fractional diffusion. J Comput Appl Math 336:408–424
Brunner H (2004) Collocation methods for Volterra integral and related functional differential equations. Cambridge monographs on applied and computational mathematics, vol 15. Cambridge University Press, Cambridge
Brunner H (2017) Volterra integral equations, volume 30 of Cambridge Monographs on Applied and Computational Mathematics. An introduction to theory and applications. Cambridge University Press, Cambridge
Brunner H, Pedas A, Vainikko G (2001) Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J Numer Anal 39(3):957–982
del Castillo-Negrete D (2006) Fractional diffusion models of nonlocal transport. Phys Plasmas 13:082308
Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection dispersion equation. Numer Methods Partial Differ Equ 22(3):558–576
Farrell PA, Hegarty AF, Miller JJH, O’Riordan E, Shishkin GI (2000) Robust computational techniques for boundary layers. Applied mathematics, vol 16. Chapman & Hall/CRC, Boca Raton
Gracia JL, O’Riordan E, Stynes M (2019) A collocation method for a two-point boundary value problem with a Riemann–Liouville–Caputo fractional derivative. In: Fifteenth international conference Zaragoza-Pau on mathematics and its applications, volume 42 of Monogr. Mat. García Galdeano. Prensas Univ. Zaragoza, Zaragoza, pp 111–125
Gracia JL, O’Riordan E, Stynes M (2020) Convergence analysis of a finite difference scheme for a two-point boundary value problem with a Riemann–Liouville–Caputo fractional derivative. BIT 60(2):411–439
Gripenberg G, Londen S-O, Staffans O (1990) Volterra integral and functional equations. Encyclopedia of mathematics and its applications, vol 34. Cambridge University Press, Cambridge
Jin B (2021) Fractional differential equations–an approach via fractional derivatives, volume 206 of applied mathematical sciences. Springer, Cham
Jin B, Zhou Z (2023) Numerical treatment and analysis of time-fractional evolution equations, volume 214 of applied mathematical sciences. Springer, Cham
Jin B, Lazarov R, Zhou Z (2016) A Petrov–Galerkin finite element method for fractional convection–diffusion equations. SIAM J Numer Anal 54(1):481–503
Köhler P (1995) Order-preserving mesh spacing for compound quadrature formulas and functions with endpoint singularities. SIAM J Numer Anal 32(2):671–686
Kopteva N (2022) Maximum principle for time-fractional parabolic equations with a reaction coefficient of arbitrary sign. Appl Math Lett 132(108209):7
Kopteva N, Stynes M (2015) An efficient collocation method for a Caputo two-point boundary value problem. BIT 55(4):1105–1123
Li Y, Ginting V (2023) On the Dirichlet BVP of fractional diffusion advection reaction equation in bounded interval: structure of solution, integral equation and approximation. J. Comput. Appl. Math. 426(115097):32
Liang H, Stynes M (2018) Collocation methods for general Caputo two-point boundary value problems. J Sci Comput 76(1):390–425
Linz P (1985) Analytical and numerical methods for Volterra equations, volume 7 of SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia
Patie P, Simon T (2012) Intertwining certain fractional derivatives. Potential Anal 36(4):569–587
Pedas A, Tamme E (2012) Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J Comput Appl Math 236(13):3349–3359
Vainikko G (1993) Multidimensional weakly singular integral equations. Lecture notes in mathematics, vol 1549. Springer, Berlin
Vainikko G (2016) Which functions are fractionally differentiable? Z Anal Anwend 35(4):465–487
Acknowledgements
The authors wish to thank Eugene O’Riordan for suggesting the decomposition of Sect. 3, which significantly aided our analysis.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of José Luis Gracia was supported by the Institute of Mathematics and Applications (IUMA), the Gobierno de Aragon (E24_23R) and the projects PID2022-141385NB-I00 and PID2022-137334NB-I00. The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grants 12171025 and NSAF-U2230402.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gracia, J.L., Stynes, M. A collocation method for an RLC fractional derivative two-point boundary value problem with a singular solution. Comp. Appl. Math. 43, 199 (2024). https://doi.org/10.1007/s40314-024-02730-6
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-024-02730-6
Keywords
- Riemann–Liouville–Caputo derivative
- Singular solution
- Comparison principle
- Piecewise polynomial collocation
- Graded mesh