Jin, B;
Lazarov, R;
Zhou, Z;
Lu, X;
(2016)
A simple finite element method for boundary value problems with a Riemann-Liouville derivative.
Journal of Computational and Applied Mathematics
, 293
pp. 94-111.
10.1016/j.cam.2015.02.058.
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Abstract
We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα-1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and L2(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm-Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.
| Type: | Article |
|---|---|
| Title: | A simple finite element method for boundary value problems with a Riemann-Liouville derivative |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.cam.2015.02.058 |
| Publisher version: | http://dx.doi.org/10.1016/j.cam.2015.02.058 |
| Language: | English |
| Additional information: | Copyright © 2015. This manuscript version is published under a Creative Commons Attribution Non-commercial Non-derivative 4.0 International licence (CC BY-NC-ND 4.0). This licence allows you to share, copy, distribute and transmit the work for personal and non-commercial use providing author and publisher attribution is clearly stated. Further details about CC BY licences are available at http://creativecommons.org/licenses/by/4.0. Access may be initially restricted by the publisher. |
| Keywords: | Finite element method; Riemann–Liouville derivative; Fractional boundary value problem; Sturm–Liouville problem; Singularity reconstruction. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1464293 |
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