A simple finite element method for boundary value problems with a Riemann–Liouville derivative

Handle URI:
http://hdl.handle.net/10754/597407
Title:
A simple finite element method for boundary value problems with a Riemann–Liouville derivative
Authors:
Jin, Bangti; Lazarov, Raytcho; Lu, Xiliang; Zhou, Zhi
Abstract:
© 2015 Elsevier B.V. All rights reserved. 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-<sup>1</sup> 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 <sup>L2</sup>(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.
Citation:
Jin B, Lazarov R, Lu X, Zhou Z (2016) A simple finite element method for boundary value problems with a Riemann–Liouville derivative. Journal of Computational and Applied Mathematics 293: 94–111. Available: http://dx.doi.org/10.1016/j.cam.2015.02.058.
Publisher:
Elsevier BV
Journal:
Journal of Computational and Applied Mathematics
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
Feb-2016
DOI:
10.1016/j.cam.2015.02.058
Type:
Article
ISSN:
0377-0427
Sponsors:
The authors are grateful to the anonymous referees for their insightful comments, which have led to improved presentation of the paper. The research of R. Lazarov was supported in parts by National Science Foundation Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). X. Lu is supported by National Natural Science Foundation of China Nos. 91230108 and 11471253. Z. Zhou was partially supported by National Science Foundation Grant DMS-1016525.
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Full metadata record

DC FieldValue Language
dc.contributor.authorJin, Bangtien
dc.contributor.authorLazarov, Raytchoen
dc.contributor.authorLu, Xiliangen
dc.contributor.authorZhou, Zhien
dc.date.accessioned2016-02-25T12:32:35Zen
dc.date.available2016-02-25T12:32:35Zen
dc.date.issued2016-02en
dc.identifier.citationJin B, Lazarov R, Lu X, Zhou Z (2016) A simple finite element method for boundary value problems with a Riemann–Liouville derivative. Journal of Computational and Applied Mathematics 293: 94–111. Available: http://dx.doi.org/10.1016/j.cam.2015.02.058.en
dc.identifier.issn0377-0427en
dc.identifier.doi10.1016/j.cam.2015.02.058en
dc.identifier.urihttp://hdl.handle.net/10754/597407en
dc.description.abstract© 2015 Elsevier B.V. All rights reserved. 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-<sup>1</sup> 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 <sup>L2</sup>(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.en
dc.description.sponsorshipThe authors are grateful to the anonymous referees for their insightful comments, which have led to improved presentation of the paper. The research of R. Lazarov was supported in parts by National Science Foundation Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). X. Lu is supported by National Natural Science Foundation of China Nos. 91230108 and 11471253. Z. Zhou was partially supported by National Science Foundation Grant DMS-1016525.en
dc.publisherElsevier BVen
dc.subjectFinite element methoden
dc.subjectFractional boundary value problemen
dc.subjectRiemann-Liouville derivativeen
dc.subjectSingularity reconstructionen
dc.subjectSturm-Liouville problemen
dc.titleA simple finite element method for boundary value problems with a Riemann–Liouville derivativeen
dc.typeArticleen
dc.identifier.journalJournal of Computational and Applied Mathematicsen
dc.contributor.institutionUCL, London, United Kingdomen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionWuhan University, Wuhan, Chinaen
kaust.grant.numberKUS-C1-016-04en
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