dc.contributor.author Jin, Bangti dc.contributor.author Lazarov, Raytcho dc.contributor.author Lu, Xiliang dc.contributor.author Zhou, Zhi dc.date.accessioned 2016-02-25T12:32:35Z dc.date.available 2016-02-25T12:32:35Z dc.date.issued 2016-02 dc.identifier.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. dc.identifier.issn 0377-0427 dc.identifier.doi 10.1016/j.cam.2015.02.058 dc.identifier.uri http://hdl.handle.net/10754/597407 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-$^{1}$ 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. dc.description.sponsorship 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. dc.publisher Elsevier BV dc.subject Finite element method dc.subject Fractional boundary value problem dc.subject Riemann-Liouville derivative dc.subject Singularity reconstruction dc.subject Sturm-Liouville problem dc.title A simple finite element method for boundary value problems with a Riemann–Liouville derivative dc.type Article dc.identifier.journal Journal of Computational and Applied Mathematics dc.contributor.institution UCL, London, United Kingdom dc.contributor.institution Texas A and M University, College Station, United States dc.contributor.institution Wuhan University, Wuhan, China kaust.grant.number KUS-C1-016-04
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