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Variational formulation of problems involving fractional order differential operators

Jin, B; Lazarov, R; Pasciak, J; Rundell, W; (2015) Variational formulation of problems involving fractional order differential operators. Mathematics of Computation , 84 (296) pp. 2665-2700. 10.1090/mcom/2960. Green open access

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Abstract

In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order α ∈ (1, 2) on the unit interval (0, 1). These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space Hα/2 0 (0, 1) but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem.

Type: Article
Title: Variational formulation of problems involving fractional order differential operators
Open access status: An open access version is available from UCL Discovery
DOI: 10.1090/mcom/2960
Publisher version: http://dx.doi.org/10.1090/mcom/2960
Language: English
Additional information: (C) 2015 American Mathematical Society. First published in Mathematics of Computation in Volume 84, Number 296, published by the American Mathematical Society.
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/1471999
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