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Variational formulation of problems involving fractional order differential operators


Authors: Bangti Jin, Raytcho Lazarov, Joseph Pasciak and William Rundell
Journal: Math. Comp. 84 (2015), 2665-2700
MSC (2010): Primary 65L60, 65N12, 65N30
DOI: https://doi.org/10.1090/mcom/2960
Published electronically: April 30, 2015
MathSciNet review: 3378843
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Abstract: In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order $ \alpha \in (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 $ H_0^{\alpha /2}(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.


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Additional Information

Bangti Jin
Affiliation: Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, California 92521
Address at time of publication: Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom
Email: bangti.jin@gmail.com

Raytcho Lazarov
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Joseph Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

William Rundell
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: lazarov,pasciak,rundell@math.tamu.edu

DOI: https://doi.org/10.1090/mcom/2960
Keywords: Fractional boundary value problem, Caputo derivative, Riemann-Liouville derivative, variational formulation, finite element method, fractional Sturm-Liouville problem
Received by editor(s): August 20, 2013
Received by editor(s) in revised form: February 12, 2014, and April 6, 2014
Published electronically: April 30, 2015
Article copyright: © Copyright 2015 American Mathematical Society