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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Variational formulation of problems involving fractional order differential operators
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by Bangti Jin, Raytcho Lazarov, Joseph Pasciak and William Rundell PDF
Math. Comp. 84 (2015), 2665-2700 Request permission

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.
References
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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
  • MR Author ID: 741824
  • Email: bangti.jin@gmail.com
  • Raytcho Lazarov
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • MR Author ID: 111240
  • Email: lazarov@math.tamu.edu
  • Joseph Pasciak
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Email: pasciak@math.tamu.edu
  • William Rundell
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Email: rundell@math.tamu.edu
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 2665-2700
  • MSC (2010): Primary 65L60, 65N12, 65N30
  • DOI: https://doi.org/10.1090/mcom/2960
  • MathSciNet review: 3378843