Variational formulation of problems involving fractional order differential operators

Bangti Jin, Raytcho Lazarov, Joseph Pasciak, William Rundell

Research output: Contribution to journalArticlepeer-review

109 Scopus citations

Abstract

In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order α ε (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.
Original languageEnglish (US)
Pages (from-to)2665-2700
Number of pages36
JournalMathematics of Computation
Volume84
Issue number296
DOIs
StatePublished - Apr 30 2015
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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