A simple finite element method for boundary value problems with a Riemann-Liouville derivative

Bangti Jin, Raytcho Lazarov, Xiliang Lu, Zhi Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

18 Citations (Scopus)

Abstract

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-1in 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, andL2(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.
Original languageEnglish
Pages (from-to)94-111
Number of pages18
JournalJournal of Computational and Applied Mathematics
Volume293
DOIs
Publication statusPublished - 11 Feb 2016
Externally publishedYes

Keywords

  • Finite element method
  • Fractional boundary value problem
  • Riemann-Liouville derivative
  • Singularity reconstruction
  • Sturm-Liouville problem

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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