A simple finite element method for boundary value problems with a Riemann-Liouville derivative Academic Article uri icon

abstract

  • 2015 Elsevier B.V. All rights reserved. We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order (3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term x- 1 in 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, and L2 (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.

published proceedings

  • JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

altmetric score

  • 0.25

author list (cited authors)

  • Jin, B., Lazarov, R., Lu, X., & Zhou, Z.

citation count

  • 25

complete list of authors

  • Jin, Bangti||Lazarov, Raytcho||Lu, Xiliang||Zhou, Zhi

publication date

  • January 2016