TY - JOUR

T1 - Variational formulation of problems involving fractional order differential operators

AU - Jin, Bangti

AU - Lazarov, Raytcho

AU - Pasciak, Joseph

AU - Rundell, William

N1 - KAUST Repository Item: Exported on 2022-06-08
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The research of B. Jin and W. Rundell was supported by US NSF Grant DMS-1319052, R. Lazarov was supported in part by US NSF Grant DMS-1016525 and J. Pasciak was supported by NSF Grant DMS-1216551. The work of all authors was supported also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2015/4/30

Y1 - 2015/4/30

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/678726

UR - http://www.ams.org/mcom/2015-84-296/S0025-5718-2015-02960-3/

UR - http://www.scopus.com/inward/record.url?scp=85000605837&partnerID=8YFLogxK

U2 - 10.1090/mcom/2960

DO - 10.1090/mcom/2960

M3 - Article

VL - 84

SP - 2665

EP - 2700

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 296

ER -