Variational formulation of problems involving fractional order differential operators

Bangti Jin; Raytcho Lazarov; Joseph Pasciak; William Rundell

doi:10.1090/mcom/2960

Variational formulation of problems involving fractional order differential operators

Bangti Jin, Raytcho Lazarov, Joseph Pasciak, William Rundell

Research output: Contribution to journal › Article › peer-review

122 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 language	English (US)
Pages (from-to)	2665-2700
Number of pages	36
Journal	Mathematics of Computation
Volume	84
Issue number	296
DOIs	https://doi.org/10.1090/mcom/2960
State	Published - Apr 30 2015
Externally published	Yes

Bibliographical note

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.

ASJC Scopus subject areas

Algebra and Number Theory
Computational Mathematics
Applied Mathematics

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title = "Variational formulation of problems involving fractional order differential operators",

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.",

author = "Bangti Jin and Raytcho Lazarov and Joseph Pasciak and William Rundell",

note = "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.",

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doi = "10.1090/mcom/2960",

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AU - Jin, Bangti

AU - Lazarov, Raytcho

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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.

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EP - 2700

JO - Mathematics of Computation

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ER -

Variational formulation of problems involving fractional order differential operators

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