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Necessary conditions for convergence of difference schemes for fractional-derivative two-point boundary value problems

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Abstract

A fractional-derivative two-point boundary value problem of the form \({\tilde{D}}^\delta u=f\) on (0, 1) with Dirichlet boundary conditions is studied. Here \({\tilde{D}}^\delta \) is a Caputo or Riemann–Liouville fractional derivative operator of order \(\delta \in (1,2)\). The discretisation of this problem by an arbitrary difference scheme is examined in detail when u or f is a polynomial. For any convergent difference scheme, it is proved rigorously that the entries of the associated matrix must satisfy certain identities. It is shown that some of these identities are not satisfied by certain well-known schemes from the research literature; this clarifies the type of problem to which these schemes can be applied successfully. The effects of the special boundary condition \(u(0)=0\) and the special right-hand-side condition \(f(0)=0\) are also investigated. This leads, under certain circumstances, to a sharpening of a recently-published finite difference scheme convergence result of two of the authors.

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Acknowledgments

We are grateful for the careful reading and helpful suggestions of the referee.

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Authors and Affiliations

  1. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Haidian District, Beijing, 100084, China

    Martin Stynes

  2. School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

    Eugene O’Riordan

  3. Department of Applied Mathematics, Torres Quevedo Building, Campus Rio Ebro, University of Zaragoza, 50018, Zaragoza, Spain

    José Luis Gracia

Authors
  1. Martin Stynes
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  2. Eugene O’Riordan
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  3. José Luis Gracia
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Corresponding author

Correspondence to Eugene O’Riordan.

Additional information

Communicated by Jan Hesthaven.

Martin Stynes: The research of this author is supported in part by the National Natural Science Foundation of China under Grant 91430216.

José Luis Gracia: The research of this author is supported in part by the Institute of Mathematics and Applications (IUMA), the projects MTM2013-40842-P and UZCUD2014-CIE-09 and the Diputación General de Aragón.

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Stynes, M., O’Riordan, E. & Gracia, J.L. Necessary conditions for convergence of difference schemes for fractional-derivative two-point boundary value problems. Bit Numer Math 56, 1455–1477 (2016). https://doi.org/10.1007/s10543-016-0602-4

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  • DOI: https://doi.org/10.1007/s10543-016-0602-4

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