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

Content Provider	Springer Nature Link
Author	Stynes, Martin Gracia, José Luis O’Riordan, Eugene
Copyright Year	2016
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.
Ending Page	1477
Page Count	23
Starting Page	1455
File Format	PDF
ISSN	00063835
e-ISSN	15729125
Journal	BIT Numerical Mathematics
Issue Number	4
Volume Number	56
Language	English
Publisher	Springer Netherlands
Publisher Date	2016-01-20
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Finite difference scheme Necessary condition Computational Mathematics and Numerical Analysis Fractional differential equations Numeric Computing Mathematics Fractional derivative Finite difference methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Mathematics Software

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