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Higher order Grünwald approximations of fractional derivatives and fractional powers of operators


Authors: Boris Baeumer, Mihály Kovács and Harish Sankaranarayanan
Journal: Trans. Amer. Math. Soc. 367 (2015), 813-834
MSC (2010): Primary 65J10, 65M12, 35R11; Secondary 47D03
DOI: https://doi.org/10.1090/S0002-9947-2014-05887-X
Published electronically: September 4, 2014
MathSciNet review: 3280028
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Abstract | References | Similar Articles | Additional Information

Abstract: We give stability and consistency results for higher order
Grünwald-type formulae used in the approximation of solutions to fractional-in-space partial differential equations. We use a new Carlson-type inequality for periodic Fourier multipliers to gain regularity and stability results. We then generalise the theory to the case where the first derivative operator is replaced by the generator of a bounded group on an arbitrary Banach space.


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Additional Information

Boris Baeumer
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin 9054, New Zealand

Mihály Kovács
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin 9054, New Zealand

Harish Sankaranarayanan
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin 9054, New Zealand

DOI: https://doi.org/10.1090/S0002-9947-2014-05887-X
Keywords: Fractional derivatives, Gr\"unwald formula, Fourier multipliers, Carlson's inequality, fractional differential equations, fractional powers of operators
Received by editor(s): January 19, 2012
Received by editor(s) in revised form: June 3, 2012
Published electronically: September 4, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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