Power functions and essentials of fractional calculus on isolated time scales
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Date
2013-08-23
Authors
Kisela, Tomáš
ORCID
0000-0002-9601-7071Advisor
Referee
Mark
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
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Abstract
This paper concerns with a recently suggested axiomatic definition of power functions on a general time scale and its consequences to fractional calculus. Besides a discussion of existence and uniqueness of such functions, we derive an efficient formula for the computation of power functions of rational orders on an arbitrary isolated time scale. It can be utilized in the introduction and evaluation of fractional sums and differences. We also deal with the Laplace transform of such fractional operators which, apart from solving of fractional difference equations, enables a more detailed comparison of our results with those in the relevant literature. Some illustrating examples (including special fractional initial value problems) are presented as well.
This paper concerns with a recently suggested axiomatic definition of power functions on a general time scale and its consequences to fractional calculus. Besides a discussion of existence and uniqueness of such functions, we derive an efficient formula for the computation of power functions of rational orders on an arbitrary isolated time scale. It can be utilized in the introduction and evaluation of fractional sums and differences. We also deal with the Laplace transform of such fractional operators which, apart from solving of fractional difference equations, enables a more detailed comparison of our results with those in the relevant literature. Some illustrating examples (including special fractional initial value problems) are presented as well.
This paper concerns with a recently suggested axiomatic definition of power functions on a general time scale and its consequences to fractional calculus. Besides a discussion of existence and uniqueness of such functions, we derive an efficient formula for the computation of power functions of rational orders on an arbitrary isolated time scale. It can be utilized in the introduction and evaluation of fractional sums and differences. We also deal with the Laplace transform of such fractional operators which, apart from solving of fractional difference equations, enables a more detailed comparison of our results with those in the relevant literature. Some illustrating examples (including special fractional initial value problems) are presented as well.
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Citation
Advances in Difference Equations. 2013, vol. 2013, issue 8, p. 1-18.
https://advancesindifferenceequations.springeropen.com/articles/10.1186/1687-1847-2013-259
https://advancesindifferenceequations.springeropen.com/articles/10.1186/1687-1847-2013-259
Document type
Peer-reviewed
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Published version
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Language of document
en