Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

•Prabhakar and related operators can be expressed as series of Riemann–Liouville operators.•Fundamental properties of Prabhakar operators are recovered from the series formulae.•The product and chain rules hold for Prabhakar fractional-calculus operators.•Fractional iteration for these operators is...

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Vydáno v:Communications in nonlinear science & numerical simulation Ročník 67; s. 517 - 527
Hlavní autoři: Fernandez, Arran, Baleanu, Dumitru, Srivastava, H.M.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 01.02.2019
Elsevier Science Ltd
Témata:
Calculus
Convergent series
Fractional calculus
Integral transforms
Iterative methods
Mathematical models
Mittag-Leffler functions
Nonlinear systems
Operators (mathematics)
Prabhakar operators
Convergent series
Mittag-Leffler functions
Prabhakar operators
Fractional calculus
ISSN:1007-5704, 1878-7274
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Popis
Shrnutí:•Prabhakar and related operators can be expressed as series of Riemann–Liouville operators.•Fundamental properties of Prabhakar operators are recovered from the series formulae.•The product and chain rules hold for Prabhakar fractional-calculus operators.•Fractional iteration for these operators is discussed. We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann–Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.
Bibliografie:ObjectType-Article-1
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ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2018.07.035