Complex Gruenwald-Letnikov, Liouville, Riemann-Liouville, and Caputo derivatives for analytic functions

The well-known Liouville, Riemann-Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Gruenwald-Letnikov derivative. Particularly, starting from a complex formulation of the Gruenwald-Letni...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2011-11, Vol.16 (11), p.4174-4182
Main Authors: Ortigueira, Manuel D, Rodriguez-Germa, Luis, Trujillo, Juan J
Format: Article
Language:English
Subjects:
Analytic functions
Computation
Computer simulation
Derivatives
Function space
Integrals
Joints
Mathematical models
Online Access:Get full text
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Summary:The well-known Liouville, Riemann-Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Gruenwald-Letnikov derivative. Particularly, starting from a complex formulation of the Gruenwald-Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann-Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann-Liouville and Caputo derivatives with the Gruenwald-Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.
ISSN:1007-5704
DOI:10.1016/j.cnsns.2011.02.022