A complex analysis approach to Atangana–Baleanu fractional calculus. (5th August 2019)

• Record Type: Journal Article
• Authors: Fernandez, Arran
• Other Names: Zeidan Dia guestEditor.; Herbst R Sheldon guestEditor.; Lu Tzon‐Tzer guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 10(2021)
• Page Start: 8070
• Page End: 8087
• Publication Date: 2019-08-05
• Subjects: analytic continuation -- complex analysis -- fractional calculus -- Mittag‐Leffler functions; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5754
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : The standard definition for the Atangana–Baleanu fractional derivative involves an integral transform with a Mittag‐Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana–Baleanu fractional integral and for iterated Atangana–Baleanu fractional differintegrals.