Fractional difference operators with discrete generalized Mittag–Leffler kernels

Thabet Abdeljawad

Chaos, Solitons & Fractals, 2019, vol. 126, issue C, 315-324

Abstract: In this article we define fractional difference operators with discrete generalized Mittag–Leffler kernels of the form (Eθ,μ¯γ(λ,t−ρ(s)) for both Riemann type (ABR) and Caputo type (ABC) cases, where AB stands for Atangana–Baleanu. Then, we employ the discrete Laplace transforms to formulate their corresponding AB−fractional sums, and prove useful and applicable versions of their semi-group properties. The action of fractional sums on the ABC type fractional differences is proved and used to solve the ABC−fractional difference initial value problems. The nonhomogeneous linear ABC fractional difference equation with constant coefficient is solved by both the discrete Laplace transforms and the successive approximation, and the Laplace transform method is remarked for the continuous counterpart. In fact, for the case μ ≠ 1, we obtain a nontrivial solution for the homogeneous linear ABC− type initial value problem with constant coefficient. The relation between the ABC and ABR fractional differences are formulated by using the discrete Laplace transform. We iterate the fractional sums of order −(θ,μ,1) to generate fractional sum-differences for which a semigroup property is proved. The nabla discrete transforms for the AB−fractional sums and the AB−iterated fractional sum-differences are calculated. Examples and remarks are given to clarify and confirm the obtained results and some of their particular cases are highlighted. Finally, the discrete extension to the higher order case is discussed.

Keywords: Discrete generalized Mittag–Leffler function; Discrete nabla Laplace transform; Convolution; AB Fractional sums; ABR Fractional difference; ABC Fractional difference; Iterated AB sum-differences; Higher order (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (11)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S096007791930222X
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:126:y:2019:i:c:p:315-324

DOI: 10.1016/j.chaos.2019.06.012

Access Statistics for this article

Chaos, Solitons & Fractals is currently edited by Stefano Boccaletti and Stelios Bekiros

More articles in Chaos, Solitons & Fractals from Elsevier
Bibliographic data for series maintained by Thayer, Thomas R. ().