A Conformable Fractional Calculus on Arbitrary Time Scales

By Nadia Benkhettou, S. Hassani, Delfim F. M. Torres. 2015

213 citations recorded by the source corpus.

AbstractA conformable time-scale fractional calculus of order α ∈]0,1] is introduced. The basictools for fractional diﬀerentiation and fractional integration are then developed. The Hilgertime-scale calculus is obtained as a particular case, by choosing α = 1.Keywords: fractional calculus, conformable operators, calculus on time scales.2010 Mathematics Subject Classiﬁcation: 26A33, 26E70. 1 Introduction Fractional calculus is nowadays one of the most intensively developing areas of mathematicalanalysis [12,14,16], including several deﬁnitions of fractional operators like Riemann–Liouville,Caputo, and Gru¨nwald–Letnikov. Operators for fractional diﬀerentiation and integration havebeen used in variousﬁelds, such as signal processing, hydraulics of dams, temperature ﬁeld problemin oil strata, diﬀusion problems, and waves in liquids and gases [5,8,15]. Here we introduce thenotion of conformable fractional derivative on a time scale T. The notion of conformable fractionalderivative in T= [0,∞) is a recent one: it was introduced in [13], then developed in [1], and iscurrently under intensive investigations [4]. In all these works, however, only the case T= [0,∞)is treated, providing a natural extension of the usual derivative. In contrast, here we introducethe conformable natural extension of the time-scale derivative. A time scale T is an arbitrarynonempty closed subset of R. It serves as a model of time. The calculus on time scales wasinitiated by Aulbach and Hilger in 1988 [3], in order to unify and generalize continuous anddiscrete analysis [10,11]. It has a tremendous potential for applications and has recently receivedmuch attention [2]. The reader interested on the subject of time scales is referred to the books [6,7].The paper is organized as follows. In Section 2, the conformable fractional derivative for func-tions deﬁned on arbitrary time scales is introduced, and the respective conformable fractionaldiﬀerential calculus developed. Then, in Section 3, we introduce the notion of conformable frac-tional integral on time scales (the α-fractional integral) and investigate some of its basic properties.We end with Section 4 of conclusion.

Subfields: Applied Mathematics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/27580e9de11e92a5531431b7fa7a566eb5731382

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Citations in the corpus

New exact solutions of Burgers’ type equations with conformable derivative (2017, unknown): “[14] introduced conformable time-scale calculus.”
Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales (2016, unknown): “In order to unify and generalize the Hilger calculus and the conformable fractional calculus on R, the authors introduce the conformable fractional calculus on time scales and study its properties in [14].”
Existence and Uniqueness of Solution for a Fractional Riemann-Liouville Initial Value Problem on Time Scales (2015, background): “For local approaches to fractional calculus on time scales we refer the reader to [5, 6, 7].”
Newly Defined Conformable Derivatives (2015, background): “This includes and generalizes the conformable time scales derivative introduced in [9].”