Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets

Abstract

Abstract. We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated, and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors.

Complex-Valued Fractional Derivatives on Time Scales

“…Our approach is, however, different from the ones available in the literature [2,3,4,5]. In particular, while in [2,3,4,5] the fractional derivative at a point is always a real number, here, in contrast, the fractional derivative at a point is, in general, a complex number. For example, the derivative of order α ∈ (0, 1] of the square function t 2 is always given by t α + (σ (t)) α , where σ (t) is the forward jump operator of the time scale, which is in general a complex number (e.g., for α = 1/2 and t < 0) and a generalization of the Hilger derivative (t 2 ) ∆ = t + σ (t).…”

“…For the definition of Gamma function on an arbitrary time scale T see [6]. Similarly to [2,3], here we introduce a new notion of fractional derivative on an arbitrary time scale T that does not involve Gamma functions. Our approach is, however, different from the ones available in the literature [2,3,4,5].…”

“…Similarly to [2,3], here we introduce a new notion of fractional derivative on an arbitrary time scale T that does not involve Gamma functions. Our approach is, however, different from the ones available in the literature [2,3,4,5]. In particular, while in [2,3,4,5] the fractional derivative at a point is always a real number, here, in contrast, the fractional derivative at a point is, in general, a complex number.…”

Time-Fractional Optimal Control of Initial Value Problems on Time Scales

“…We adopt a recent notion of fractional derivative on time scales introduced in [24], which is based on the notion of fractional integral on time scales T. This is in contrast with [22,23,25], where first a notion of fractional differentiation on time scales is introduced and only after that, with the help of such concept, the fraction integral is defined. The classical gamma and beta functions are used.…”

“…Another approach originate from the inverse Laplace transform on time scales [18]. After such pioneer work, the study of fractional calculus on time scales developed in a popular research subject: see [20,22,23,25,42] and the more recent references [2,19,21,38,41].…”