Complex-Valued Fractional Derivatives on Time Scales

Bayour, Benaoumeur; Torres, Delfim F. M.

doi:10.1007/978-3-319-32857-7_8

Cited by 10 publications

(8 citation statements)

References 12 publications

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“…It is clear that if λ = 1 and p(t) = t, then the new derivative coincides with the standard Hilger derivative (i.e., Definition 2 reduces to Definition 1). Our first result shows, in particular, that for T = R and λ = 1, we obtain from Definition 2 the structural derivative (2). (…”

Section: Structural Derivatives On Time Scalesmentioning

confidence: 70%

Structural derivatives on time scales

Bayour¹,

Torres²

2018

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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We introduce the notion of structural derivative on time scales. The new operator of differentiation unifies the concepts of fractal and fractional order derivative and is motivated by lack of classical differentiability of some self-similar functions. Some properties of the new operator are proved and illustrated with examples. words and phrases. Hausdorff derivative of a function with respect to a fractal measure; structural and fractal derivatives; self-similarity; time scales; Hilger derivative of non-integer order.This paper is in final form and no version of it will be submitted for publication elsewhere.

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Section: Structural Derivatives On Time Scalesmentioning

confidence: 70%

Structural derivatives on time scales

Bayour¹,

Torres²

2018

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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“…The study of fractional calculus on time scales was initiated with the papers [10,11,12] and is now under strong development: see, e.g., [13,14,15,25,32]. Here, inspired by the results of [3,8], we introduce new nabla fractional operators of variable order on isolated time scales.…”

Section: Fractional Sums and Differences Of Variable Ordermentioning

confidence: 99%

Variable Order Mittag–Leffler Fractional Operators on Isolated Time Scales and Application to the Calculus of Variations

Abdeljawad

Mert

Torres

2019

Studies in Systems, Decision and Control

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We introduce new fractional operators of variable order on isolated time scales with Mittag-Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main results give fractional integration by parts formulas and necessary optimality conditions of Euler-Lagrange type.Keywords: fractional calculus on isolated time scales; variable order operators with Mittag-Leffler kernels; fractional sums and differences of variable order; summation by parts; variational principles on isolated time scales. MSC 2010: 26A33; 26E70; 49K05. * This is a preprint of a paper whose final and definite form is with Springer, as a chapter book.

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“…The first local notion of fractional derivative for functions defined on completely arbitrary time scales seems to have been introduced in 2015 [13]. Such results were then extended to the nabla and nonsymmetric cases in [14], while the possibility of a fractional derivative to be a complex number is addressed in [12]. In [7,8,9], the authors study the existence of solutions for fractional differential equations including the Caputo-Fabrizio derivative, that is, they concentrate on systems with continuous time.…”

Section: Introductionmentioning

confidence: 99%

“…12 (See[31]). Given a time scale T unbounded from above, we denoteS C (T) := T) := {λ ∈ R : ∀T ∈ T ∃t ∈ T, t > T : 1 + µ(t)λ = 0} .The set of exponential stability for the time scale T is then defined by S(T) := S C (T) ∪ S R (T).…”

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Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales

Mozyrska

Torres

Wyrwas

2019

Nonlinear Analysis: Hybrid Systems

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Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly continuous and partly discrete, i.e., hybrid time scales, one gets new fractional operators. We concentrate on the behavior of solutions to initial value problems with the Caputo-Fabrizio fractional delta derivative on an arbitrary time scale. In particular, the exponential stability of linear systems is studied. A necessary and sufficient condition for the exponential stability of linear systems with the Caputo-Fabrizio fractional delta derivative on time scales is presented. By considering a suitable fractional dynamic equation and the Laplace transform on time scales, we also propose a proper definition of Caputo-Fabrizio fractional integral on time scales. Finally, by using the Banach fixed point theorem, we prove existence and uniqueness of solution to a nonlinear initial value problem with the Caputo-Fabrizio fractional delta derivative on time scales.

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Complex-Valued Fractional Derivatives on Time Scales

Abstract: We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.

Cited by 10 publications

References 12 publications

Structural derivatives on time scales

Structural derivatives on time scales

Variable Order Mittag–Leffler Fractional Operators on Isolated Time Scales and Application to the Calculus of Variations

Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales

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