A Truly Conformable Calculus on Time Scales

Bayour, Benaoumeur; Hammoudi, Ahmed; Torres, Delfim F. M.

doi:10.48550/arxiv.1705.08928

Cited by 2 publications

(3 citation statements)

References 8 publications

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“…The theory of fractional differential equations on time scales, specifically the questions of existence, uniqueness of solutions and their stability, is a recent research topic of great importance, see [11,28] and references therein. Here, we studied deterministic fractional operators on time scales with Caputo-Fabrizio type kernels.…”

Section: Discussionmentioning

confidence: 99%

“…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. Existence and uniqueness results for fractional initial value problems on arbitrary time scales are investigated in [15,32] and the so-called conformable case is studied in [11,16]. Ortigueira et al introduced fractional derivatives on arbitrary time scales through convolution [30].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Bahaa

Torres

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann-Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann-Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.

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A Truly Conformable Calculus on Time Scales

Cited by 2 publications

References 8 publications

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

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

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

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