Fractional derivatives with no-index law property: Application to chaos and statistics

Atangana, Abdon; Gómez‐Aguilar, J.F.

doi:10.1016/j.chaos.2018.07.033

Cited by 307 publications

(103 citation statements)

References 25 publications

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“…Parallel to increasing research involving the Caputo-Fabrizio fractional operator [3,12,45,135,136] (see the analysis in [73]), there are critical articles against it [63,113,142] written from the position of the classical fractional calculus with singular power-law memory kernel. Since these critiques are purely mathematical exercises [63,142] and the examples are in the narrow field of signal processing [113] and to some extent touch some formal rheological relationship [63], they, generally, do not show the physical origins of exponential memory kernels and stress the attention on the mathematical constructions only.…”

Section: Caputo-fabrizio Fractional Operator: Emerging Criticismmentioning

confidence: 99%

Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

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This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

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Section: Caputo-fabrizio Fractional Operator: Emerging Criticismmentioning

confidence: 99%

Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

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“…The results of equations (15)- (17) are especially important in the theory of the AB model. They demonstrate that the AB differintegral operators do not obey an index law, which fact has been discussed at more length in papers such as [4,6,7]. It is also important to be aware, contrary to some claims seen in the literature, that the AB differintegral operators are commutative.…”

Section: Atangana-baleanu Fractional Calculusmentioning

confidence: 79%

“…The analytic continuation of the AB integral defined by (7) is much simpler to manage than that of the AB derivatives, since this time we can simply consider RL integrals rather than Mittag-Leffler series.…”

Section: Ab Integrals and The Iterated Ab Modelmentioning

confidence: 99%

A complex analysis approach to Atangana–Baleanu fractional calculus

Fernandez¹

2019

Math Methods in App Sciences

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The standard definition for the Atangana-Baleanu fractional derivative involves an integral transform with a Mittag-Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana-Baleanu fractional integral and for iterated Atangana-Baleanu fractional differintegrals.

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“…However, this apparent limitation allows to describe more appropriate real world problems. () Atangana and Gómez proposed the work entitled “Decolonization of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena.” In this work, several examples of noncommutative and nonassociative problems were presented. Also, the authors justify why the fractional derivatives with nonsingular kernel are needed to describe those physical problems.…”

Section: Resultsmentioning

confidence: 99%

Determination of supercapacitor parameters based on fractional differential equations

Hidalgo‐Reyes

Gómez‐Aguilar

Escobar-Jiménez

et al. 2019

Circuit Theory & Apps

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In this paper, we estimate the parameter values of a fractional-order model of supercapacitors involving fractional derivatives of Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu and fractional conformable derivative in the Liouville-Caputo sense. We present the exact solution of the considered model using the properties of the Laplace transform operator together with the convolution theorem. They developed numerical simulations using each one of the fractional derivatives; the results were compared graphically with experimental data obtained from different supercapacitors using standard laboratory equipment. The nonlocal parameters involved in the equivalent electrical circuit for the supercapacitor model are recalculated for each fractional derivative using a particle swarm optimization algorithm for generating optimal solutions. KEYWORDS exponential decay-law, fractional calculus, fractional conformable derivative, Mittag-Leffler function, power-law, supercapacitor model Int J Circ Theor Appl. 2019;47:1225-1253.wileyonlinelibrary.com/journal/cta

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Fractional derivatives with no-index law property: Application to chaos and statistics

Cited by 307 publications

References 25 publications

Response functions in linear viscoelastic constitutive equations and related fractional operators

Response functions in linear viscoelastic constitutive equations and related fractional operators

A complex analysis approach to Atangana–Baleanu fractional calculus

Determination of supercapacitor parameters based on fractional differential equations

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