The Chronicles of Fractional Calculus

Machado, J. A. Tenreiro; Kiryakova, Virginia

doi:10.1515/fca-2017-0017

Cited by 134 publications

(61 citation statements)

References 133 publications

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“…In 1860s, Riemann and Liouville obtained now celebrated definitions of fractional operators, by extending the Cauchy integral formula. Such fractional operators have a major role in practical problems [3]. In particular, Heymans and Podlubny have shown that it is possible to attribute a physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives on the field of viscoelasticity, which is more appropriate than standard initial conditions [4].…”

Section: Introductionmentioning

confidence: 99%

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

Zouiten

Boutoulout

Torres

2018

Axioms

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Section: Introductionmentioning

confidence: 99%

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

Zouiten

Boutoulout

Torres

2018

Axioms

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“…The Cauchy-type problem for equation (31), in which the initial condition is given at the time τ = 0 by A(0), has solution that can be represented in the form…”

Section: Violation Of Semi-group Property: Non-markovian Dynamicsmentioning

confidence: 99%

“…An effective and powerful tool for describing memory effects is the fractional calculus of derivatives and integrals of non-integer orders [22,23,24,25,26]. These operators have a long history of more than three hundred years [27,28,29,30,31]. We should note that the characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order [32,33,34,35,36].…”

Section: Introductionmentioning

confidence: 99%

Dirac particle with memory: Proper time non-locality

Tarasov

2020

Physics Letters A

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A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described. PACS 45.10.Hj Perturbation and fractional calculus methods PACS 03.65.Yz Decoherence; open systems; quantum statistical methods PACS 11.90.+t Other topics in general theory of fields and particles

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“…However, upon experimentation on a low-power 415-V permanent magnet synchronous motor (PMSM) used in a high-rise elevator, it was discovered that the insulation resistance dynamics do not follow (1.4) nor (1.5). In fact, upon using fractional calculus [24,27], it was assumed that the model should appropriately follow the fractional-order transfer function (FOTF)…”

Section: Thermal Class Rating Minimum Pimentioning

confidence: 99%

Novel Polarization Index Evaluation Formula and Fractional-Order Dynamics in Electric Motor Insulation Resistance

Gonzalez¹,

Petráš

Ortigueira

2018

FCAA

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One of the common test metrics prescribed by IEEE Std 43 for testing motor insulation is the Polarization Index (P.I.) which evaluates the “goodness” of the machine’s insulation resistance by getting the ratio of the insulation resistance measured upon reaching t2 > 0 minutes (IRt2) from t1 > 0 minutes (IRt2) for t2 > t1 > 0, after applying a DC step voltage. However, such definition varies from different manufacturers and operators despite of decades of research in this area because the values of t1 and t2 remain to be uncertain. It is hypothesized in this paper that the main cause of having various P.I. definitions in literature is due to the lack of understanding of the electric motor’s dynamics at a systems level which is usually assumed to follow the dynamics of the exponential function. As a result, we introduce in this paper the fractional dynamics of an electric motor insulation resistance that could be represented by fractional-order model and where the resistance follows the property of a Mittag-Leffler function rather than an exponential function as observed on the tests done on a 415-V permanent magnet synchronous motor (PMSM). As a result, a new PMSM health measure called the Three-Point Polarization Index (3PPI) is proposed.

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The Chronicles of Fractional Calculus

Cited by 134 publications

References 133 publications

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

Dirac particle with memory: Proper time non-locality

Novel Polarization Index Evaluation Formula and Fractional-Order Dynamics in Electric Motor Insulation Resistance

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