Generalized Langevin Equation and the Prabhakar Derivative

Sandev, Trifce

doi:10.3390/math5040066

Cited by 64 publications

(46 citation statements)

References 42 publications

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“…We recall that the Mittag-Leffler function is known to be very significant in fractional calculus [33,34,49,45], and its properties have been exhaustively studied in this connection [22,23]. Prabhakar operators in particular have already found applications in diffusion, relaxation, and stochastic processes [44,17,18,41].…”

Section: Introductionmentioning

confidence: 99%

Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

Fernandez

Băleanu

Srivástava

2019

Communications in Nonlinear Science and Numerical Simulation

122

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We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann-Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.

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Section: Introductionmentioning

confidence: 99%

Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

Fernandez

Băleanu

Srivástava

2019

Communications in Nonlinear Science and Numerical Simulation

122

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“…Besides, it is worth noting that fractional calculus plays a central role also in many other fields of science, see e.g. [8][9][10][11] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Scott-Blair models with time-varying viscosity

Colombaro

Garra

Giusti

et al. 2018

Applied Mathematics Letters

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In a recent paper, Zhou et al. [12] studied the time-dependent properties of Glass Fiber Reinforced Polymers composites by employing a new rheological model with a time-dependent viscosity coefficient. This rheological model is essentially based on a generalized Scott-Blair body with a time-dependent viscosity coefficient. Motivated by this study, in this note we suggest a different generalization of the Scott-Blair model based on the application of Caputo-type fractional derivatives of a function with respect to another function. This new mathematical approach can be useful in viscoelasticity and diffusion processes in order to model systems with time-dependent features. In this paper we also provide the general solution of the creep experiment for our improved Scott-Blair model together with some explicit examples and illuminating plots.

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“…Finally, it is also worth mentioning that a study of the generalized Langevin Equation with both the Prabhakar and tempered Prabhakar kernels as memory kernels has been carried out by T. Sandev in [131].…”

Section: Stochastic Processes and Diffusionmentioning

confidence: 99%

A Practical Guide to Prabhakar Fractional Calculus

Giusti

Colombaro

Garra

et al. 2020

Fract Calc Appl Anal

146

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The Mittag-Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

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Generalized Langevin Equation and the Prabhakar Derivative

Cited by 64 publications

References 42 publications

Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

Scott-Blair models with time-varying viscosity

A Practical Guide to Prabhakar Fractional Calculus

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