Fractional relaxation with time-varying coefficient

Garra, Roberto; Giusti, Andrea; Mainardi, Francesco; Pagnini, Gianni

doi:10.2478/s13540-014-0178-0

Cited by 54 publications

(49 citation statements)

References 21 publications

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“…Here,

^{C} {((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ}

is called a regularized Caputo‐like counterpart hyper‐Bessel operator of order 0<β<1. From Garra et al, we have the following formula:

^{C} {((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ} sans-serifv false(sans-serift false) = {((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ} sans-serifv false(sans-serift false) - \frac{sans-serifv false(0 false) t^{- normalβ false(1 - normalα false)}}{{false(1 - normalα false)}^{- normalβ} normalΓ false(1 - normalβ false)},

where

{((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ}

is called the hyper‐Bessel operator , which was given by Dimovski . Since , we see that the study of comes from the definition of hyper‐Bessel operator.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Tuan

Huynh

Băleanu

et al. 2019

Math Methods in App Sciences

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In this paper, we consider an inverse problem of recovering the initial value for a generalization of time‐fractional diffusion equation, where the time derivative is replaced by a regularized hyper‐Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill‐posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

show abstract

“…Here,

^{C} {((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ}

is called a regularized Caputo‐like counterpart hyper‐Bessel operator of order 0<β<1. From Garra et al, we have the following formula:

^{C} {((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ} sans-serifv false(sans-serift false) = {((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ} sans-serifv false(sans-serift false) - \frac{sans-serifv false(0 false) t^{- normalβ false(1 - normalα false)}}{{false(1 - normalα false)}^{- normalβ} normalΓ false(1 - normalβ false)},

where

{((), t^{α} \frac{\partial}{\partial sans-serift})}^{normalβ}

is called the hyper‐Bessel operator , which was given by Dimovski . Since , we see that the study of comes from the definition of hyper‐Bessel operator.…”

Section: Introductionmentioning

confidence: 99%

“…Here, C ( t α t ) β is called a regularized Caputo-like counterpart hyper-Bessel operator of order 0 < β < 1. From Garra et al, 1 we have the following formula:…”

Section: Introductionmentioning

confidence: 99%

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Tuan

Huynh

Băleanu

et al. 2019

Math Methods in App Sciences

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“…The second particular case is associated with the well-known "stretched" exponential function and is given by With an analogous generalization purpose, Garra et al [14], using the general theory of the hyper-Bessel operator [3], solved a generalized relaxation equation which was called fractional differential equation with time-variant coefficient.…”

Section: Fractional Kinetic Equationsmentioning

confidence: 99%

“…The mathematical investigation of relaxation processes in dielectrics has been conducted with the use of fractional calculus tools [11][12][13][14]. The properties of dielectric materials are usually represented by the empirical susceptibility functions as they have been initially modeled by Debye (D), then by the Cole brothers (C-C) and, later, in the works of Cole-Davidson (C-D) and Havriliak-Negami (H-N).…”

Section: Introductionmentioning

confidence: 99%

Relaxation Equations: Fractional Models

E¹,

Rosa

2015

J Phys Math

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The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order γ, 0<γ≤1. In order to solve the generalized equations, the Laplace transform methodology is introduced and the corresponding solutions are then presented, in terms of Mittag-Leffler functions. In the case in which the derivative's order is γ = 1, the traditional relaxation functions are recovered. Finally, we presented some 2D graphs of these function.

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“…[16][17][18][19] Also, they are used in modeling of chemical processes, control theory, electromagnetism, thermodynamics, and many other problems in physics, engineering, and the various works cited therein. [20][21][22][23] The spring-mass-viscodamper system is a dynamic system that provides a stage of minimal complexity to essentially study all the phenomena of mechanical vibrations. In this context, some researches concerning classical mechanics introduced fractional-order derivatives, for example, Kutay and Ozaktas 24 investigated the relationship of the Fourier transform of fractional order to harmonic oscillation.…”

Section: Introductionmentioning

confidence: 99%

Experimental evaluation of viscous damping coefficient in the fractional underdamped oscillator

Escalante-Martínez

Gómez-Aguilar

Calderón-Ramón

et al. 2016

Advances in Mechanical Engineering

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In this article, we obtain the viscous damping coefficient b theoretically and experimentally in the spring-mass-viscodamper system. The calculation is performed to obtain the quasi-period t. The influence of the viscosity of the fluid and the damping coefficient is analyzed using three fluids, water, edible oil, and gasoline engine oil SAE 10W-40. These processes exhibit temporal fractality and non-local behaviors. Our general non-local damping model incorporates fractional derivatives of Caputo type in the range (0, 1, and the viscous damping coefficients are determined in terms of the inverse Mittag-Leffler function. The classical models are recovered when the order of the fractional derivatives is equal to 1.

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Fractional relaxation with time-varying coefficient

Cited by 54 publications

References 21 publications

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Relaxation Equations: Fractional Models

Experimental evaluation of viscous damping coefficient in the fractional underdamped oscillator

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