Search results for "Fractional Calculus"
showing 10 items of 128 documents
Applications and Implications of Fractional Dynamics for Dielectric Relaxation
2012
This article summarizes briefly the presentation given by the author at the NATO Advanced Research Workshop on “Broadband Dielectric Spectroscopy and its Advanced Technological Applications”, held in Perpignan, France, in September 2011. The purpose of the invited presentation at the workshop was to review and summarize the basic theory of fractional dynamics (Hilfer, Phys Rev E 48:2466, 1993; Hilfer and Anton, Phys Rev E Rapid Commun 51:R848, 1995; Hilfer, Fractals 3(1):211, 1995; Hilfer, Chaos Solitons Fractals 5:1475, 1995; Hilfer, Fractals 3:549, 1995; Hilfer, Physica A 221:89, 1995; Hilfer, On fractional diffusion and its relation with continuous time random walks. In: Pekalski et al. …
Fractional-Order Thermal Energy Transport for Small-Scale Engineering Devices
2014
Fractional-order thermodynamics has proved to be an efficient tool to describe several small-scale and/or high-frequency thermodynamic processes, as shown in many engineering and physics applications. The main idea beyond fractional-order physics and engineering relies on replacing the integer-order operators of classical differential calculus with their real-order counterparts. In this study, the authors aim to extend a recently proposed physical picture of fractional-order thermodynamics to a generic 3D rigid heat conductor where the thermal energy transfer is due to two phenomena: a short-range heat flux ruled by stationary and nonstationary transport equations, and a long-range thermal …
Non-linear viscoelastic behavior of polymer melts interpreted by fractional viscoelastic model
2016
Very recently, researchers dealing with constitutive law pertinent viscoelastic materials put forward the successful idea to introduce viscoelastic laws embedded with fractional calculus, relating the stress function to a real order derivative of the strain function. The latter consideration leads to represent both, relaxation and creep functions, through a power law function. In literature there are many papers in which the best fitting of the peculiar viscoelastic functions using a fractional model is performed. However there are not present studies about best fitting of relaxation function and/or creep function of materials that exhibit a non-linear viscoelastic behavior, as polymer melt…
Self-referenced phase reconstruction proposal of Ghz bandwidth non-periodical optical pulses by in-fiber semi-differintegration
2011
Abstract We propose two new techniques able to retrieve the phase profile of a given temporal optical pulse based on the use of in-fiber semi-differintegral operators, where by semi-differintegration we mean either a 0.5th-order differentiation or integration. In both cases, the signal's temporal phase can be obtained by simple dividing two temporal intensity profiles, namely the intensities of the input and output pulses of a spectrally shifted semi-differintegral operator. In both cases, we obtained simple analytical expressions for the phase profile. The techniques are self-referenced and well-suited for real-time applications. We numerically prove the viability of these proposals.
Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity
2022
AbstractWe study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under “fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index $$\alpha$$ α . We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Laplacian (corresponding to Lévy index $$\alpha =2$$ α = 2 ), the soliton is unstable, even infinitesimal diffe…
Elastic waves propagation in 1D fractional non-local continuum
2008
Aim of this paper is the study of waves propagation in a fractional, non-local 1D elastic continuum. The non-local effects are modeled introducing long-range central body interactions applied to the centroids of the infinitesimal volume elements of the continuum. These non-local interactions are proportional to a proper attenuation function and to the relative displacements between non-adjacent elements. It is shown that, assuming a power-law attenuation function, the governing equation of the elastic waves in the unbounded domain, is ruled by a Marchaud-type fractional differential equation. Wave propagation in bounded domain instead involves only the integral part of the Marchaud fraction…
Electrical analogous in viscoelasticity
2014
In this paper, electrical analogous models of fractional hereditary materials are introduced. Based on recent works by the authors, mechanical models of materials viscoelasticity behavior are firstly approached by using fractional mathematical operators. Viscoelastic models have elastic and viscous components which are obtained by combining springs and dashpots. Various arrangements of these elements can be used, and all of these viscoelastic models can be equivalently modeled as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed models are validated by using modal analysis. Moreover, a comparison with numerical expe…
Fractional-order theory of heat transport in rigid bodies
2014
Abstract The non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53] , Mongiovi and Zingales, 2013 [54] ) that thermal energy transport is due to two phenomena: ( i ) A short-range heat flux ruled by a local transport equation; ( ii ) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law …
Innovative modeling of Tuned Liquid Column Damper motion
2015
Abstract In this paper a new model for the liquid motion within a Tuned Liquid Column Damper (TLCD) device is developed, based on the mathematical tool of fractional calculus. Although the increasing use of these devices for structural vibration control, it is shown that existing model does not always lead to accurate prediction of the liquid motion. A better model is then needed for accurate simulation of the behavior of TLCD systems. As regards, it has been demonstrated how correctly including the first linear liquid sloshing mode, through the equivalent mechanical analogy well established in literature, produces numerical results that highly match the corresponding experimental ones. Sin…
Fractional visco-elastic systems under normal white noise
2011
In this paper an original method is presented to compute the stochastic response of singledegree- of-freedom structural systems with viscoelastic fractional damping. The key-idea stems from observing that, based on a few manipulations involving an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion can be reverted to a coupled linear system involving additional degrees of freedom, the number of which depends on the discretization adopted for the fractional derivative operator. The method applies for fractional damping of arbitrary order a (0 < α < 1). For most common input correlation functions, including a Gaussian white noise, …