1,721,140 research outputs found
A comment on some new definitions of fractional derivative
No full textAfter reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann–Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo and Fabrizio cannot be regarded as fractional. Besides, we also observe that the one suggested by Atangana and Baleanu is indeed fractional, but it is ultimately related to the ordinary Riemann–Liouville and Caputo fractional operators. All these statements are then further supported by a precise analysis of differential equations involving the aforementioned operators. To further strengthen our narrative, we also show that these new operators do not add any new insight to the linear theory of viscoelasticity when employed in the constitutive equation of the Scott–Blair model
On infinite order differential operators in fractional viscoelasticity
No full textIn this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the recently developed Bessel models of linear viscoelasticity that, for short times, behave like fractional Maxwell bodies of order 1/2
Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation
No full textIn this paper, after a brief review of the general theory of dispersive waves in dissipative media, we present a complete discussion of the dispersion relations for both the ordinary and the time-fractional Cattaneo-Maxwell heat equations. Consequently, we provide a complete characterization of the group and phase velocities for these two cases, together with some non-trivial remarks on the nature of wave dispersion in fractional models
A Computational Approach to Exponential-Type Variable-Order Fractional Differential Equations
Full text linkWe investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterization of such operators is performed in the Laplace domain, it is necessary to resort to accurate numerical methods to derive the corresponding behaviours in the time domain. In this regard, we develop a computational procedure to solve variable-order fractional differential equations of this novel class. Furthermore, we provide some numerical experiments to show the effectiveness of the proposed technique
On infinite series concerning zeros of Bessel functions of the first kind
No full textA relevant result independently obtained by Rayleigh and Sneddon on an identity on series involving the zeros of Bessel functions of the first kind is derived by an alternative method based on Laplace transforms. Our method leads to a Bernstein function of time, expressed by Dirichlet series, that allows us to recover the Rayleigh-Sneddon sum. We also consider another method arriving at the same result based on a relevant formula by Calogero. Moreover, we also provide an electrical example in which this sum results to be extremely useful in order to recover the analytical expression for the response of the system to a certain external input
A dynamic viscoelastic analogy for fluid-filled elastic tubes
No full textIn this paper we evaluate the dynamic effects of the fluid viscosity for fluid filled elastic tubes in the framework of a linear uni-axial theory. Because of the linear approximation, the effects on the fluid inside the elastic tube are taken into account according to the Womersley theory for a pulsatile flow in a rigid tube. The evolution equations for the response variables are derived by means of the Laplace transform technique and they all turn out to be the very same integro-differential equation of the convolution type. This equation has the same structure as the one describing uni-axial waves in linear viscoelastic solids characterized by a relaxation modulus or by a creep compliance. In our case, the analogy is connected with a peculiar viscoelastic solid which exhibits creep properties similar to those of a fractional Maxwell model (of order 1Â /Â 2) for short times, and of a standard Maxwell model for long times. The present analysis could find applications in biophysics concerning the propagation of pressure waves within large arteries
The role of collapsed matter in the decay of black holes
Full text linkWe try to shed some light on the role of matter in the final stages of black hole evaporation from the fundamental frameworks of classicalization and the black-to-white hole bouncing scenario. Despite being based on very different grounds, these two approaches attempt at going beyond the background field method and treat black holes as fully quantum systems rather than considering quantum field theory on the corresponding classical manifolds. They also lead to the common prediction that the semiclassical description of black hole evaporation should break down and the system be disrupted by internal quantum pressure, but they both arrive at this conclusion neglecting the matter that formed the black hole. We instead estimate this pressure from the bootstrapped description of black holes, which allows us to express the total Arnowitt–Deser–Misner mass in terms of the baryonic mass still present inside the black hole. We conclude that, although these two scenarios provide qualitatively similar predictions for the final stages, the corpuscular model does not seem to suggest any sizeable deviation from the semiclassical time scale at which the disruption should occur, unlike the black-to- white hole bouncing scenario. This, in turn, makes the phenomenology of corpuscular black holes more subtle from an astrophysical perspective
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