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Michele Caputo

Bio: Michele Caputo is an academic researcher from Texas A&M University. The author has contributed to research in topics: Fractional calculus & Constitutive equation. The author has an hindex of 25, co-authored 120 publications receiving 8461 citations. Previous affiliations of Michele Caputo include Marconi University & Sapienza University of Rome.


Papers
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Journal ArticleDOI
TL;DR: In this paper, a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges has been investigated by introducing fractional derivatives in the stressstrain relation, and a rigorous proof of the formulae to be used in obtaining the analytic expression of Q is given.
Abstract: Summary Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10-2-107 cis, although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression for a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stressstrain relation. Since the aim of this research is also to contribute to elucidating the dissipating mechanism in the Earth free modes, we shall treat the dissipation in the free, purely torsional, modes of a shell. The dissipation in a plane wave will also be treated. The theory is checked with the new values determined for the Q of spheroidal free modes of the Earth in the range between 10 and 5 min integrated with the Q of Rayleigh waves in the range between 5 and 0.6 min. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an aluminium rod in the range between lo-’ and 10-3s. In both checks the theory represents the observed phenomena very satisfactorily. The time derivative which enters the stress-strain relation in both cases is of order 0.15. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. In a homogeneous isotropic elastic field the elastic properties of the substance are specified by a description of the strains and stresses in a limited portion of the field since the strains and stresses are linearly related by two parameters which describe the elastic properties of the field. If the elastic field is not homogeneous nor isotropic the properties of the field are specified in a similar manner by a larger number of parameters which also depend on the position.

3,372 citations

01 Jan 2015
TL;DR: In this article, the authors present a new definition of fractional derivative with a smooth kernel, which takes on two different representations for the temporal and spatial variable, for which it is more convenient to work with the Fourier transform.
Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

1,972 citations

Journal ArticleDOI
01 Apr 1971

726 citations

Journal ArticleDOI
TL;DR: The model of dissipation based on memory introduced by Caputo is generalized and checked with experimental dissipation curves of various materials.
Abstract: The model of dissipation based on memory introduced by Caputo is generalized and checked with experimental dissipation curves of various materials

725 citations

Journal ArticleDOI
TL;DR: In this paper, the analytic expression of a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges was obtained by introducing fractional derivatives in the stress strain relation.
Abstract: Laboratory experiments and field observations indicate that tlie Q of many non ferromagnetic inorganic solids is almost frequency independent in the range 10' to 10~2 cps; although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression of a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stress strain relation. Since the aim of this research is to also contribute to elucidating the dissipating mechanism in the earth free modes, we shall treat the cases of dissipation in the free purely torsional modes of a shell and the purely radial vibration of a solid sphere. The theory is checked with the new values determined for the Q of the spheroidal free modes of the earth in the range between 10 and 5 minutes integrated with the Q of the Railegh waves in the range between 5 and 0.6 minutes. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an alluminimi rod, in the range between 10-5 and 10-3 seconds. In both clicks the theory represents the observed phenomena very satisfactory.

515 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, the authors discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order, and investigate the dependence of the solution on the order of the differential equation and on the initial condition.

3,047 citations

Journal ArticleDOI
TL;DR: In this article, a fractional-order PI/sup/spl lambda/D/sup /spl mu/controller with fractionalorder integrator and fractional order differentiator is proposed.
Abstract: Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI/sup /spl lambda//D/sup /spl mu//-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller for both open- and closed-loops. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI/sup /spl lambda//D/sup /spl mu//-controllers is given.

2,479 citations

Journal ArticleDOI
TL;DR: In this article, a new fractional derivative with non-local and no-singular kernel was proposed and applied to solve the fractional heat transfer model, and some useful properties of the new derivative were presented.
Abstract: In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.

2,364 citations

Dissertation
01 Oct 1948
TL;DR: In this article, it was shown that a metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.
Abstract: IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

2,042 citations

01 Jan 2015
TL;DR: In this article, the authors present a new definition of fractional derivative with a smooth kernel, which takes on two different representations for the temporal and spatial variable, for which it is more convenient to work with the Fourier transform.
Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

1,972 citations