24,354 research outputs found
On the Zitterbewegung Transient Regime in a Coarse-Grained Space-Time
In the present contribution, by studying a fractional version of Dirac's
equation for the electron, we show that the phenomenon of Zitterbewegung in a
coarse-grained medium exhibits a transient oscillatory behavior, rather than a
purely oscillatory regime, as it occurs in the integer case, . Our
result suggests that, in such systems, the Zitterbewegung-type term related to
a trembling motion of a quasiparticle is tamed by its complex interactions with
other particles and the medium. This can justify the difficulties in the
observation of this interesting phenomenon. The possibility that the
Zitterbewegung be accompanied by a damping factor supports the viewpoint of
particle substructures in Quantum Mechanics.Comment: 11 pages, 1 figure. This paper has been published in J. Adv. Phys. 7
(2015) 144
Fractional Order Version of the HJB Equation
We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB)
equation for fractional order dynamical systems in which a generalized
performance index is considered for the related optimal control problem. Owing
to the nonlocality of the fractional order operators, the classical HJB
equation, in the usual form, does not hold true for fractional problems.
Effectiveness of the proposed technique is illustrated through a numerical
example.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN
1555-1423, CODEN: JCNDDM. Submitted 28-June-2018; Revised 15-Sept-2018;
Accepted 28-Oct-201
A comment on some new definitions of fractional derivative
After 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.Comment: 10 pages, 1 figure, to appear in Nonlinear Dynamics, comment adde
Weyl and Marchaud derivatives: a forgotten history
In this paper we recall the contribution given by Hermann Weyl and Andr\'e
Marchaud to the notion of fractional derivative. In addition we discuss some
relationships between the fractional Laplace operator and Marchaud derivative
in the perspective to generalize these objects to different fields of the
mathematics.Comment: arXiv admin note: text overlap with arXiv:1705.00953 by other author
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