251 research outputs found
An Historical Perspective on Fractional Calculus in Linear Viscoelasticity
The article provides an historical survey of the early contributions on the
applications of fractional calculus in linear viscoelasticty. The period under
examination covers four decades, since 1930's up to 1970's and authors are from
both Western and Eastern countries. References to more recent contributions may
be found in the bibliography of the author's book. This paper reproduces, with
Publisher's permission, Section 3.5 of the book: F. Mainardi, Fractional
Calculus and Waves in Linear Viscoelasticity, Imperial College Press - London
and World Scienti?c - Singapore, 2010.Comment: 6 page
Fractional Calculus in Wave Propagation Problems
Fractional calculus, in allowing integrals and derivatives of any positive
order (the term "fractional" kept only for historical reasons), can be
considered a branch of mathematical physics which mainly deals with
integro-differential equations, where integrals are of convolution form with
weakly singular kernels of power law type. In recent decades fractional
calculus has won more and more interest in applications in several fields of
applied sciences. In this lecture we devote our attention to wave propagation
problems in linear viscoelastic media. Our purpose is to outline the role of
fractional calculus in providing simplest evolution processes which are
intermediate between diffusion and wave propagation. The present treatment
mainly reflects the research activity and style of the author in the related
scientific areas during the last decades.Comment: 33 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1008.134
A note on the equivalence of fractional relaxation equations to differential equations with varying coefficients
In this note we show how a initial value problem for a relaxation process
governed by a differential equation of non-integer order with a constant
coefficient may be equivalent to that of a differential equation of the first
order with a varying coefficient. This equivalence is shown for the simple
fractional relaxation equation that points out the relevance of the
Mittag-Leffler function in fractional calculus. This simple argument may lead
to the equivalence of more general processes governed by evolution equations of
fractional order with constant coefficients to processes governed by
differential equations of integer order but with varying coefficients. Our main
motivation is to solicit the researchers to extend this approach to other areas
of applied science in order to have a more deep knowledge of certain phenomena,
both deterministic and stochastic ones, nowadays investigated with the
techniques of the fractional calculus.Comment: 6 pqages 4 figure
Fractional Cable Model for Signal Conduction in Spiny Neuronal Dendrites
The cable model is widely used in several fields of science to describe the
propagation of signals. A relevant medical and biological example is the
anomalous subdiffusion in spiny neuronal dendrites observed in several studies
of the last decade. Anomalous subdiffusion can be modelled in several ways
introducing some fractional component into the classical cable model. The
Chauchy problem associated to these kind of models has been investigated by
many authors, but up to our knowledge an explicit solution for the signalling
problem has not yet been published. Here we propose how this solution can be
derived applying the generalized convolution theorem (known as Efros theorem)
for Laplace transforms. The fractional cable model considered in this paper is
defined by replacing the first order time derivative with a fractional
derivative of order of Caputo type. The signalling problem is
solved for any input function applied to the accessible end of a semi-infinite
cable, which satisfies the requirements of the Efros theorem. The solutions
corresponding to the simple cases of impulsive and step inputs are explicitly
calculated in integral form containing Wright functions. Thanks to the
variability of the parameter , the corresponding solutions are expected
to adapt to the qualitative behaviour of the membrane potential observed in
experiments better than in the standard case .Comment: arXiv admin note: substantial text overlap with arXiv:1702.0533
On the fractional Poisson process and the discretized stable subordinator
The fractional Poisson process and the Wright process (as discretization of
the stable subordinator) along with their diffusion limits play eminent roles
in theory and simulation of fractional diffusion processes. Here we have
analyzed these two processes, concretely the corresponding counting number and
Erlang processes, the latter being the processes inverse to the former.
Furthermore we have obtained the diffusion limits of all these processes by
well-scaled refinement of waiting times and jumpsComment: 30 pages, 4 figures. A preliminary version of this paper was an
invited talk given by R. Gorenflo at the Conference ICMS2011, held at the
International Centre of Mathematical Sciences, Pala-Kerala (India) 3-5
January 2011, devoted to Prof Mathai on the occasion of his 75 birthda
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