IRIS Università degli Studi di Bolognahttps://cris.unibo.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Thu, 26 Nov 2020 15:04:08 GMT2020-11-26T15:04:08Z10681Becker and Lomnitz rheological models: A comparisonhttp://hdl.handle.net/11585/125294Titolo: Becker and Lomnitz rheological models: A comparison
Abstract: The viscoelastic material functions for the Becker and the Lomnitz rheological models, sometimes employed to describe the transient flow of rocks, are studied and compared. Their creep functions, which are known in a closed form, share a similar time dependence and asymptotic behavior. This is also found for the relaxation
functions, obtained by solving numerically a Volterra equation of the second kind. We show that the two rheologies constitute a clear example of broadly similar creep and relaxation patterns associated with neatly distinct retardation spectra, for which analytical expressions are available.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11585/1252942012-01-01T00:00:00ZAn historical perspective on fractional calculus in linear viscoelasticityhttp://hdl.handle.net/11585/128287Titolo: An historical perspective on fractional calculus in linear viscoelasticity
Abstract: The article provides an historical survey of 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 Scienific-Singapore, 2010.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11585/1282872012-01-01T00:00:00ZA Renewal process of Mittag-Leffler typehttp://hdl.handle.net/11585/17812Titolo: A Renewal process of Mittag-Leffler type
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/178122004-01-01T00:00:00ZFox H functions in fractional diffusionhttp://hdl.handle.net/11585/17712Titolo: Fox H functions in fractional diffusion
Abstract: The H functions, introduced by Fox in 1961,
are special functions of a very general nature,
which allow one to treat several phenomena including
anomalous diffusion in a unified and elegant framework.
In this paper we express the fundamental solutions of
the Cauchy problem for the space-time fractional diffusion
equation in terms of proper Fox H functions, based on
their Mellin-Barnes integral representations.
We pay attention to the particular cases of space-fractional,
time-fractional and neutral-fractional diffusion.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/11585/177122005-01-01T00:00:00ZA fractional generalization of the Poisson processeshttp://hdl.handle.net/11585/18121Titolo: A fractional generalization of the Poisson processes
Abstract: It is our intention to provide via fractional calculus
a generalization of the pure and
compound Poisson processes,
which are known to play a fundamental role in renewal theory,
without and with reward, respectively.
We first recall the basic renewal theory including its
fundamental concepts like waiting time between events,
the survival probability, the counting function.
If the waiting time is exponentially distributed we
have a Poisson process, which is Markovian.
However, other waiting time distributions are also relevant
in applications, in particular such ones with a fat tail
caused by a power law decay of its density.
In this context we analyze a non-Markovian
renewal process with a waiting time distribution
described by the Mittag-Leffler function.
This distribution, containing the exponential as
particular case, is shown to play a fundamental role
in the infinite thinning procedure of a generic renewal
process governed by a power-asymptotic waiting time.
We then consider the renewal theory with reward
that implies a random walk subordinated to a renewal process.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/181212004-01-01T00:00:00ZThe functions of the Wright-type in fractional calculushttp://hdl.handle.net/11585/130458Titolo: The functions of the Wright-type in fractional calculus
Abstract: Here we provide a survey of the high transcendental functions
related to the Wright special function.
Like the functions of the Mittag-Leffler type, the functions of
the Wright type are known to play fundamental roles in various
applications of the fractional calculus. This is mainly due to the
fact that they are interrelated with the Mittag-Leffler functions
through a Laplace transformation.
We start providing the definitions in the complex plane for the
general Wright function and for two special cases that we call
auxiliary functions. Then we devote particular attention to the
auxiliary functions in the real field, because they admit a probabilisticinterpretation related to the fundamental solutions of
certain evolution equations of fractional order. These equations
are fundamental to understand phenomena of anomalous diffusion
or intermediate between diffusion and wave propagation.
At the end we add some historical and bibliographical notes.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/11585/1304582010-01-01T00:00:00ZGeneralized fractional master equation for self-similar stochastic processes modelling anomalous diffusionhttp://hdl.handle.net/11585/128289Titolo: Generalized fractional master equation for self-similar stochastic processes modelling anomalous diffusion
Abstract: The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and non-stationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erd´elyi-Kober fractional diffusion, that describes
the evolution of the marginal distribution of the so-called generalized grey Brownian motion.
This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and
depends on two real parameters. The class includes the fractional Brownian motion, the time fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function known also as Mainardi function emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11585/1282892012-01-01T00:00:00ZThe Role of Salvatore Pincherle in the Development of Fractional Calculushttp://hdl.handle.net/11585/128286Titolo: The Role of Salvatore Pincherle in the Development of Fractional Calculus
Abstract: We revisit two contributions by Salvatore Pincherle (Professor of Mathematics at the University of Bologna from 1880 to 1928) published (in Italian) in 1888 and 1902 in order to point out his possible role in the development of Fractional Calculus.
Fractional Calculus is that branch of mathematical analysis
dealing with pseudo-differential operators interpreted as integrals and derivatives of non-integer order. Even if the former contribution (published in two notes on Accademia dei Lincei) on generalized hypergeomtric functions does not concern Fractional Calculus it contains the first example in the literature of the use of the so
called Mellin–Barnes integrals. These integrals will be proved to be a fundamental task to deal with all higher transcendental functions including the Meijer and Fox functions introduced much later. In particular, the solutions of differential equations of fractional order are suited to be expressed in terms of these integrals. In the second paper (published on Accademia delle Scienze di Bologna), the author is interested to insert in the framework of his operational theory the notion of derivative of non integer order that appeared in those times not yet well established. Unfortunately,
Pincherle’s foundation of Fractional Calculus seems still ignored.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11585/1282862012-01-01T00:00:00ZCreep, relaxation and viscosity properties for basic fractional models in rheology”,http://hdl.handle.net/11585/102583Titolo: Creep, relaxation and viscosity properties for basic fractional models in rheology”,
Abstract: The purpose of this paper is twofold: from one side we attempt to
provide a general introduction to the viscoelastic models constructed via fractional calculus and from the other side we intend to analyze the basic fractional models as far as their creep, relaxation and viscosity properties are considered. The basic
models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as Kelvin-Voigt, Maxwell, Zener, Ant-Zener and Burgers. For each fractional model we provide plots of the creep compliance, relaxation modulus and efficient
viscosity in non dimensional form in terms of a suitable time scale for different values of the order of fractional derivative. We also discuss the role of the order of fractional derivative in modifying the properties of the classical models.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/11585/1025832011-01-01T00:00:00ZUncoupled continuous-time random walks:
Solution and limiting behavior of the master equationhttp://hdl.handle.net/11585/18102Titolo: Uncoupled continuous-time random walks:
Solution and limiting behavior of the master equation
Abstract: A detailed study is presented for a large class of
uncoupled continuous-time random walks (CTRWs).
The master equation is solved for the Mittag-Leffler
survival probability.
The properly scaled diffusive limit of the master equation
is taken and its relation with the fractional diffusion
equation is discussed.
Finally, some common objections found in the
literature are thoroughly reviewed.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/181022004-01-01T00:00:00Z