Papers by Francesco Mainardi
arXiv (Cornell University), Apr 17, 2010
In the present review we survey the properties of a transcendental function of the Wright type, n... more In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as timefractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the M-Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast type.
Computers & Mathematics with Applications, 2017
In seismic forward modeling and fluid-flow simulations, sources and initial conditions are two ap... more In seismic forward modeling and fluid-flow simulations, sources and initial conditions are two approaches to initiate the perturbation of the medium in order to compute synthetic seismograms and pore-pressure maps of a reservoir, respectively. Assuming delta functions in time and space, source and initial particle velocity are equivalent in the first case (wave equation), while in the second case (diffusion equation) source and diffusion field are equivalent. The differential equation based on fractional derivatives unifies both cases but those equivalences break down when the order of the derivative is not a natural number. A simulation example, based on the fractional wave equation, illustrates the implementation of the source with a band-limited time history on a numerical mesh. In this case, the implementation of the initial condition requires a numerical calculation since the medium is heterogeneous. Body forces and stress sources (e.g., earthquakes) can easily be related in uniform media. A 1D example shows such a relation.
Mathematicians in Bologna 1861–1960, 2011
We revisit two contributions by Salvatore Pincherle (Professor of Mathematics at the University o... more 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.
Physica A: Statistical Mechanics and its Applications, 2002
We present a variety of models of random walk, discrete in space and time, suitable for simulatin... more We present a variety of models of random walk, discrete in space and time, suitable for simulating random variables whose probability density obeys a space-time fractional di usion equation.
Springer Monographs in Mathematics, 2014
Zeitschrift für Analysis und ihre Anwendungen, 1999
After an outline of W. Feller's inversion of the (later so called) Feller potential operators and... more After an outline of W. Feller's inversion of the (later so called) Feller potential operators and the presentation of the semigroups thus generated, we interpret the two-level difference scheme resulting from the Grünwald-Letnikov discretization of fractional derivatives as a random walk model discrete in space and time. We show that by properly scaled transition to vanishing space and time steps this model converges to the continuous Markov process that we view as a generalized diffusion process. By re-interpretation of the proof we get a discrete probability distribution that lies in the domain of attraction of the corresponding stable Levy distribution. By letting only the time-step tend to zero we get a random walk model discrete in space but continuous in time. Finally, we present a random walk model for the timeparametrized Cauchy probability density.
Pure and Applied Geophysics, 2002
Kjartansson's constant-Q model is solved in the time-domain using a new modeling algorithm based ... more Kjartansson's constant-Q model is solved in the time-domain using a new modeling algorithm based on fractional derivatives. Instead of time derivatives of order 2, Kjartansson's model requires derivatives of order 2c, with 0 < c < 1=2, in the dilatation-stress formulation. The derivatives are computed with the Gru¨nwald-Letnikov and central-difference approximations, which are finite-difference extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries. A synthetic cross-well seismic experiment illustrates the capabilities of this novel modeling algorithm.
Fractional Calculus and Applied Analysis, 2011
This paper presents a brief overview of the life story and professional career of Prof. R. Gorenf... more This paper presents a brief overview of the life story and professional career of Prof. R. Gorenflo — a well-known mathematician, an expert in the field of Differential and Integral Equations, Numerical Mathematics, Fractional Calculus and Applied Analysis, an interesting conversational partner, an experienced colleague, and a real friend. Especially his role in the modern Fractional Calculus and its applications is highlighted.
Journal of Computational and Applied Mathematics, 2005
The H functions, introduced by Fox in 1961, are special functions of a very general nature, which... more 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.
Journal of Computational and Applied Mathematics, 2000
We review a variety of fractional evolution processes (so deÿned being governed by equations of f... more We review a variety of fractional evolution processes (so deÿned being governed by equations of fractional order), whose solutions turn out to be related to Mittag-Le er-type functions. The chosen equations are the simplest of the fractional calculus and include the Abel integral equations of the second kind, which are relevant in typical inverse problems, and the fractional di erential equations, which govern generalized relaxation and oscillation phenomena.
Journal of Computational and Applied Mathematics, 2000
The time-fractional di usion-wave equation is obtained from the classical di usion or wave equati... more The time-fractional di usion-wave equation is obtained from the classical di usion or wave equation by replacing the ÿrst-or second-order time derivative by a fractional derivative of order (0 ¡ 62). Using the similarity method and the method of the Laplace transform, it is shown that the scale-invariant solutions of the mixed problem of signalling type for the time-fractional di usion-wave equation are given in terms of the Wright function in the case 0 ¡ ¡ 1 and in terms of the generalized Wright function in the case 1 ¡ ¡ 2. The reduced equation for the scale-invariant solutions is given in terms of the Caputo-type modiÿcation of the Erdà elyi-Kober fractional di erential operator.
Journal of Computational and Applied Mathematics, 2007
After sketching the basic principles of renewal theory and recalling the classical Poisson proces... more After sketching the basic principles of renewal theory and recalling the classical Poisson process, we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of Mittag-Leffler and of Wright type. We compare these three processes with each other.
Journal of Computational and Applied Mathematics, 2010
The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as... more The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of Doppler and pressure broadenings is known as the Voigt profile function. Because of its convolution integral representation, the Voigt profile can be interpreted as the probability density function of the sum of two independent random variables with Gaussian density (due to the Doppler effect) and Lorentzian density (due to the pressure effect). Since these densities belong to the class of symmetric Lévy stable distributions, a probabilistic generalization is proposed as the convolution of two arbitrary symmetric Lévy densities. We study the case when the widths of the considered distributions depend on a scale-factor τ that is representative of spatial inhomogeneity or temporal non-stationarity. The evolution equations for this probabilistic generalization of the Voigt function are here introduced and interpreted as generalized diffusion equations containing two Riesz space-fractional derivatives, thus classified as space-fractional diffusion equations of double order.
Applied Mathematics and Computation, 2003
We revisit the Cauchy problem for the time-fractional diffusion equation, which is obtained from ... more We revisit the Cauchy problem for the time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order b 2 ð0; 2. By using the Fourier-Laplace transforms the fundamentals solutions (Green functions) are shown to be high transcendental functions of the Wright-type that can be interpreted as spatial probability density functions evolving in time with similarity properties. We provide a general representation of these functions in terms of Mellin-Barnes integrals useful for numerical computation.
Fractional Calculus and Applied Analysis, 2008
In this expository paper we first survey the method of Mellin-Barnes
integrals to represent the ... more In this expository paper we first survey the method of Mellin-Barnes
integrals to represent the \alpha stable Le'vy distributions in probability theory
(0 < \alpha <= 2). These integrals are known to be useful for obtaining
convergent and asymptotic series representations of the corresponding probability
density functions. The novelty concerns the convolution between two stable
probability densities of different Levy index, which turns to be a
probability law of physical interest, even if it is no longer stable and self-similar.
A particular but interesting case of convolution is obtained combining the
Cauchy-Lorentz density (\alpha = 1) with the Gaussian density (\alpha = 2) that
yields the so-called Voigt profile. Our machinery can be applied to derive
the fundamental solutions of space-fractional diffusion equations of two
orders.
arXiv (Cornell University), Jan 7, 2007
The partial differential equation of Gaussian diffusion is generalized by using the time-fraction... more The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of time orders we provide the fundamental solution, that is still a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative except for the single order case where the two approaches turn to be equivalent. We consider with some detail two cases of order distribution: the double-order and the uniformly distributed order. For these cases we exhibit plots of the corresponding fundamental solutions and their variance, pointing out the remarkable difference between the two approaches for small and large times.
Studies of Nonlinear Phenomena in Life Science, 1999
From the point of view of the general theory of the hyper-Bessel operators, we consider a particu... more From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument �
Journal of Mathematical Physics
A generalization of the canonical coherent states of a quantum harmonic oscillator has been perfo... more A generalization of the canonical coherent states of a quantum harmonic oscillator has been performed by requiring the conditions of normalizability, continuity in the label, and resolution of the identity operator with a positive weight function. Relying on this approach, in the present scenario, coherent states are generalized over the canonical or finite dimensional Fock space of the harmonic oscillator. A class of generalized coherent states is determined such that the corresponding distributions of the number of excitations depart from the Poisson statistics according to combinations of stretched exponential decays, power laws, and logarithmic forms. The analysis of the Mandel parameter shows that the generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations, based on the realization of determined constraints. Mittag-Leffler and Wright generalized coherent states are analyzed as particular cases.
Chaos, Solitons & Fractals
In this paper we consider a modified fractional Maxwell model based on the application of Hadamar... more In this paper we consider a modified fractional Maxwell model based on the application of Hadamard-type fractional derivatives. The model is physically motivated by the fact that we can take into account at the same time memory effects and the time-dependence of the viscosity coefficient. We obtain an ultra-slow relaxation response whose explicit analytic form is given by the Mittag-Leffler function with a logarithmic argument. We show graphically the main properties of this relaxation response, also with the asymptotic behaviour. This paper has been published in Chaos, Solitons and Fractals,
Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics
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Papers by Francesco Mainardi
integrals to represent the \alpha stable Le'vy distributions in probability theory
(0 < \alpha <= 2). These integrals are known to be useful for obtaining
convergent and asymptotic series representations of the corresponding probability
density functions. The novelty concerns the convolution between two stable
probability densities of different Levy index, which turns to be a
probability law of physical interest, even if it is no longer stable and self-similar.
A particular but interesting case of convolution is obtained combining the
Cauchy-Lorentz density (\alpha = 1) with the Gaussian density (\alpha = 2) that
yields the so-called Voigt profile. Our machinery can be applied to derive
the fundamental solutions of space-fractional diffusion equations of two
orders.
integrals to represent the \alpha stable Le'vy distributions in probability theory
(0 < \alpha <= 2). These integrals are known to be useful for obtaining
convergent and asymptotic series representations of the corresponding probability
density functions. The novelty concerns the convolution between two stable
probability densities of different Levy index, which turns to be a
probability law of physical interest, even if it is no longer stable and self-similar.
A particular but interesting case of convolution is obtained combining the
Cauchy-Lorentz density (\alpha = 1) with the Gaussian density (\alpha = 2) that
yields the so-called Voigt profile. Our machinery can be applied to derive
the fundamental solutions of space-fractional diffusion equations of two
orders.