arXiv subjects

Roberto Garra

Publications and source records attributed to Roberto Garra.

36 records · Page 2Linked to original sources

arXiv2018

The fractional Dodson diffusion equation: a new approach

In this paper, after a brief review of the general theory concerning regularized derivatives and integrals of a function with respect to another function, we provide a peculiar fractional generalization of the $(1+1)$-dimensional Dodson's diffusion equation. For the latter we then compute the fundamental solution, which turns out to be expressed in terms of an M-Wright function of two variables. Then, we conclude the paper providing a few interesting results for some nonlinear fractional Dodson-like equations.

math-ph↗

arXiv2017

A note on Hadamard fractional differential equations with varying coefficients and their applications in probability

In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions.

math.PR↗

arXiv2017

The Prabhakar or three parameter Mittag--Leffler function: theory and application

The Prabhakar function (namely, a three parameter Mittag-Leffler function) is investigated. This function plays a fundamental role in the description of the anomalous dielectric properties in disordered materials and heterogeneous systems manifesting simultaneous nonlocality and nonlinearity and, more generally, in models of Havriliak-Negami type. After reviewing some of the main properties of the function, the asymptotic expansion for large arguments is investigated in the whole complex plane and, with major emphasis, along the negative semi-axis. Fractional integral and derivative operators of Prabhakar type are hence considered and some nonlinear heat conduction equations with memory involving Prabhakar derivatives are studied.

math-ph↗

arXiv2017

Fractional Herglotz Variational Principles with Generalized Caputo Derivatives

We obtain Euler-Lagrange equations, transversality conditions and a Noether-like theorem for Herglotz-type variational problems with Lagrangians depending on generalized fractional derivatives. As an application, we consider a damped harmonic oscillator with time-depending mass and elasticity, and arbitrary memory effects.

math.OC↗

arXiv2017

A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus

We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter $ν\in (0,1]$, the logarithmic creep law known in rheology as Lomnitz law (obtained for $ν=1$). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics.

math.CV↗

arXiv2016

Nonlinear heat conduction equations with memory: physical meaning and analytical results

We study nonlinear heat conduction equations with memory effects within the framework of the fractional calculus approach to the generalized Maxwell-Cattaneo law. Our main aim is to derive the governing equations of heat propagation, considering both the empirical temperature-dependence of the thermal conductivity coefficient (which introduces nonlinearity) and memory effects, according to the general theory of Gurtin and Pipkin of finite velocity thermal propagation with memory. In this framework, we consider in detail two different approaches to the generalized Maxwell-Cattaneo law, based on the application of long-tail Mittag-Leffler memory function and power law relaxation functions, leading to nonlinear time-fractional telegraph and wave-type equations. We also discuss some explicit analytical results to the model equations based on the generalized separating variable method and discuss their meaning in relation to some well-known results of the ordinary case.

math-ph↗

arXiv2016

Confinement of a hot temperature patch in the modified SQG model

In this paper we study the time evolution of a temperature patch in $R^2$ according to the modified Surface Quasi-Geostrophic Equation (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation.

math.AP↗

arXiv2015

Fractional diffusions with time-varying coefficients

This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators $L$ and converting their fractional power $L^α$ into Erdélyi--Kober fractional integrals. We study also probabilistic properties of the r.v.'s whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed r.v.'s.

math.PR↗

arXiv2014

Correlated fractional counting processes on a finite time interval

We present some correlated fractional counting processes on a finite time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to zero, the univariate distributions coincide with the ones of the space-time fractional Poisson process in Orsingher and Polito (2012). On the other hand, when we consider the time fractional Poisson process, the multivariate finite dimensional distributions are different from the ones presented for the renewal process in Politi et al. (2011). Another case concerns a class of fractional negative binomial processes.

math.PR↗

arXiv2014

Hilfer-Prabhakar Derivatives and Some Applications

We present a generalization of Hilfer derivatives in which Riemann--Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Further, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

math.PR↗

arXiv2014

Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications

In this paper we discuss some exact results related to the fractional Klein--Gordon equation involving fractional powers of the D'Alembert operator. By means of a space-time transformation, we reduce the fractional Klein--Gordon equation to a fractional hyper-Bessel-type equation. We find an exact analytic solution by using the McBride theory of fractional powers of hyper-Bessel operators. A discussion of these results within the framework of linear dispersive wave equations is provided. We also present exact solutions of the fractional Klein-Gordon equation in the higher dimensional cases. Finally, we suggest a method of finding travelling wave solutions of the nonlinear fractional Klein-Gordon equation with power law nonlinearities.

math.AP↗

arXiv2014

Fractional Klein-Gordon equations and related stochastic processes

This paper presents finite-velocity random motions driven by fractional Klein-Gordon equations of order $α\in (0,1]$. A key tool in the analysis is played by the McBride's theory which converts fractional hyper-Bessel operators into Erdelyi-Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein-Gordon equation. The distribution of the fractional telegraph process for $α= 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein-Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations $B(n,α)$ (with uniformly-distributed orientation) are considered. The parameter $n$ of $B(n,α)$ is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of directions. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to $N$-dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler-Poisson-Darboux equation to which our theory applies.

math.PR↗

arXiv2014

State-dependent Fractional Point Processes

The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{ν_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0<ν_k\leq 1$ depending on the number $k$ of events occurred up to time $t$. We are able to obtain explicitely the Laplace transform of $p_k^{ν_k}(t)$ and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on $ν_k$ differs from that constructed from the fractional state equations (in the case $ν_k = ν$, for all $k$, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally we consider the fractional birth process governed by equations with state-dependent fractionality.

math.PR↗

arXiv2013

Analytic solutions of fractional differential equations by operational methods

We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional derivative. Furthermore, operational relations between ordinary and fractional differentiation are shown and discussed in detail. Finally, a last example concerns the application of the method to the study of a fractional Poisson process.

math.AP↗

arXiv2013

Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity

In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the time dependent viscosity by using an integro-differential term and therefore generalizing the classical equation of a Newtonian viscous fluid. A possible useful choice, in this framework, is to use a rheology based on stress/strain relation generalized by fractional calculus modelling. This is a model that can be used in applied problems, taking into account a power law time variability of the viscosity coefficient. We find analytic solutions of initial value problems in an unbounded and bounded domain. Furthermore, we discuss the explicit solution in a meaningful particular case.

math.AP↗

arXiv2013

Coupled systems of fractional equations related to sound propagation: analysis and discussion

In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in the constitutive equations. By doing so, we embrace a vast phenomenology, including subdiffusive, superdiffusive and also memoryless processes like classical diffusions. From a mathematical point of view, we study systems of coupled fractional equations, leading to fractional diffusion equations or to equations with sequential fractional derivatives. In this framework we also propose a method to solve partial differential equations with sequential fractional derivatives by analysing the corresponding coupled system of equations.

math.AP↗

arXiv2013

On Some Operators Involving Hadamard Derivatives

In this paper we introduce a novel Mittag--Leffler-type function and study its properties in relation to some integro-differential operators involving Hadamard fractional derivatives or Hyper-Bessel-type operators. We discuss then the utility of these results to solve some integro-differential equations involving these operators by means of operational methods. We show the advantage of our approach through some examples. Among these, an application to a modified Lamb--Bateman integral equation is presented.

math.AP↗

arXiv2011

A note on fractional linear pure birth and pure death processes in epidemic models

In this note we highlight the role of fractional linear birth and linear death processes recently studied in \citet{sakhno} and \citet{pol}, in relation to epidemic models with empirical power law distribution of the events. Taking inspiration from a formal analogy between the equation of self consistency of the epidemic type aftershock sequences (ETAS) model, and the fractional differential equation describing the mean value of fractional linear growth processes, we show some interesting applications of fractional modelling to study \textit{ab initio} epidemic processes without the assumption of any empirical distribution. We also show that, in the frame of fractional modelling, subcritical regimes can be linked to linear fractional death processes and supercritical regimes to linear fractional birth processes. Moreover we discuss a simple toy model to underline the possible application of these stochastic growth models to more general epidemic phenomena such as tumoral growth.

math.PR↗