arXiv subjects

R. Garra

Publications and source records attributed to R. Garra.

At least 19 recordsLinked to original sources

arXiv2026

On the role of Giovanni Giorgi in the history of operational methods to mathematical-physics problems

In this paper we discuss the historical role played by Prof. Giovanni Giorgi (1871-1950) in the development of operational methods in mathematical physics. In the literature, the analysis of the scientific contributions of Giorgi is discussed in many historical papers mainly about the $MKS\Omega$-system and its contributions in the field of electrical engineering. Starting from the obituary written by Prof. Dario Graffi (1905-1990), here we analyze in detail the contributions given by Giorgi in mathematical physics, especially in the framework of the studies started by Heaviside (1850-1925) in order to obtain a symbolic representation of the solutions of differential equations emerging in physical problems. \\ Moreover, we underline the little known contribution of Giorgi to the analysis of derivatives of any real order, namely fractional derivatives, according to the present notation. \\ The main aim of this note is to underline the historical relevance of Giorgi in the mathematical foundations of operational methods in relation to the solutions of concrete problems emerging in applied physics.

math.HO↗

arXiv2023

On variable-order fractional linear viscoelasticity

We discuss a generalisation of fractional linear viscoelasticity based on Scarpi's approach to variable-order fractional calculus. After reviewing the general mathematical framework, we introduce the variable-order fractional Maxwell model as a simple example for our analysis. We then provide some physical considerations for the fractionalisation procedure and on the choice of the transition functions. Lastly, we compute the material functions for the considered model and evaluate them numerically for exponential-type and Mittag-Leffler-type order functions.

math-ph↗

arXiv2023

A note on differential equations of logistic type

Logistic equations play a pivotal role in the study of any non linear evolution process exhibiting growth and saturation. The interest for the phenomenology, they rule, goes well beyond physical processes and cover many aspects of ecology, population growth, economy...According to such a broad range of applications, there are different forms of functions and distributions which are recognized as generalized logistics. Sometimes they are obtained by fitting procedures. Therefore, criteria might be needed to infer the associated non linear differential equations, useful to guess "hidden" evolution mechanisms. In this article we analyze different forms of logistic functions and use simple means to reconstruct the differential equation they satisfy. Our study includes also differential equations containing non standard forms of derivative operators, like those of the Laguerre type.

math.CA↗

arXiv2023

Exact solutions for the fractional nonlinear Boussinesq equation

We investigate the existence of exact solutions in closed form to a fractional version of the nonlinear Boussinesq equation for groundwater flow through an unconfined aquifer. We show this fractional equation appears naturally when the classical nonlinear Darcy's law is replaced by a space-fractional one. After a physical discussion on the fractional model, we give several exact solutions in closed form for special choices of initial and boundary data. We provide solutions for steady and unsteady cases, by considering both classical and fractional derivatives in time.

math.AP↗

arXiv2023

G-fractional diffusion on bounded domains in $\mathbb{R}^d$

In this paper we study $g$-fractional diffusion on bounded domains in $\mathbb{R}^d$ with absorbing boundary conditions. We show the explicit representation of the solution and then we study the first passage time distribution, showing the dependence on the particular choice of the function $g$. Then, we specialize the analysis to the interesting case of a rectangular domain. Finally we briefly discuss the connection of this general theory with the physical application to the so-called fractional Dodson diffusion model recently discussed in the literature.

math.AP↗

arXiv2022

On G-fractional diffusion models in bounded domains

In the recent literature, the g-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive regimes. In this paper we study the problem of g-fractional diffusion in a bounded domain with absorbing boundaries. We find the explicit solution for the initial-boundary value problem and we study the first passage time distribution and the mean first-passage time (MFPT). An interestin outcome is the proof that with a particular choice of the function $g$ it is possible to obtain a finite MFPT, differently from the anomalous diffusion described by a fractional heat equation involving the classical Caputo derivative.

cond-mat.stat-mech↗

arXiv2020

Some applications of Wright functions in fractional differential equations

In this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to obtain some new particular solutions for nonlinear fractional partial differential equations.

math.CA↗

arXiv2020

A note on generalized fractional diffusion equations on Poincarè half plane

In this paper we study generalized time-fractional diffusion equations on the Poincarè half plane $\mathbb{H}_2^+$. The time-fractional operators here considered are fractional derivatives of a function with respect to another function, that can be obtained by starting from the classical Caputo-derivatives essentially by means of a deterministic change of variable. We obtain an explicit representation of the fundamental solution of the generalized-diffusion equation on $\mathbb{H}_2^+$ and provide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion. We finally include an explicit result regarding the non-linear case admitting a separating variable solution.

math-ph↗

arXiv2020

On the relation between non-homogeneous fractional Burgers equations and time-dependent harmonic oscillator

In this paper we discuss the relation between non-homogeneous nonlinear fractional diffusive equations and the Schrodinger equation with time-dependent harmonic potential. It is well known that the Cole-Hopf transformation allows to linearize non-homogeneous nonlinear diffusive equations (NHNDEs) into a Schrodinger-type equation with time-dependent potential. We first discuss the utility of the results about time-dependent harmonic oscillator to build explicit solution of such non-homogeneous nonlinear partial differential equations. In particular, we recall that starting from a trial polynomial solution of the NHNDE, it is possible to construct other solutions by using linear invariants of the Schrodinger equation with time-dependent potential. Finally we apply these results to find explicit solutions to a novel non-homogeneous fractional Burgers-type equation.

nlin.SI↗

arXiv2019

Cyclic random motions with orthogonal directions

A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$. We obtain in both cases the explicit conditional distributions of the position of the moving particle when the number of switches of directions is fixed. The explicit unconditional distributions are also obtained and are expressed in terms of Bessel functions. The governing equations are derived and given as products of D'Alembert operators. The limiting form of the equations is provided in the Euclidean space $\mathbb{R}^d$ and takes the form of a heat equation with infinitesimal variance $1/d$.

math.PR↗

arXiv2016

Application of the fractional conservation of mass to Gas Flow diffusivity equation in heterogeneous porous media

In this paper we reconsider the classical nonlinear diffusivity equation of real gas in an heterogenous porous medium in light of the recent studies about the generalized fractional equation of conservation of mass. We first recall the physical meaning of the fractional conservation of mass recently studied by Wheatcraft and Meerschaert (2008) and then consider the implications in the classical model of diffusion of a real gas in a porous medium. Then we show that the obtained equation can be simply linearized into a classical space-fractional diffusion equation, widely studied in the literature. We also consider the case of a power-law pressure-dependence of the permeability coefficient. In this case we provide some useful exact analytical results. In particular, we are able to find a Barenblatt-type solution for a space-fractional Boussinesq equation, arising in this context.

physics.geo-ph↗

arXiv2016

On the effect of a two-rocks boundary on the propagation of nonlinear transients of temperature and pressure in deformable porous rocks

We here analyze the propagation of transients of fluid-rock temperature and pressure through a thin boundary layer, where a steady trend is present, between two adjacent homogeneous rocks. We focus on the effect of convection on transients crossing such thin layer. In comparison with early models where this boundary was assumed a sharp mathematical plane separating the two rocks, here we show a realistic analysis of such boundary layer that implies a novel nonlinear model. Its solutions describe large amplitude, quick and sharp transients characterized by a novel drift and variations of the signal amplitude, leading to a nonlinear wave propagation. Possible applications are in volcanic, hydrologic, hydrothermal systems as well as for deep oil drilling. In addition, this formalism could easily be generalized for the case of a signal arriving in a rock characterized by a steady trend of pressure and/or temperature. These effects, being proportional to the initial conditions, can also give velocity variations not particularly important. A further heuristic model has therefore been analyzed, i.e. assuming a pressure dependent rock permeability. In this way, a remarkable increase of the system velocities is obtained.

physics.geo-ph↗

arXiv2016

Random motions with space-varying velocities

Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position $\mathcal{T}(t)$, $t>0$, of the moving particle. Also the case of a non-homogeneous Poisson process (with rate $λ= λ(t)$) governing the changes of direction is analyzed in three specific cases. For the special case $λ(t)= α/t$ we obtain a random motion related to the Euler-Poisson-Darboux (EPD) equation which generalizes the well-known case treated e.g. in Foong and Van Kolck (1992), Garra and Orsingher (2016) and Rosencrans (1973). A EPD--type fractional equation is also considered and a parabolic solution (which in dimension $d=1$ has the structure of a probability density) is obtained. Planar random motions with space--varying velocities and infinite directions are finally analyzed in Section 5. We are able to present their explicit distributions and for polynomial-type velocity structures we obtain the hyper and hypo-elliptic form of their support (of which we provide a picture).

math.PR↗

arXiv2016

Some probabilistic properties of fractional point processes

This paper studies the first hitting times of generalized Poisson processes $N^f(t)$, related to Bernstein functions $f$. For the space-fractional Poisson processes, $N^α(t)$, $t>0$ (corresponding to $f= x^α$), the hitting probabilities $P\{T_k^α<\infty\}$ are explicitly obtained and analyzed. The processes $N^f(t)$ are time-changed Poisson processes $N(H^f(t))$ with subordinators $H^f(t)$ and here we study $N\left(\sum_{j=1}^n H^{f_j}(t)\right)$ and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form $N(|\mathcal{G}_{H,ν}(t)|)$ where $\mathcal{G}_{H,ν}(t)$ are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space-time Poisson process is no longer a renewal process.

math.PR↗

arXiv2015

Random flights related to the Euler-Poisson-Darboux equation

This paper is devoted to the analysis of random motions on the line and in the space R^d (d > 1) performed at finite velocity and governed by a non-homogeneous Poisson process with rate λ(t). The explicit distributions p(x,t) of the position of the randomly moving particles are obtained solving initial-value problems for the Euler- Poisson-Darboux equation when λ(t) = α/t, t > 0. We consider also the case where λ(t) = λcoth λt and λ(t) = λtanh λt, where some Riccati differential equations emerge and the explicit distributions are obtained for d = 1. We also examine planar random motions with random velocities by projecting random flights in R^d onto the plane. Finally the case of planar motions with four orthogonal directions is considered and the corresponding higher-order equations with time-varying coefficients obtained.

math.PR↗

arXiv2015

Fractional telegraph equation, Mittag-Leffler function, Hilfer derivative, Hadamard fractional derivative, Riesz-Feller space-fractional derivative

In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided.

math.PR↗

arXiv2015

Fractional relaxation and fractional oscillation models involving Erdelyi-Kober integrals

We consider fractional relaxation and fractional oscillation equations involving Erdelyi-Kober integrals. In terms of Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing Riemann-Liouville integrals with Erdelyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.

math.NA↗

arXiv2014

Finite velocity planar random motions driven by inhomogeneous fractional Poisson distributions

In this paper we study finite velocity planar random motions with an infinite number of possible directions, where the number of changes of direction is randomized by means of an inhomogeneous fractional Poisson distribution. We first discuss the properties of the distributions of the generalized fractional inhomogeneous Poisson process. Then we show that the explicit probability law of the planar random motions where the number of changes of direction is governed by this fractional distribution can be obtained in terms of Mittag-Leffler functions. We also consider planar random motions with random velocities obtained from the projection of random flights with Dirichlet displacements onto the plane, randomizing the number of changes of direction with a suitable adaptation of the fractional Poisson distribution.

math.PR↗