arXiv subjects

Francesco Mainardi

Publications and source records attributed to Francesco Mainardi.

At least 19 recordsLinked to original sources

arXiv2026

Energy dissipation in linear viscoelasticity

We compare the specific dissipation $Q^{-1}\left( \omega \right) $ for the Becker, Lomnitz, and Lambert models in linear viscoelasticity. Graphically, the specific dissipation of these models behaves similarly as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$. Furthermore, we obtain analytic formulas for the asymptotic behavior of $Q^{-1}\left( \omega \right) $ in the Becker and Lomnitz models as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$, where the latter limit coincides with the specific dissipation of the Maxwell model. We also derive a novel closed-form expression for the specific dissipation in the generalized Becker model, $Q_{\nu }^{-1}\left( \omega \right) $, when the parameter $\nu \in \left( 0,1\right] $ is rational. When $\nu \rightarrow 0$, we recover the specific dissipation of the Maxwell model, whereas when $\nu =1$, we recover that of the original Becker model. In addition, we derive two new closed-form expressions for the specific dissipation for the extended Jeffreys--Lomnitz model $Q_{\alpha }^{-1}\left( \omega \right) $, the first being valid for $\alpha \in \left( 0,1\right] $, and the second for $\alpha \in \left( -\infty ,1\right] $. When $\alpha \rightarrow 0$, we recover the specific dissipation of the Lomnitz model, whereas when $\alpha =1$, we recover that of the Maxwell model. Finally, we conclude that the asymptotic behavior of $Q_{\alpha }^{-1}\left( \omega \right) $ as $\omega \rightarrow +\infty $ coincides with the specific dissipation of the Maxwell model. Keywords:

math-ph↗

arXiv2026

Delta-pulse solution in Zener viscoelastic model

We derive the integral representation of the solution for the propagation of a delta-pulse (impulsive wave) in a semi-infinite, homogeneous, linear viscoelastic medium governed by the Zener model. Starting from the Bromwich integral representation of the response function, we obtain a closed-form integral representation by analytically inverting the relevant Laplace transforms. The result is expressed in terms of modified Bessel functions of the first kind and Macdonald functions of half-integer order, and is shown to reduce to the known Maxwell model solution in the appropriate limit. As an independent computational approach, we derive the steepest descent path (SDP) associated with the phase function of the Bromwich integral, characterizing its saddle points and showing that the SDP can be expressed explicitly as the zero locus of a sixth-degree polynomial in the imaginary part of the complex variable. The two methods are compared numerically for several values of the model parameters, confirming their agreement. While the integral representation provides analytical insight into the structure of the solution, the steepest descent method requires no explicit inversion of the Laplace transform and may therefore prove especially valuable in more general viscoelastic settings where a closed-form integral representation is not available.

math-ph↗

arXiv2026

Pulse waves in the viscoelastic Kelvin-Voigt model: a revisited approach

We calculate the mechanical response $r(x,t$) of an initially quiescent semi-infinite homogeneous medium to a pulse applied at the origin, and this is achieved within the framework of the Kelvin-Voigt model. Although this problem has been extensively studied in the literature because of its wide range of applications -- particularly in seismology -- here, we present a solution in a novel integral form. This integral solution avoids the numerical computation of the solution in terms of the inverse Laplace transform; that is, numerical integration in the complex plane. In particular, we derive integral form expressions for both delta-pulse and step-pulse excitations which are simpler and more computationally efficient than those previously reported in the literature. Furthermore, the obtained expressions allow us to obtain simple asymptotic formulas for $r(x,t$ as $x,t \to 0,\infty$ for both step- and delta-type pulses.

math.GM↗

arXiv2025

Transient waves in linear dispersive media with dissipation: an approach based on the steepest descent path

In the study of linear dispersive media it is of primary interest to gain knowledge of the impulse response of the material. The standard approach to compute the response involves a Laplace transform inversion, i.e., the solution of a Bromwich integral, which can be a notoriously troublesome problem. In this paper we propose a novel approach to the calculation of the impulse response, based on the well assessed method of the steepest descent path, which results in the replacement of the Bromwich integral with a real line integral along the steepest descent path. In this exploratory investigation, the method is explained and applied to the case study of the Klein- Gordon equation with dissipation, for which analytical solutions of the Bromwich integral are available, as to compare the numerical solutions obtained by the newly proposed method to exact ones. Since the newly proposed method, at its core, consists in replacing a Laplace transform inverse with a potentially much less demanding real line integral, the method presented here could be of general interest in the study of linear dispersive waves in presence of dissipation, as well as in other fields in which Laplace transform inversion come into play.

math.GM↗

arXiv2025

The Tautochrone of Huygens and Abel: From Constructive Geometry to Fractional Calculus

In this paper, we explore the connections between Christiaan Huygens and Niels Henrik Abel through the tautochrone problem. The problem -- determining the curve along which a particle descends under gravity in the same time, regardless of its starting point -- has been a central topic at the intersection of physics, geometry, and analysis. Though these two major figures are separated by nearly two centuries, they approached the problem in radically different ways. While Huygens proposed a physical solution based on geometric construction, Abel approached the problem within the analytic framework of integral equations, employing a procedure that can be seen as anticipating and paving the way for the development of differential calculus of arbitrary order. This contrast highlights a broader historical narrative: the transformation of mathematical thinking from constructive geometry to abstract analysis.

math.HO↗

arXiv2025

On the Laplace transforms of derivatives of special functions with respect to parameters

This article is devoted to derivation of the Laplace transforms of the derivatives with respect to parameters of certain special functions, namely, the Mittag-Leffler type, Wright and Le Roy type functions. These formulas show interconnection of these functions and lead to better understanding of their behaviour on the real line. These formulas are represented in the convoluted form and reconstructed in a more suitable form by using Efros theorem

math.GM↗

arXiv2024

A comparative view of Becker, Lomnitz, and Lambert linear viscoelastic models

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model.

physics.class-ph↗

arXiv2024

Some fractional integral and derivative formulas revisited

In the most common literature about fractional calculus, we find that $_{a}D_{t}^{\alpha }f\left( t\right) =\,_{a}I_{t}^{-\alpha }f\left( t\right) $ is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of $_{a}I_{t}^{\alpha }f\left( t\right) $ and $_{a}D_{t}^{\alpha }f\left( t\right) $. In this sense, we prove that $_{0}D_{t}^{\alpha }f\left( t\right) =\,_{0}I_{t}^{-\alpha }f\left( t\right) $ is true for $f\left( t\right) =t^{\nu -1}\log t$, and $f\left( t\right) =e^{\lambda t}$, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for $_{-\infty }D_{t}^{\alpha }\left\vert t\right\vert ^{-\delta }$ and $_{-\infty }I_{t}^{\alpha }\left\vert t\right\vert ^{-\delta }$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that $_{-\infty }D_{t}^{\alpha }\left\vert t\right\vert ^{-\delta }=\,_{-\infty }I_{t}^{-\alpha }\left\vert t\right\vert ^{-\delta }$ holds true

math.GM↗

arXiv2024

On differentiation with respect to parameters of the functions of the Mittag-Leffler type

The formal term-by-term differentiation with respect to parameters is demonstrated to be legitimate for the Mittag-Leffler type functions. The justification of differentiation formulas is made by using the concept of the uniform convergence. This approach is applied to the Mittag-Leffler function depending on two parameters and, additionally, for the $3$-parametric Mittag-Leffler functions (namely, for the Prabhakar function and the Le Roy type functions), as well as for the $4$-parametric Mittag-Leffler function (and, in particular, for the Wright function). The differentiation with respect to the involved parameters is discussed also in case those special functions which are represented via the Mellin-Barnes integrals.

math.GM↗

arXiv2024

Calculation of the Relaxation Modulus in the Andrade Model by Using the Laplace Transform

In the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model $G_{\alpha }\left( t\right) $ for the case of rational parameter \mbox{$\alpha =m/n\in (0,1)$} in terms of Mittag--Leffler functions from its Laplace transform $\tilde{G}_{\alpha }\left( s\right) $. It turns out that the expression obtained can be rewritten in terms of Rabotnov functions. Moreover, for the original parameter $\alpha =1/3$ in the Andrade model, we obtain an expression in terms of Miller-Ross functions. The asymptotic behaviours of $G_{\alpha }\left( t\right) $ for $t\rightarrow 0^{+}$ and $t\rightarrow +\infty $ are also derived applying the Tauberian theorem. The analytical results obtained have been numerically checked by solving the Volterra integral equation satisfied by $G_{\alpha }\left( t\right) $ by using a successive approximation approach, as well as computing the inverse Laplace transform of $\tilde{G}_{\alpha }\left( s\right) $ by using Talbot's method.

physics.class-ph↗

arXiv2022

Differentiation of the Wright functions with respect to parameters and other results

In this survey we discuss derivatives of the Wright functions (of the first and the second kind) with respect to parameters. Differentiation of these functions leads to infinite power series with coefficient being quotients of the digamma (psi) and gamma functions. Only in few cases it is possible to obtain the sums of these series in a closed form. Functional form of the power series resembles those derived for the Mittag-Leffler functions. If the Wright functions are treated as the generalized Bessel functions, differentiation operations can be expressed in terms of the Bessel functions and their derivatives with respect to the order. It is demonstrated that in many cases it is possible to derive the explicit form of the Mittag-Leffler functions by performing simple operations with the Laplace transforms of the Wright functions. The Laplace transform pairs of the both kinds of the Wright functions are discussed for particular values of the parameters. Some transform pairs serve to obtain functional limits by applying the shifted Dirac delta function.

math.GM↗

arXiv2022

Truncated generalized coherent states

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 distribution of the number of excitations departs from the Poisson statistics according to combinations of stretched exponential decays, power laws and logarithmic forms. The analysis of the Mandel parameter shows that these generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations for small values of the label, according to determined properties. The statistics is uniquely sub-Poissonian for large values of the label. As particular cases, truncated Wright generalized coherent states exhibit uniquely non-classical properties, differently from the truncated Mittag-Leffler generalized coherent states.

quant-ph↗

arXiv2022

A note on a modified fractional Maxwell model

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.

math.AP↗

arXiv2022

Wright functions of the second kind and Whittaker functions

In the framework of higher transcendental functions the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker functions in some special cases of fractional order. In addition, we point out two erroneous representations in the literature.

math.GM↗

arXiv2022

The tempered space-fractional Cattaneo equation

We consider the time-fractional Cattaneo equation involving the tempered Caputo space-fractional derivative. We find the characteristic function of the related process and we explain the main differences with previous stochastic treatments of the time-fractional Cattaneo equation.

math.PR↗

arXiv2021

Variable-order fractional calculus: a change of perspective

Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative.

math.CA↗

arXiv2021

The Bateman Functions Revisited After 90 Years -- A Survey of Old and New Results

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to known in the literature generalizations of these functions and to the Bateman-integral function.

math.GM↗

arXiv2021

On the asymptotic of Wright functions of the second kind

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book {\it Fractional Calculus and Waves in Linear Viscoelasticity}, (2010)], $$ F_σ(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-nσ)}~,\quad M_σ(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-nσ+1-σ)}\quad(0<σ<1)$$ for $x\to\pm\infty$ are presented. The situation corresponding to the limit $σ\to1^-$ is considered, where $M_σ(x)$ approaches the Dirac delta function $δ(x-1)$. Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as $σ\to 1^-$.

math.CA↗