arXiv subjects

Ivano Colombaro

Publications and source records attributed to Ivano Colombaro.

At least 19 recordsLinked to original sources

arXiv2026

A Representation Theorem for Fractional Viscoelastic Networks

We prove a representation theorem for finite series-parallel viscoelastic networks composed of springs and spring-pots of a common order $0<α<1$. The associated complex shear modulus, expressed in an appropriately chosen auxiliary variable, is rational and admits a finite partial-fraction representation with simple poles on the negative real axis and negative residues. Its reciprocal admits an analogous representation with positive residues and a possible additional pole at the origin. Inverting the corresponding Laplace transforms gives finite fractional Prony representations of the relaxation modulus and creep compliance with non-negative amplitudes. These results provide a rigorous proof of a corrected version of the Mainardi-Spada conjecture for this network class. A six-element network illustrates the construction.

math-ph↗

arXiv2026

A canonical two-scale Sonine fractional calculus induced by the Tricomi function

We introduce a Tricomi-type generalized fractional calculus in the Sonine kernel framework. The key result is that the Tricomi branch is a Stieltjes function in the admissible parameter range, so its reciprocal is a complete Bernstein function. This fact induces a Sonine fractional calculus together with the canonical Tricomi integral and the associated Riemann--Liouville-type and Caputo-type derivatives. We also prove that, within the Kummer class, the Tricomi branch is the unique Stieltjes representative, once the natural asymptotic normalization is fixed, and we derive the corresponding Lévy--Khintchine representation, Volterra formulation, and scalar Cauchy problem. A distinctive feature of the resulting operators is the emergence of two independent asymptotic orders.

math-ph↗

arXiv2026

Passive two-plateau relaxation from Tricomi confluent hypergeometric kernels

Anomalous relaxation with memory spectra arises in disordered solids, soft matter, biological tissues and electrochemical interfaces. Fractional-order models capture broad power-law behaviour efficiently, but they can obscure spectral structure and are not always convenient for passive realisation or finite-dimensional simulation. We introduce a non-fractional passive framework based on the Tricomi confluent hypergeometric function, combined with a bounded Moebius normalisation that enforces prescribed low-frequency and high-frequency plateaux while preserving a broad dispersive transition. The resulting family contains the Debye and Cole-Cole responses as exact subcases, while extending them to asymmetric two-plateau dispersive laws with independently tunable low- and high-frequency exponents. For an admissible parameter range, we prove that the bounded block admits a Stieltjes representation with nonnegative spectral density, implying complete monotonicity, passivity, causality and compatibility with standard circuit and state-space descriptions. Building on this structure, we derive a passive Gauss-Stieltjes discretisation leading to Foster-type rational approximations and first-order state-space realisations with positive poles and residues. Numerical experiments show convergence of these finite-dimensional approximations across moderate-memory and long-tail regimes, enabling passive reduced-order representations of broad-memory responses. The framework is then validated on broadband dielectric data and battery electrochemical impedance spectra. In tissues, multi-block Tricomi mixtures improve complex-domain fitting accuracy relative to classical Cole-Cole baselines while preserving interpretable modal structure.

math-ph↗

arXiv2026

On mathematical characterization of a Bessel functions-based passive element in electronic circuits

Modeling relaxation phenomena in complex media is central to understanding multiscale dynamics in materials science, bioengineering and condensed matter physics. Existing fractional-order models, while flexible, sometimes lack physical interpretability, closed-form time-domain expressions, and compatibility with physically realizable architectures. In this work, we propose a novel passive element whose impedance and admittance are defined analytically via modified Bessel functions of first kind, through the electro-mechanical analogy. This approach preserves key physical properties such as analyticity, passivity, BIBO (bounded-input, bounded-output) stability and monotonicity, while enabling the direct use of its time-domain representation in simulations and system modeling. As an application, we demonstrate that this model accurately captures the broadband dispersive behavior of biological tissues, offering a physically grounded and tractable alternative to fractional-order formulations.

physics.comp-ph↗

arXiv2025

On exterior-algebraic quaternions with application to Maxwell equations

Within the framework of exterior algebra, the concept of time-like quaternions has been previously established. This paper advances beyond the existing structure by elucidating the procedure for constructing time-like quaternions with the feature of complex scalar terms. We delineate on the crucial role of these extended definition of quaternions in formulating Maxwell equations, having properly defined a pure quaternion containing the components of the classical electric and magnetic fields. Through a formal introduction, we describe the approach followed to acquiring time-like quaternions, characterized by having complex scalar term, and their significant relationship with the derivation of Maxwell equations. This topic not only underscores the mathematical intricacies of quaternionic algebra, but also highlights its profound implications in the description of fundamental electromagnetic phenomena.

physics.gen-ph↗

arXiv2023

Time-like definition of quaternions in exterior algebra

A formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of quaternions by means of the properties arising from exterior algebra and calculus. As an application, it is also illustrated how rotations may be written in terms of quaternions, in accordance with definition provided in exterior algebra.

physics.gen-ph↗

arXiv2022

Energy dissipation in viscoelastic Bessel media

We investigate the specific attenuation factor for the Bessel models of viscoelasticity. We find that the quality factor for this class can be expressed in terms of Kelvin functions and that its asymptotic behaviours confirm the analytical results found in previous studies for the rheological properties of these models.

math-ph↗

arXiv2021

On the Angular Momentum and Spin of Generalized Electromagnetic Field for $r$-Vectors in $(k,n)$ Space-Time Dimensions

This paper studies the relativistic angular momentum for the generalized electromagnetic field, described by $r$-vectors in $(k,n)$ space-time dimensions, with exterior-algebraic methods. First, the angular-momentum tensor is derived from the invariance of the Lagrangian to space-time rotations (Lorentz transformations), avoiding the explicit need of the canonical tensor in Noether's theorem. The derivation proves the conservation law of angular momentum for generic values of $r$, $k$, and $n$. Second, an integral expression for the flux of the tensor across a $(k+n-1)$-dimensional surface of constant $\ell$-th space-time coordinate is provided in terms of the normal modes of the field; this analysis is a natural generalization of the standard analysis of electromagnetism, i. e. a three-dimensional space integral at constant time. Third, a brief discussion on the orbital angular momentum and the spin of the generalized electromagnetic field, including their expression in complex-valued circular polarizations, is provided for generic values of $r$, $k$, and $n$.

math-ph↗

arXiv2021

An Exterior Algebraic Derivation of the Euler-Lagrange Equations from the Principle of Stationary Action

In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler-Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler-Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler--Lagrange equations.

math-ph↗

arXiv2021

Generalized Maxwell equations for exterior-algebra multivectors in $(k,n)$ space-time dimensions

This paper presents an exterior-algebra generalization of electromagnetic fields and source currents as multivectors of grades $r$ and $r-1$ respectively in a flat space-time with $n$ space and $k$ time dimensions. Formulas for the Maxwell equations and the Lorentz force for arbitrary values of $r$, $n$, and $k$ are postulated in terms of interior and exterior derivatives, in a form that closely resembles their vector-calculus analogues. These formulas lead to solutions in terms of potentials of grade $r-1$, and to conservation laws in terms of a stress-energy-momentum tensor of rank 2 for any values of $r$, $n$, and $k$, for which a simple explicit formula is given. As an application, an expression for the flux of the stress-energy-momentum tensor across an $(n+k-1)$-dimensional slice of space-time is given in terms of the Fourier transform of the potentials. The abstraction of Maxwell equations with exterior calculus combines the simplicity and intuitiveness of vector calculus, as the formulas admit explicit expressions, with the power of tensors and differential forms, as the formulas can be given for any values of $r$, $n$, and $k$.

math-ph↗

arXiv2021

An exterior-algebraic derivation of the symmetric stress-energy-momentum tensor in flat space-time

This paper characterizes the symmetric rank-2 stress-energy-momentum tensor associated with fields whose Lagrangian densities are expressed as the dot product of two multivector fields, e. g., scalar or gauge fields, in flat space-time. The tensor is derived by a direct application of exterior-algebraic methods to deal with the invariance of the action to infinitesimal space-time translations; this direct derivation avoids the use of the canonical tensor. Formulas for the tensor components and for the tensor itself are derived for generic values of the multivector grade $s$ and of the number of time and space dimensions, $k$ and $n$, respectively. A simple, coordinate-free, closed-form expression for the interior derivative (divergence) of the symmetric stress-energy-momentum tensor is also obtained: this expression provides a natural generalization of the Lorentz force that appears in the context of the electromagnetic theory. Applications of the formulas derived in this paper to the cases of generalized electromagnetism, Proca action, Yang-Mills fields and conformal invariance are briefly discussed.

physics.gen-ph↗

arXiv2020

An Introduction to Space-Time Exterior Calculus

The basic concepts of exterior calculus for space-time multivectors are presented: interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux respectively are derived. As an application, it is shown how the exterior-calculus space-time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms.

math-ph↗

arXiv2020

A practical guide to Prabhakar fractional calculus

The Mittag-Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

math.CA↗

arXiv2018

Scott-Blair models with time-varying viscosity

In a recent paper, Zhou et al. [Mech Time-Depend Mater (2017) 21: 151] studied the time-dependent properties of Glass Fiber Reinforced Polymers composites by employing a new rheological model with a time-dependent viscosity coefficient. This rheological model is essentially based on a generalized Scott-Blair body with a time-dependent viscosity coefficient. Motivated by this study, in this note we suggest a different generalization of the Scott-Blair model based on the application of Caputo-type fractional derivatives of a function with respect to another function. This new mathematical approach can be useful in viscoelasticity and diffusion processes in order to model systems with time-dependent features. In this paper we also provide the general solution of the creep experiment for our improved Scott-Blair model together with some explicit examples and illuminating plots.

math-ph↗

arXiv2018

Storage and dissipation of energy in Prabhakar viscoelasticity

In this paper, after a brief review of the physical notion of quality factor in viscoelasticity, we present a complete discussion of the attenuation processes emerging in the Maxwell-Prabhakar model, recently developed by Giusti and Colombaro. Then, taking profit of some illuminating plots, we discuss some potential connections between the presented model and the modern mathematical modelling of seismic processes.

physics.geo-ph↗

arXiv2017

Prabhakar-like fractional viscoelasticity

The aim of this paper is to present a linear viscoelastic model based on Prabhakar fractional operators. In particular, we propose a modification of the classical fractional Maxwell model, in which we replace the Caputo derivative with the Prabhakar one. Furthermore, we also discuss how to recover a formal equivalence between the new model and the known classical models of linear viscoelasticity by means of a suitable choice of the parameters in the Prabhakar derivative. Moreover, we also underline an interesting connection between the theory of Prabhakar fractional integrals and the recently introduced Caputo-Fabrizio differential operator.

math-ph↗

arXiv2017

On transient waves in linear viscoelasticity

The aim of this paper is to present a comprehensive review of method of the wave-front expansion, also known in the literature as the Buchen-Mainardi algorithm. In particular, many applications of this technique to the fundamental models of both ordinary and fractional linear viscoelasticity are thoroughly presented and discussed.

math-ph↗

arXiv2017

On the propagation of transient waves in a viscoelastic Bessel medium

In this paper we discuss the uniaxial propagation of transient waves within a semi-infinite viscoelastic Bessel medium. First, we provide the analytic expression for the response function of the material as we approach the wave-front. To do so, we take profit of a revisited version of the so called Buchen-Mainardi algorithm. Secondly, we provide an analytic expression for the long time behavior of the response function of the material. This result is obtained by means of the Tauberian theorems for the Laplace transform. Finally, we relate the obtained results to a peculiar model for fluid-filled elastic tubes.

math-ph↗