290 citations · 1.8k across the 21 of their papers we have counts for
22 papers
On some inequalities for the two-parameter Mittag-Leffler function in the complex plane
Roberto Garrappa, Stefan Gerhold, Marina Popolizio +1
For the two-parameter Mittag-Leffler function with and we consider the question whether and are comparable on the whole c…
Renewal processes linked to fractional relaxation equations with variable order
Luisa Beghin, Lorenzo Cristofaro, Roberto Garrappa
We introduce and study here a renewal process defined by means of a time-fractional relaxation equation with derivative order varying with time . In particular, we u…
A computational approach to exponential-type variable-order fractional differential equations
Roberto Garrappa, Andrea Giusti
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation…
On the fractional Laplacian of variable order
Eric Darve, Marta D'Elia, Roberto Garrappa +2
We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential. Our definition holds for values of the…
Variable-order fractional calculus: a change of perspective
Roberto Garrappa, Andrea Giusti, Francesco Mainardi
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…
On the Kuzmin model in fractional Newtonian gravity
Andrea Giusti, Roberto Garrappa, Geneviève Vachon
Fractional Newtonian gravity, based on the fractional generalization of Poisson's equation for Newtonian gravity, is a novel approach to Galactic dynamics aimed at providing an alt…
Stability of fractional-order systems with Prabhakar derivatives
Roberto Garrappa, Eva Kaslik
Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showi…
On a generalized three-parameter Wright function of the Le Roy type
Roberto Garrappa, Sergei Rogosin, Francesco Mainardi
Recently S. Gerhold and R. Garra-F. Polito independently introduced a new function related to the special functions of Mittag-Leffler family. This function is a generalization of t…
Why fractional derivatives with nonsingular kernels should not be used
Kai Diethelm, Roberto Garrappa, Andrea Giusti +1
In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-…
On initial conditions for fractional delay differential equations
Roberto Garrappa, Eva Kaslik
Derivatives of fractional order are introduced in different ways: as left-inverse of the fractional integral or by generalizing the limit of the difference quotient defining intege…
A practical guide to Prabhakar fractional calculus
Andrea Giusti, Ivano Colombaro, Roberto Garra +4
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…
Trapezoidal methods for fractional differential equations: theoretical and computational aspects
Roberto Garrappa
The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss comput…
Some results on the complete monotonicity of the Mittag-Leffler functions of Le Roy type
K. Górska, A. Horzela, R. Garrappa
The paper by R. Garrappa, S. Rogosin, and F. Mainardi, entitled {\em On a generalized three-parameter Wright function of the Le Roy type} and published in [Fract. Calc. Appl. Anal.…
Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations
Roberto Garrappa
In the paper titled "New numerical approach for fractional differential equations" by A. Atangana and K.M. Owolabi [Math. Model. Nat. Phenom., 13(1), 2018], it is presented a metho…
Computing the matrix Mittag-Leffler function with applications to fractional calculus
Roberto Garrappa, Marina Popolizio
The computation of the Mittag-Leffler (ML) function with matrix arguments, and some applications in fractional calculus, are discussed. In general the evaluation of a scalar functi…
The Prabhakar or three parameter Mittag--Leffler function: theory and application
Roberto Garra, Roberto Garrappa
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 pr…
Models of dielectric relaxation based on completely monotone functions
Roberto Garrappa, Francesco Mainardi, Guido Maione
The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts,…
On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics
Francesco Mainardi, Roberto Garrappa
The three parameters Mittag--Leffler function (often referred as the Prabhakar function) has important applications, mainly in physics of dielectrics, in describing anomalous relax…
On Volterra functions and Ramanujan integrals
Roberto Garrappa, Francesco Mainardi
Volterra functions were introduced at the beginning of the twentieth century as solutions of some integral equations of convolution type with logarithmic kernel. Since then, few au…
Numerical evaluation of two and three parameter Mittag-Leffler functions
Roberto Garrappa
The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method f…
Suppressing chaos in discontinuous systems of fractional order by active control
Marius-F Danca, Roberto Garrappa
In this paper, a chaos control algorithm for a class of piece-wise continuous chaotic systems of fractional order, in the Caputo sense, is proposed. With the aid of Filippov's conv…
Exponential quadrature rules for linear fractional differential equations
Roberto Garrappa, Marina Popolizio
This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solut…