Roberto Garra

Hilfer-Prabhakar Derivatives and Some Applications

Abstract 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. Furthermore, we show some applications of these generalized Hilfer–Prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations, and in difference–differential equations governing the dynamics of generalized renewal stochastic processes.

Fractional relaxation with time-varying coefficient

From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.

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.

A fractional variational approach to the fractional basset-type equation

In this paper we discuss an application of fractional variational calculus to the Basset-type fractional equations. It is well known that the unsteady motion of a sphere immersed in a Stokes fluid is described by an integro-differential equation involving derivative of real order. Here we study the inverse problem, i.e. we consider the problem from a Lagrangian point of view in the framework of fractional variational calculus. In this way we find an application of fractional variational methods to a classical physical model, finding a Basset-type fractional equation starting from a Lagrangian depending on derivatives of fractional order.

The Prabhakar or three parameter Mittag–Leffler function: Theory and application

Abstract 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.

STATE-DEPENDENT FRACTIONAL POINT PROCESSES

3 Abstract. The aim of this paper is the analysis of the fractional Poisson process where the state probabilities p k k ptq, t ¥ 0, are governed by time-fractional equations of order 0 † k ⁄ 1 depending on the numberk of events occurred up to timet. We are able to obtain explicitely the Laplace transform of p k k ptq and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on k diers from that constructed from the fractional state equations (in the case k , for all k, they coincide with the time-fractional Poisson process). We also introduce a dierent 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.

Analytic solutions of fractional differential equations by operational methods

Abstract 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.

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

Abstract 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 ∈ (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.

Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

Abstract We consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of the Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing the Riemann-Liouville integrals with Erdélyi-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 the 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.

Fractional Klein-Gordon Equations and Related Stochastic Processes

This paper presents finite-velocity random motions driven by fractional Klein–Gordon equations of order