Federico Polito

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.

The space-fractional Poisson process

In this paper, we introduce the space-fractional Poisson process whose state probabilities pkα(t), t≥0, α∈(0,1], are governed by the equations (d/dt)pkα(t)=−λα(1−B)αpkα(t), where (1−B)α is the fractional difference operator found in the time series analysis. We explicitly obtain the distributions pkα(t), the probability generating functions Gα(u,t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space–time-fractional Poisson process of which we give the explicit distribution.

Fractional pure birth processes

We consider a fractional version of the classical nonlinear birth process of which the Yule–Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the difference-differential equations which govern the probability law of the process with the Dzherbashyan–Caputo fractional derivative. We derive the probability distribution of the number N�(t) of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation N�(t) = N(T2�(t)), where N(t) is the classical generalized birth process and T2�(t) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed.

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.

Simulation and estimation for the fractional Yule process

In this paper, we propose some representations of a generalized linear birth process called fractional Yule process (fYp). We also derive the probability distributions of the random birth and sojourn times. The inter-birth time distribution and the representations then yield algorithms on how to simulate sample paths of the fYp. We also attempt to estimate the model parameters in order for the fYp to be usable in practice. The estimation procedure is then tested using simulated data as well. We also illustrate some major characteristics of fYp which will be helpful for real applications.

On a fractional linear birth–death process

In this paper, we introduce and examine a fractional linear birth–death process N�(t), t > 0, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities p �(t), t > 0, k � 0. We present a subordination relationship connecting N�(t), t > 0, with the classical birth–death process N(t), t > 0, by means of the time process T2�(t), t > 0, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability p �(t) and the state probabilities

Some properties of Prabhakar-type fractional calculus operators

In this paper we study some properties of the Prabhakar integrals and derivatives and of some of their extensions such as the regularized Prabhakar derivative or the Hilfer--Prabhakar derivative. Some Opial- and Hardy-type inequalities are derived. In the last section we point out on some relationships with probability theory.

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.

On the integral of fractional Poisson processes

In this paper, we consider the Riemann–Liouville fractional integral Nα,ν(t)=1Γ(α)∫0t(t−s)α−1Nν(s)ds, where Nν(t), t≥0, is a fractional Poisson process of order ν∈(0,1], and α>0. We give the explicit bivariate distribution Pr{Nν(s)=k,Nν(t)=r}, for t≥s, r≥k, the mean ENα,ν(t) and the variance VarNα,ν(t). We study the process Nα,1(t) for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on N1,1(t) are presented in the last section where also distributional properties of the integrated Poisson process (including the representation as random sums) is derived. The integral of powers of the Poisson process is examined and its connections with generalized harmonic numbers are discussed.

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.