Analytic solutions of fractional differential equations by operational methods

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

Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity

In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the time dependent viscosity by using an integro-differential term and therefore generalizing the classical equation of a Newtonian viscous fluid. A possible useful choice, in this framework, is to use a rheology based on stress/strain relation generalized by fractional calculus modelling. This is a model that can be used in applied problems, taking into account a power law time variability of the viscosity coefficient. We find analytic solutions of initial value problems in an unbounded and bounded domain. Furthermore, we discuss the explicit solution in a meaningful particular case

Random flights governed by Klein-Gordon-type partial differential equations

In this paper we study random flights in R^d with displacements possessing Dirichlet distributions of two different types and uniformly oriented. The randomization of the number of displacements has the form of a generalized Poisson process whose parameters depend on the dimension d. We prove that the distributions of the point X(t) and Y(t), t \geq 0, performing the random flights (with the first and second form of Dirichlet intertimes) are related to Klein-Gordon-type partial differential equations. Our analysis is based on McBride theory of integer powers of hyper-Bessel operators. A special attention is devoted to the three-dimensional case where we are able to obtain the explicit form of the equations governing the law of X(t) and Y(t). In particular we show that that the distribution of Y(t) satisfies a telegraph-type equation with time-varying coefficients

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 note on Hadamard fractional differential equations with varying coefficients and their applications in probability

In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions

Correlated fractional counting processes on a finite time interval

We present some correlated fractional counting processes on a finite time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to zero, the univariate distributions coincide with the ones of the space-time fractional Poisson process in Orsingher and Polito (2012). On the other hand, when we consider the time fractional Poisson process, the multivariate finite dimensional distributions are different from the ones presented for the renewal process in Politi et al. (2011). Another case concerns a class of fractional negative binomial processes

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

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 $\nu \in (0,1]$, the logarithmic creep law known in rheology as Lomnitz law (obtained for $\nu=1$). 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.Comment: 15 pages, 2 figures, to appear in Chaos, Solitons and Fractals (2017