Numerical evaluation of two and three parameter Mittag-Leffler functions

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 for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.Comment: Accepted for publication in SIAM Journal on Numerical Analysi

Evaluation of generalized Mittag-Leffler functions on the real line

This paper addresses the problem of the numerical computation of generalized Mittag-Leffler functions with two parameters, with applications in fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are investigated and combined with quadrature rules for the numerical integration. An in-depth error analysis is carried out to select suitable contour's parameters, depending on the parameters of the Mittag-Leffler function, in order to achieve any fixed accuracy. We present numerical experiments to validate theoretical results and some computational issues are discussed

On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics

The three parameters Mittag-Leffler function (often referred to as the Prabhakar function) has important applications, mainly in physics of dielectrics, in describing anomalous relaxation of non-Debye type. This paper concerns with the investigation of the conditions, on the characteristic parameters, under which the function is locally integrable and completely monotonic; these properties are essential for the physical feasibility of the corresponding models. In particular the classical Havriliak–Negami model is extended to a wider range of the parameters. The problem of the numerical evaluation of the three parameters Mittag-Leffler function is also addressed and three different approaches are discussed and compared. Numerical simulations are hence used to validate the theoretical findings and present some graphs of the function under investigation (lavoro effettuato nell'ambito di ricerche finanziate dall'INdAM)

Fractional prabhakar derivative and applications in anomalous dielectrics: A numerical approach

Fractional integrals and derivatives based on the Prabhakar function are useful to describe anomalous dielectric properties of materials whose behaviour obeys to the Havriliak–Negami model. In this work some formulas for defining these operators are described and investigated. A numerical method of product-integration type for solving differential equations with the Prabhakar derivative is derived and its convergence properties are studied. Some numerical experiments are presented to validate the theoretical results

On Volterra functions and Ramanujan integrals

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 authors have studied this family of functions and faced with the problem of providing a clear understanding of their asymptotic behavior for small and large arguments. This paper reviews some of the most important results on Volterra functions and in particular collects, into a quite general framework, several results on their asymptotic expansions; these results turn out to be useful not only for the full understanding of the behavior of the Volterra functions but also for their numerical computation. The connections with integrals of Ramanujan type, which have several important applications, are also discussed

Computing the matrix Mittag-Leffler function with applications to fractional calculus

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 function in matrix arguments may require the computation of derivatives of possible high order depending on the matrix spectrum. Regarding the ML function, the numerical computation of its derivatives of arbitrary order is a completely unexplored topic; in this paper we address this issue and three different methods are tailored and investigated. The methods are combined together with an original derivatives balancing technique in order to devise an algorithm capable of providing high accuracy. The conditioning of the evaluation of matrix ML functions is also studied. The numerical experiments presented in the paper show that the proposed algorithm provides high accuracy, very often close to the machine precision

On the Kuzmin model in fractional Newtonian gravity

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 alternative to the dark matter paradigm through a non-local modification of Newton's theory. We provide an in-depth discussion of the gravitational potential for the Kuzmin disk within this new approach. Specifically, we derive an integral and a series representation for the potential, we verify its asymptotic behavior at large scales, and we provide illuminating plots of the resulting equipotential surfaces.Comment: 13 pages, 8 figure

Renewal processes linked to fractional relaxation equations with variable order

We introduce and study here a renewal process de fined by means of a time-fractional relaxation equation with derivative order \alpha(t) varying with time t \geq 0. In particular, we use the operator introduced by Scarpi in the Seventies (see [23]) and later reformulated in the regularized Caputo sense in [4], inside the framework of the so-called general fractional calculus. The model obtained extends the well-known time-fractional Poisson process of fi xed order \alpha \in (0,1) and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the interarrival times) with respect to time. The variable order renewal process is proved to fall outside the usual subordinated representation, since it can not be simply defi ned as a Poisson process with random time (as happens in the standard fractional case). Finally a related continuous-time random walk model is analysed and its limiting behavior established

Models of dielectric relaxation based on completely monotone functions

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, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order