365 research outputs found
Alternative numerical computation of one-sided Levy and Mittag-Leffler distributions
We consider here the recently proposed closed form formula in terms of the
Meijer G-functions for the probability density functions of
one-sided L\'evy stable distributions with rational index , with
. Since one-sided L\'evy and Mittag-Leffler distributions are known
to be related, this formula could also be useful for calculating the
probability density functions of the latter. We show, however,
that the formula is computationally inviable for fractions with large
denominators, being unpractical even for some modest values of and . We
present a fast and accurate numerical scheme, based on an early integral
representation due to Mikusinski, for the evaluation of and
, their cumulative distribution function and their derivatives
for any real index . As an application, we explore some
properties of these probability density functions. In particular, we determine
the location and value of their maxima as functions of the index . We
show that and correspond,
respectively, to the one-sided L\'evy and Mittag-Leffler distributions with
shortest maxima. We close by discussing how our results can elucidate some
recently described dynamical behavior of intermittent systems.Comment: 6 pages, 5 figures. New references added, final version to appear in
PRE. Numerical code available at http://vigo.ime.unicamp.br/dist
Local discontinuous Galerkin methods for fractional ordinary differential equations
This paper discusses the upwinded local discontinuous Galerkin methods for
the one-term/multi-term fractional ordinary differential equations (FODEs). The
natural upwind choice of the numerical fluxes for the initial value problem for
FODEs ensures stability of the methods. The solution can be computed element by
element with optimal order of convergence in the norm and
superconvergence of order at the downwind point of each
element. Here is the degree of the approximation polynomial used in an
element and () represents the order of the one-term
FODEs. A generalization of this includes problems with classic 'th-term
FODEs, yielding superconvergence order at downwind point as
. The underlying mechanism of the
superconvergence is discussed and the analysis confirmed through examples,
including a discussion of how to use the scheme as an efficient way to evaluate
the generalized Mittag-Leffler function and solutions to more generalized
FODE's.Comment: 17 pages, 7 figure
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
Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics
We revisit the Mittag-Leffler functions of a real variable , with one, two
and three order-parameters , as far as their Laplace
transform pairs and complete monotonicty properties are concerned. These
functions, subjected to the requirement to be completely monotone for ,
are shown to be suitable models for non--Debye relaxation phenomena in
dielectrics including as particular cases the classical models referred to as
Cole-Cole, Davidson-Cole and Havriliak-Negami. We show 3D plots of the response
functions and of the corresponding spectral distributions, keeping fixed one of
the three order-parameters.Comment: 22 pages, 6 figures, Second Revised Versio
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
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