141 research outputs found
Studies on generalized Yule models
We present a generalization of the Yule model for macroevolution in which,
for the appearance of genera, we consider point processes with the order
statistics property, while for the growth of species we use nonlinear
time-fractional pure birth processes or a critical birth-death process.
Further, in specific cases we derive the explicit form of the distribution of
the number of species of a genus chosen uniformly at random for each time.
Besides, we introduce a time-changed mixed Poisson process with the same
marginal distribution as that of the time-fractional Poisson process.Comment: Published at https://doi.org/10.15559/18-VMSTA125 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
On the Integral of Fractional Poisson Processes
In this paper we consider the Riemann--Liouville fractional integral
, where , , is a
fractional Poisson process of order , and . We give
the explicit bivariate distribution , for , , the mean and the
variance . We study the
process for which we are able to produce explicit
results for the conditional and absolute variances and means. Much more
involved results on 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 generalised
harmonic numbers is 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
Discussion on the paper "On Simulation and Properties of the Stable Law" by L. Devroye and L. James
We congratulate the authors for the interesting paper. The reading has been
really pleasant and instructive. We discuss briefly only some of the
interesting results given in Devroye and James "On simulation and properties of
the stable law", 2014 with particular attention to evolution problems. The
contribution of the results collected in the paper is useful in a more wide
class of applications in many areas of applied mathematics
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
Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions
We present the stochastic solution to a generalized fractional partial
differential equation involving a regularized operator related to the so-called
Prabhakar operator and admitting, amongst others, as specific cases the
fractional diffusion equation and the fractional telegraph equation. The
stochastic solution is expressed as a L\'evy process time-changed with the
inverse process to a linear combination of (possibly subordinated) independent
stable subordinators of different indices. Furthermore a related SDE is derived
and discussed
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 of individuals at an
arbitrary time . We also present an interesting representation for the
number of individuals at time , in the form of the subordination relation
, where is the
classical generalized birth process and 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.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ235 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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