Nonlocal and Fractional Operators

We study some features of the transient probability distribution of a fractional \(M/M/\infty \) queueing system. Such model is constructed as a suitable time-changed birth-death process. The fractional differential-difference problem is studied for the corresponding probability distribution and a fractional partial differential equation is obtained for the generating function. Finally, the interpretation of the system as an actual \(M/M/\infty \) queue and as a M/M/1 queue with responsive server is given and some conditioned virtual waiting times are studied.

We shall examine the fractional generalization of the eigenvalue problem of Schrödinger’s equation for one dimensional problems in connection with Lévy stable probability distributions. The corresponding Sturm–Liouville (SL) problem for the fractional Schrödinger equation is formulated and solved on \(\mathbb {R}\) satisfying natural Dirichlet boundary conditions. The eigenvalues and eigenfunctions are computed in a numerical Sinc approximation applied to the Riesz–Feller representation of Schrödinger’s generalized equation. We demonstrate that the eigenvalues for a fractional operator approach deliver the well known eigenvalues of the integer order Schrödinger equation and are consistent with analytic WKB estimations. We can also confirm the conjecture that only for skewness parameters \(\theta =0\) the eigenvalues are real quantities and thus relevant in quantum mechanics. However, for skewness parameters \(\theta \ne 0\), the Sinc approach yields complex eigenvalues with related complex eigenfunctions, and a fortiori, real probability densities.

We consider the problem of describing the dynamics of a test particle moving in a thermal bath using the stochastic differential equations. We briefly recall the stochastic approach to the Brownian based on the statistical properties of collision theory for a gas of elastic particles and the molecular chaos hypothesis. The mathematical formulation of the Brownian motion leads to the formulation of the Ornstein-Uhlenbeck equation that provides a stationary solution consistent with the Maxwell-Boltzmann distribution. According to the stochastic thermodynamics, we assume that the stochastic differential equations allow to describe the transient states of the test particle dynamics in a thermal bath and it extends their application to the study of the non-equilibrium statistical physics. Then we consider the problem of the dynamics of a test massive particle in a non homogeneous thermal bath where a gradient of temperature is present. We discuss as the existence of a local thermodynamics equilibrium is consistent with a Stratonovich interpretation of the stochastic differential equations with a multiplicative noise. The stochastic model applied to the test particle dynamics implies the existence of a long transient state during which the particle shows a net drift toward the cold region of the system. This effect recalls the thermophoresis phenomenon performed by large molecule in a solution in response to a macroscopic temperature gradient and it can be explained as an effect of the non-locality character of the collision interactions between the test particle and the thermal bath particles. To validate the stochastic model assumptions we analyze the simulation results of the 2-dimensional hard sphere gas obtained by using an event-based computer code, that solves exactly the sphere dynamics. The temperature gradient is simulated by the presence of two reflecting boundary conditions at different temperature. The simulations suggest that existence of a local thermodynamic equilibrium is justified and highlight the presence of a drift in the average dynamics of an ensemble of massive particles. The results of the paper could be relevant for the applications of stochastic dynamical systems to the non-equilibrium statistical physics that is a key issue for the Complex Systems Physics.

The dielectric susceptibility of a wide class of dielectric materials like magnetized laboratory and astrophysical plasmas, which are non local in space, characterizes an integral relation between the polarization P and the electric field E of the propagating electromagnetic perturbation. The electromagnetic fields in such dielectric media are described by fractional differential equations with space derivatives of non-integer order. In this paper an attempt to derive the fractional differential equation from the Maxwell equation system for the quasi-longitudinal waves propagating in an unmagnetized plasma like the Lower Hybrid (LH) waves (also useful in the Thermonuclear Fusion Research domain) or the Langmuir waves is outlined. Extrapolation of the method can also been considered for magnetized plasma. A one-dimensional example of fractional wave equation is given and new family of analytical solutions has been found.

In this paper we reconsider the classical nonlinear diffusivity equation of real gas in an heterogenous porous medium in light of the recent studies about nonlocal space-fractional generalizations of diffusion models. The obtained equation can be simply linearized into a classical space-fractional diffusion equation, widely studied in the literature. We consider the case of a power-law pressure-dependence of the permeability coefficient. In this case we provide some useful new exact analytical results. In particular, we are able to find a Barenblatt-type solution for a space-fractional Boussinesq equation, arising in this context.

Using the concepts and formalism of different families of Hermite polynomials, we discuss here some generalizations of polynomials belonging to the Bernoulli class, and we also show how to represent the action of the operators involving fractional derivatives. In particular, by using the method of generating function, we introduce generalized Bernoulli polynomials by operating in their generating function with the formalism of the two-variable Hermite polynomials. In addition, we extend some operational techniques in order to derive different forms of Bernoulli numbers and polynomials. Finally, we explore some general properties of generalized Bernoulli polynomials, focusing on their extension to the 2D case, and we introduce a family of polynomials strictly related to the Hermite polynomials in order to compute the effect of fractional operators on a given function.

We modify ETAS models by replacing the Pareto-like kernel proposed by Ogata with a Mittag-Leffler type kernel. Provided that the kernel decays as a power law with exponent \(\beta + 1 \in (1,2]\), this replacement has the advantage that the Laplace transform of the Mittag-Leffler function is known explicitly, leading to simpler calculation of relevant quantities.

This work deals with the results and the simulations obtained for the time-fractional diffusion-wave equation, i.e. a diffusion-like linear integro partial differential equation containing a pseudo-differential operator interpreted as a fractional derivative in time. The data function (initial signal) is provided by a box-function and the solutions are so obtained by a convolution of the Green function with the initial data function. The relevance of the topic lies in the possibility of describing physical processes that interpolates between the different responses of the diffusion and waves equations, equipped with a physically realistic initial signal. Here two problems are considered where the use of the Laplace transform in the analysis of the problems has lead since 1990s to special functions of the Wright type.

In this paper, we deal with some recent and old results, concerning fractional operators, obtained via the extension technique. This approach is particularly fruitful for exploiting some of those well known properties, true for the local operators obtained via the extension approach, for deducing some parallel results about the underlaying nonlocal operators.

In the last two decades, the theoretical advancement of the point processes witnessed an important and deep interconnection with the fractional calculus. It was also found that the stable subordinator plays a vital role in this connection. The survey intends to present recent results on the fractional versions of point processes. We will also discuss generalization attempted by several authors in this direction. Finally, we present some plots and simulations of the well-known fractional Poisson process of Laskin (2003).

We investigate a generalization of the randomly accelerated motion obtained by iterated integration of the telegraph signal. We give the exact and explicit expression for the cumulative distribution function, conditionally on the number n of Poisson events, when n is sufficiently small. The unconditional mean value and variance are also obtained.

We review the Probabilistic Domain Decomposition (PDD) method for the numerical solution of linear and nonlinear Partial Differential Equation (PDE) problems. This Domain Decomposition (DD) method is based on a suitable probabilistic representation of the solution given in the form of an expectation which, in turns, involves the solution of a Stochastic Differential Equation (SDE). While the structure of the SDE depends only upon the corresponding PDE, the expectation also depends upon the boundary data of the problem. The method consists of three stages: (i) only few values of the sought solution are solved by Monte Carlo or Quasi-Monte Carlo at some interfaces; (ii) a continuous approximation of the solution over these interfaces is obtained via interpolation; and (iii) prescribing the previous (partial) solutions as additional Dirichlet boundary conditions, a fully decoupled set of sub-problems is finally solved in parallel. For linear parabolic problems, this is based on the celebrated Feynman-Kac formula, while for semilinear parabolic equations requires a suitable generalization based on branching diffusion processes. In case of semilinear transport equations and the Vlasov-Poisson system, a generalization of the probabilistic representation was also obtained in terms of the Method of Characteristics (characteristic curves). Finally, we present the latest progress towards the extension of the PDD method for nonlocal fractional operators. The algorithm notably improves the scalability of classical algorithms and is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Numerical examples conducted in 1D and 2D, including some for the KPP equation and Plasma Physics, are given.

In this contribution we show that fractional diffusion emerges from a simple Markovian Gaussian random walk when the medium displays a power-law heterogeneity. Within the framework of the continuous time random walk, the heterogeneity of the medium is represented by the selection, at any jump, of a different time-scale for an exponential survival probability. The resulting process is a non-Markovian non-Gaussian random walk. In particular, for a power-law distribution of the time-scales, the resulting random walk corresponds to a time-fractional diffusion process. We relates the power-law of the medium heterogeneity to the fractional order of the diffusion. This relation provides an interpretation and an estimation of the fractional order of derivation in terms of environment heterogeneity. The results are supported by simulations.

In this article, we formulate two kinds of time fractional derivatives of the Caputo type with order \(\alpha \) in fractional Sobolev spaces and prove that they are isomorphisms between the corresponding Sobolev space of order \(\alpha \) and the \(L^2\)-space. On the basis of such fractional derivatives, we formulate initial value problems for time fractional ordinary differential equations and prove the well-posedness.