Non-Markovian diffusion equations and processes: analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.Comment: 43 pages, 19 figures, in press on Physica A (2008

Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation

In this paper, after a brief review of the general theory of dispersive waves in dissipative media, we present a complete discussion of the dispersion relations for both the ordinary and the time-fractional Cattaneo-Maxwell heat equations. Consequently, we provide a complete characterization of the group and phase velocities for these two cases, together with some non-trivial remarks on the nature of wave dispersion in fractional models.Comment: 18 pages, 7 figure

Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy

Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality

Two-particle anomalous diffusion: Probability density functions and self-similar stochastic processes

Two-particle dispersion is investigated in the context of anomalous diffusion. Two different modeling approaches related to time subordination are considered and unified in the framework of self-similar stochastic processes. By assuming a single-particle fractional Brownian motion and that the two-particle correlation function decreases in time with a power law, the particle relative separation density is computed for the cases with time subordination directed by a unilateral M-Wright density and by an extremal Lévy stable density. Looking for advisable mathematical properties (for instance, the stationarity of the increments), the corresponding selfsimilar stochastic processes are represented in terms of fractional Brownian motions with stochastic variance, whose profile is modelled by using the M-Wright density or the Lévy stable density

Variable-order fractional calculus: A change of perspective

Several approaches to the formulation of a fractional theory of calculus of “variable order” have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative

Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion

Boundary value problems for the diffusion equation of the variable order in differential and difference settings

Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case. The credibility of the obtained results is verified by performing numerical calculations for a test problem.Comment: 19 pages. Presented at the 4-th IFAC Workshop on Fractional Differentiation and Its Applications, Badajoz, Spain, October 18-20, 201

Recent history of fractional calculus

This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date

Systemisation of knowledge for the conservation and cultural development uf piedmont's mosaic heritage

Mosaics, in all their possible variants of form, material and location, can and must be recognised within the definition of Architectural Heritage. A further examination also reveals that mosaics are fully included within the definition of Cultural Heritage and, so, constitute part of the CH of a territorial area. In the absence of specific regulations, studies have been carried out in relation to source data and scheduling instruments at national and regional level with a view to devising a schedule model specifically for the mosaic, so that it is no longer regarded as an archaeological finding in its own right, but as a systematic element. These have been compared with other local situations, in Italy and abroad, which need unambiguous parameters for standardisation. These operations pass unavoidably through the identification of parameters, metadata, final users and methods through which the project could be developed in the future. Of no small importance is the diversified input of specific and inter-disciplinary skills, which are necessary for a correct cataloguing of resources; that means determining the obligatory fields and structuring the various headings, devising also appropriate key words. The cataloguing procedure is fundamental in the process for an effective cultural development of Piedmont's mosaic heritage. More precisely, it becomes an element in a structure for multi-level querying of the Territorial Information System, devised in particular for visualising data relating to files on interactive support, but also for a web-GIS configuration