Propagation Speed of the Maximum of the Fundamental Solution to the Fractional Diffusion-Wave Equation

In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order $1 \le \alpha \le 2$ is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the time fractional diffusion-wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion-wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce plots are discussed.Comment: 22 pages 6 figures. This paper has been presented by F. Mainardi at the International Workshop: Fractional Differentiation and its Applications (FDA12) Hohai University, Nanjing, China, 14-17 May 201

Random-time processes governed by differential equations of fractional distributed order

We analyze here different types of fractional differential equations, under the assumption that their fractional order $\nu \in (0,1]$ is random\ with probability density $n(\nu).$ We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process $N(t),t>0.$\ We prove that, for a particular (discrete) choice of $n(\nu)$, it leads to a process with random time, defined as $N(\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}(t)),t>0.$ The distribution of the random time argument $\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}(t)$ can be expressed, for any fixed $t$, in terms of convolutions of stable-laws. The new process $N(\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}})$ is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of $N(\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}})$, as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see \cite{Vib}%). In view of the previous results it is natural to consider diffusion-type fractional equations of distributed order. We present here an approach to their solutions in terms of composition of the Brownian motion $B(t),t>0$ with the random time $\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}$. We thus provide an alternative to the constructions presented in Mainardi and Pagnini \cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order case.Comment: 26 page

An Historical Perspective on Fractional Calculus in Linear Viscoelasticity

The article provides an historical survey of the early contributions on the applications of fractional calculus in linear viscoelasticty. The period under examination covers four decades, since 1930's up to 1970's and authors are from both Western and Eastern countries. References to more recent contributions may be found in the bibliography of the author's book. This paper reproduces, with Publisher's permission, Section 3.5 of the book: F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press - London and World Scienti?c - Singapore, 2010.Comment: 6 page

Fractional Calculus in Wave Propagation Problems

Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.Comment: 33 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1008.134

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

A note on the equivalence of fractional relaxation equations to differential equations with varying coefficients

In this note we show how a initial value problem for a relaxation process governed by a differential equation of non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying coefficient. This equivalence is shown for the simple fractional relaxation equation that points out the relevance of the Mittag-Leffler function in fractional calculus. This simple argument may lead to the equivalence of more general processes governed by evolution equations of fractional order with constant coefficients to processes governed by differential equations of integer order but with varying coefficients. Our main motivation is to solicit the researchers to extend this approach to other areas of applied science in order to have a more deep knowledge of certain phenomena, both deterministic and stochastic ones, nowadays investigated with the techniques of the fractional calculus.Comment: 6 pqages 4 figure

AN ECONOMETRIC ANALYSIS OF FACTORS AFFECTING TROPICAL AND SUBTROPICAL DEFORESTATION

In most developing countries deforestation has reached alarming rates. In view of their relevance for the local economy (e.g., as a source of foreign exchange earnings and supply of fuelwood), an adequate management of forest resources should be pursued. In these economies forest exploitation and land conversion have often been seen as a temporary solution to structural problems. In this way, however, the same problems are even aggravated in the long run. The study first reviews recent explanations of tropical deforestation: a distinction is drawn between areas of substantial agreement on the one hand, and discordant results and interpretations on the other. In the main part of the analysis, based on cross-country data for the 1980s, regression models incorporating different sets of determinants of deforestation are applied. Compared to previous studies, the analysis tries to better account for the sequential timing of some of these determinants. Different patterns are identified among country groups, according to specific features of economic activities, macroeconomic and political environments, and climatic conditions.Land Economics/Use, Resource /Energy Economics and Policy,