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Plenary Talk

ICFDA'14 plenary talk:

Igor Podlubny

Igor Podlubny

Title: The Evolution of Generalized Differentiation

Abstract: Historical and conceptual development of integration and differentiation of non-integer order will be discussed.

Yuri Luchko

Yury Luchko

Title: Selected topics in fractional transport equations

Abstract: Fractional transport processes are for sure among most popular topics in FC and its applications and include in particular fractional diffusion, fractional diffusion-wave processes, and fractional wave propagation. In this talk, some remarkable mathematical and physical properties of the fractional transport equations and their solutions are revisited. From the mathematical viewpoint, the symmetry groups of scaling transformations of the fractional transport equations and their group-invariant solutions, the maximum principle for the fractional transport equations and its applications, as well as the role of the Mellin integral transform technique for the analytical treatment of the fractional transport equations with the constant coefficients are considered. Physical aspects include a discussion of the probabilistic interpretation of solutions to the Cauchy problems for the fractional transport equations, some different concepts for the propagation velocities of the fractional transport processes, and their entropy and the entropy production rates.

Rudolf Hilfer

Title: Time Flow and the Foundations of Nonequilibrium Statistical Mechanics

Abstract: The time evolution of macroscopic states (or mixtures) for classical and quantum many body systems in statistical physics need not correspond to a translation group or semigroup. Instead convolution semigroups appears generically. The presentation will discuss the implications of this result for the foundations of nonequilibrium statistical physics as well as possible applications to experiment.

Enzo Orsingher

Enzo Orsingher

Title: Random flights governed by fractional D'Alembert operators

Abstract: We consider the fractional equations of the form (1)
$$ \left({\partial^2 \over \partial t^2} - c^2 {\partial^2 \over \partial x^2}\right)^\alpha u = \lambda u, 0 < \alpha < 1$$ obtained by taking the fractional power of the D'Alembert operator in the Klein-Gordon equation. For \( \alpha = 1 \) this equation is related to the classical telegraph process. The analysis of (1) can be done by applying the McBride theory on fractional powers of hyper-Bessel operators of which some hints are provided. The representation of fractional powers in terms of Erdelyi-Kober integrals (generalizing the classical Riemann-Liouville integrals) is the main tool of our analysis. We are able to construct fractional telegraph processes \( {T^\alpha}(t), 0 < \alpha < 1\), whose distributions are related to (1). The distribution obtained coincides for \( \alpha = 1 \) to the well-known law of the telegraph process and its conditional distributions coincide to those derived by using the order statistics approach. The same tools permit us to obtain the fractional planar random flight, the conditional and unconditional laws and the inhomogenous equation governing the distribution. Hints are given for fractional random flights in Euclidean spaces of dimension \( d > 2 \). Within the framework of the theory presented it si possible to derive the explicit equations governing the law of random flights with Dirichlet-distributed displacements and with a fractional Poisson number of changes of direction.

Jose Antonio Tenreiro Machado

Jose Antonio Tenreiro Machado

Title:The Persistence of Memory

Abstract: This paper analyses several natural and man-made complex phenomena in the perspective of dynamical systems. Such phenomena are often characterized by the absence of a characteristic length-scale, long range correlations and persistent memory, which are features also associated to fractional order systems. For each system, the output, interpreted as a manifestation of the system dynamics, is analysed by means of the Fourier transform. The amplitude spectrum is approxi- mated by a Power Law function and the parameters interpreted as an underlying signature of the system dynamics. The complex systems under analysis are then compared in a global perspective in order to unveil and visualize hidden similarities among them.

Semiplenary Talk

ICFDA'14 semiplenary talk:

Mark M Meerschaert

Mark M. Meerschaert

Title: Tempered Fractional Calculus

Abstract: Fractional derivatives and integrals are convolutions with a power law. Including an exponential term leads to tempered fractional derivatives and integrals. Tempered stable processes are the limits of random walk models where the power law probability of long jumps is tempered by an exponential factor. These random walks converge to tempered stable stochastic process limits, whose probability densities solve tempered fractional diffusion equations. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. A tempered Grunwald-Letnikov formula provides the basis for finite difference methods to solve tempered diffusion equations. Tempered fractional Brownian motion, the tempered fractional integral or derivative of a Brownian motion, is a new stochastic process that can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data, consistent with the Davenport spectrum.

Todd Freeborn

Title: Fractional-Order Circuits: State-of-the Art Design and Applications

Abstract: Fractional-Order Circuits have lately gained increased attention as they show a wide range of applications ranging from modeling of biomedical tissues and organs, modeling of super-capacitors and fuel cells to the design of fractional-step analog filters and ultra-high-frequency oscillators. The purpose of this talk is to offer a survey of the state-of-the art in Fractional-Order Circuit Design and applications highlighting the fast progress that has been achieved in the fabrication of fractional capacitors and the diagnosis of a number of diseases using noninvasive electrical devices based on fractional-order circuit models.

Teodor Atanackovic

Teodor Atanackovic

Title: Linear viscoelasticity of real and complex fractional order

Abstract: Although one of the first area where fractional derivatives of real order are used, the application of complex order fractional derivatives in linear viscoleasticity is not often considered. In the presentation we shall review the real order fractional derivatives linear viscoelasticity and then consider the extension to the complex order derivatives case. Special attention will be paid to the mathematical and thermodynamical restrictions. Several examples will be treated in detail. The presentation is joint work with Sanja Konjik, S. Pilipovic and D. Zorica.

Giuseppe Nunnari

Giuseppe Nunnari

Title: Evidences of Self-Organized Criticality in Volcanology

Abstract: The aim of this talk is to provide evidences about the Self-Organized Criticality (SOC) of volcanic systems. In particular, after a review of literature results, concerning the seismicity in volcanic areas and the high frequency acoustic emissions, we present some results concerning the inter-time power law distribution of eruptions and explosive activity on Mt Etna and Stromboli and about lava fountains at Mt. Etna. The power laws experimentally evaluated from considered data set are reported. Results, even far to be exhaustive, may encourage some other scientists to consider SOC as a unifying approach also in the field of volcanology.

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