Showing papers in "Communications in Applied and Industrial Mathematics in 2014"
TL;DR: In this article, the correlation function for time-changed L evy processes has been studied in the context of continuous time random walks, where the second-order correlation function of a continuous-time random walk is defined.
Abstract: Time-changed L evy processes include the fractional Poisson process, and the scaling limit of a continuous time random walk. They are obtained by replacing the deterministic time variable by a positive non-decreasing random process. The use of time-changed processes in modeling often requires the knowledge of their second order properties such as the correlation function. This paper provides the explicit expression for the correlation function for time-changed L evy processes. The processes used to model random time include subordinators and inverse subordinators, and the time-changed L evy processes include limits of continuous time random walks. Several examples useful in applications are discussed.
68 citations
TL;DR: The pioneering role of the British scientist, Professor G.W.Scott Blair, in the creation of the application of fractional modelling in rheology is discussed in this article.
Abstract: The article shows the pioneering role of the British scientist, Professor G.W.Scott Blair, in the creation of the application of fractional modelling in rheology. Discussion of his results is presented. His approach is highly recognized by the rheological society and is adopted and generalized by his successors. Further development of this branch of Science is briefly described in this article too.
43 citations
TL;DR: In this paper, a multi-dimensional fractional wave equation that describes propaga- tion of damped waves is introduced and analyzed, and new integral representations of the fundamental solution of the multidimensional wave equa- tion are presented.
Abstract: In this paper, a multi-dimensional fractional wave equation that describes propaga- tion of damped waves is introduced and analyzed. In contrast to the fractional diffusion- wave equation, the fractional wave equation contains fractional derivatives of the same order α, 1 ≤ α ≤ 2 both in space and in time. This feature is a decisive factor for inher- iting some crucial characteristics of the wave equation, such as, a constant phase velocity of the damped waves which is now described by the fractional wave equation. Some new integral representations of the fundamental solution of the multi-dimensional wave equa- tion are presented. In the one- and three-dimensional cases, the fundamental solution is obtained in explicit form in terms of elementary functions. In the one-dimensional case, the fundamental solution is shown to be a spatial probability density function evolving in time. However, for the dimensions greater than one, the fundamental solution can be negative, and therefore, does not allow a probabilistic interpretation. To illustrate the analytical findings, results of numerical calculations and numerous plots are presented.
40 citations
TL;DR: In this paper, the authors studied the asymptotic behavior of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories.
Abstract: In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We show that these sums converge in law to the integral of the weight process with respect to the Brownian motion when the observation distances goes to zero. The result is obtained with the help of fractional calculus showing the power of this technique. This study, though interesting by itself, is motivated by an error found in the proof of Theorem 4 in: J. M. Corcuera, D. Nualart, and J. H. C. Woerner, Power variation of some integral fractional processes, Bernoulli, vol. 12, no. 4, pp. 713–735, 2006.
28 citations
TL;DR: Veretennikova and Kolokoltsov as discussed by the authors studied a controlled continuous time random walk and their position-dependent extensions and derived the optimal payoff function for their scaling limits.
Abstract: In the article we study a controlled Continuous Time Random Walk and their position-dependent extensions. We heuristically derive the optimal payoff function equa- tions for their scaling limits. The general equation for the corresponding optimal payoff of the limiting process may be called a fractional Hamilton Jacobi Bellman equation. This paper is an improved version of the preprint: M. Veretennikova and V. Kolokoltsov. Controlled continuous time ran- dom walks and fractional Hamilton Jacobi Bellman equations. Arxiv, http://arxiv.org/abs/1203.6333, 2012.
27 citations
TL;DR: In this paper, the invariant subspace method has been used to find exact solutions to nonlinear time-fractional partial differential equations by separating variables, including Caputo derivatives and Mittag-Leffler functions.
Abstract: In this paper we study some cases of time-fractional nonlinear dispersive equations (NDEs) involving Caputo derivatives, by means of the invariant subspace method. This method allows to find exact solutions to nonlinear time-fractional partial differential equations by separating variables. We first consider a third order time-fractional NDE that admits a four-dimensional invariant subspace and we find a similarity solution. We also study a fifth order NDE. In this last case we find a solution involving Mittag-Leffler functions. We finally observe that the invariant subspace method permits to find explicit solutions for a wide class of nonlinear dispersive time-fractional equations.
19 citations
TL;DR: In this article, a medium consisting of a region 0 < x < L and a region L < X < ∞ is considered, where heat conduction in one region is described by the equation with the Caputo time-fractional derivative of order α, whereas in another region, the problem is solved under conditions of perfect contact, i.e., when the temperatures at the contact point and the heat fluxes through the contact points are the same for both regions.
Abstract: A medium consisting of a region 0 < x < L and a region L < x < ∞ is considered. Heat conduction in one region is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another region is described by the equation with the time derivative of the order β. The problem is solved under conditions of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The solution valid for small values of time is expressed in terms of the Mittag-Leffler function and the Mainardi function. Several particular cases are considered and illustrated graphically.
13 citations
TL;DR: The M-Wright/Mainardi function as mentioned in this paper is the Green function of the time-fractional diffusion equation in the Caputo sense, and its history is briefly reported with its history since the first appearing in conferences and proceedings to the publication in peer-reviewed journals.
Abstract: Definition and main properties of the M-Wright/Mainardi function, which is the Green function of the time-fractional diffusion equation in the Caputo sense, are briefly reported with its history since the first appearing in conferences and proceedings to the publication in peer-reviewed journals.
10 citations
TL;DR: In this article, the interpretation of Ulysses and Voyager spacecrafts data on solar cosmic rays fluxes is studied, where the authors assume that particles move around turbulent magnetic field lines over spiral curves and see a one-dimensional walk with a finite speed.
Abstract: The work is devoted to the interpretation of Ulysses and Voyager spacecrafts data on solar cosmic rays fluxes. Particles move around turbulent magnetic field lines over spiral curves. Considering only their projections on these lines, we see a one-dimensional walk with a finite speed. Taking an inverse power law for the free path distribution leads to the differential equation with partial derivatives of fractional orders. Its solution gives the propagator which provides coincidence with observed data not only in the asymptotical region as obtained by Perri and Zimbardo (2007), but in the intermediate one as well.
5 citations
TL;DR: The identification of the parameters in a model of immune competition is treated in this article, where an approach of inverse problems toward the identification from measurements of densities of cells population is used.
Abstract: The identification of the parameters in a model of immune competition is treated in this paper. More precisely, an approach of inverse problems toward the identification from measurements of densities of cells population is used. The inverse problem is transferred into a parametric optimization problem using the nonlinear identification approach with a Least Square objective function. Global optimization techniques are pursued and a design procedure for global robust optimization is developed using the so-called Kriging method, optimization approaches are used to determine the global robust optimum of a surrogate model.
4 citations
TL;DR: This paper deals with the modeling of learning dynamics in a large system of interacting entities based on the kinetic theory on active particles that allows the derivation of specific models and of numerical simulations related to real systems.
Abstract: This paper deals with the modeling of learning dynamics in a large system of interacting entities. The mathematical approach is based on the kinetic theory on active particles. Their microscopic state is modeled by a scalar variable called activity, which is assumed to be heterogeneously distributed among the particles. Nonlinear interactions lead to collective phenomena of learning. The structure allows the derivation of specific models and of numerical simulations related to real systems.
TL;DR: In this paper, the fractal dimension of a vortex tangle is investigated and an alternative random walk toy model is proposed for very low temperature, which is valid also for finite temperature.
Abstract: This paper deals with the fractal dimension of a superfluid vortex tangle. It extends a previous model [J. Phys. A: Math. Theor. 43, 205501 (2010)] (which was proposed for very low temperature), and it proposes an alternative random walk toy model, which is valid also for finite
temperature. This random walk model combines a recent Nemirovskii’s proposal, and a simple modelization of a self-similar structure of vortex loops (mimicking the geometry of the loops of several sizes which compose the tangle). The fractal dimension of the vortex tangle is then related
to the exponents describing how the vortex energy per unit length changes with the length scales, for which we take recent proposals in the bibliography. The range between 1.35 and 1.75 seems the most consistent one.
TL;DR: Explicit forms of the minimum free energy and the corresponding rate of dissipation for general histories of strain are derived and discussed within the context of fractional derivative models of materials with memory.
Abstract: Explicit forms of the minimum free energy and the corresponding rate of dissipation, for general histories of strain, are derived and discussed within the context of fractional derivative models of materials with memory. Simple formulae are also given for sinusoidal and exponential histories.
TL;DR: In this article, a linear axial dispersion model involving two parameters, Peclet number (Pe) and retardation coefficient (Rd), has been presented, and model equations have been solved analytically, using Laplace transform.
Abstract: A linear axial dispersion model involving two parameters, Peclet number (Pe) and retardation coefficient (Rd) has been presented. Model equations have been solved analytically, using Laplace transform. Solution has been obtained in terms of complementary error function. Two limiting cases of perfect mixing and perfect displacement have also been discussed. Verification of model is carried out by using the experimental data of a washing cell.
Abstract: A short bio of Professor Francesco Mainardi is presented. Among other reported facts, the start of his professional career on Fractional Calculus in Bologna during the '70s with Michele Caputo is reminded together with the fruitful collaboration since mid '90s with Rudolf Gorenflo from Berlin.
TL;DR: A selection of papers presented at the workshop "Fractional calculus, probability and non-local operators: applications and recent developments" as discussed by the authors is presented in this issue of CAIM Vol. 6 (1).
Abstract: This issue of CAIM Vol. 6 (1) contains a first selection of papers presented at: "Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments", a workshop organised on the occasion of the retirement of Francesco Mainardi and held in Bilbao, Basque Country — Spain, on November 2013.