Volume 18, Issue 2

The non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

This paper presents a frequency-domain methodology for robust systemcontrol design, and its application in airpath control of a real turbocharged Diesel engine with intake throttle and exhaust gas re-circulation system. The objective is to find a robust control-system which reduces pollutant emission. The three inputs (throttle, wastegate and EGR valves) and two outputs (airflow and boost pressure) of the airpath make the system complex. AMulti InputMulti Output (MIMO) fractional order approach called CRONE (French abbreviation which means robust control of a non integer order) is used to design a non-square control-system for this non-linear plant. Firstly, a multisine signal is used for the system-identification, the airpath inputs are excited around all operating points in order to find a linear model of the plant. Then, the frequency responses of a defined nominal plant and experimental plant are taken into account to design the MIMO control-system using the CRONE shaping approach which permits to maintain performance and robust stability around a wide set of operating points. Finally, pollutant emission results from a driving cycle in test-bench show the relevance of the proposed approach.

Fractional Bochner-Levy-Riesz diffusion arises from ordinary diffusion by replacing the Laplacean with a noninteger power of itself. Bochner- Levy-Riesz diffusion as a mathematical model leads to nonlocal boundary value problems. As a model for physical transport processes it seems to predict phenomena that have yet to be observed in experiment.

The mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. This paper discusses the application of nonlocal diffusion theory to specify well-posed fractional diffusion equations on bounded domains.

We investigate the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to coupled integral boundary conditions. The nonsingular and singular cases are studied.

In this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

A new method to solve non-linear fractional-order differential equations involving delay has been presented. Applications to a variety of problems demonstrate that the proposed method is more accurate and time efficient compared to existing methods. A detailed error analysis has also been given.

Finite difference methods for approximating fractional derivatives are often analyzed by determining their order of consistency when applied to smooth functions, but the relationship between this measure and their actual numerical performance is unclear. Thus in this paper several wellknown difference schemes are tested numerically on simple Riemann-Liouville and Caputo boundary value problems posed on the interval [0, 1] to determine their orders of convergence (in the discrete maximum norm) in two unexceptional cases: (i) when the solution of the boundary-value problem is a polynomial (ii) when the data of the boundary value problem is smooth. In many cases these tests reveal gaps between a method’s theoretical order of consistency and its actual order of convergence. In particular, numerical results show that the popular shifted Gr¨unwald-Letnikov scheme fails to converge for a Riemann-Liouville example with a polynomial solution p(x), and a rigorous proof is given that this scheme (and some other schemes) cannot yield a convergent solution when p(0)≠ 0.

The paper discusses asymptotic stability conditions for the linear fractional difference equation ∇ α y(n) + a∇ β y(n) + by(n) = 0 with real coefficients a, b and real orders α > β > 0 such that α/β is a rational number. For given α, β, we describe various types of discrete stability regions in the (a, b)-plane and compare them with the stability regions recently derived for the underlying continuous pattern D α x(t) + aD β x(t) + bx(t) = 0 involving two Caputo fractional derivatives. Our analysis shows that discrete stability sets are larger and their structure much more rich than in the case of the continuous counterparts.

In this paper we study several natural and man-made complex phenomena in the perspective of dynamical systems. For each class of phenomena, the system outputs are time-series records obtained in identical conditions. The time-series are viewed as manifestations of the system behavior and are processed for analyzing the system dynamics. First, we use the Fourier transform to process the data and we approximate the amplitude spectra by means of power law functions. We interpret the power law parameters as a phenomenological signature of the system dynamics. Second, we adopt the techniques of non-hierarchical clustering and multidimensional scaling to visualize hidden relationships between the complex phenomena. Third, we propose a vector field based analogy to interpret the patterns unveiled by the PL parameters.

In the context of fractional calculus (FC), this paper is devoted to model thermal processes in trees based on the measurement of the temperature difference (ΔT) between sensors located above and below a heater inserted in the tree trunk. By evaluating several temperature curves taken from real trees of different species, the current approach shows that the temperature in each probe can be successfully described by the two-parameter Mittag- Leffler function Eα,β. Then, a simple methodology is followed to derive a novel expression of the heat-pulse velocity (v) as a function of ΔT and the parameter α of the mentioned Eα,β function. Experimental results are given to validate the goodness of the current proposal.

Discussions under this title were held during a special session in frames of the International Conference “Fractional Differentiation and Applications” (ICFDA ’14) held in Catania (Italy), 23-25 June 2014, see details at http://www.icfda14.dieei.unict.it/. Along with the presentations made during this session, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of this special session was to continue the useful traditions from the first conferences on the Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, to discuss them and try to find ways to resolve.