Fractional Calculus: Quo Vadimus? (Where Are We Going?)

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.

A Review of Operational Matrices and Spectral Techniques for Fractional Calculus

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.

Fractional Calculus: Application in modeling and control

This contribution introduces the fractional calculus (FC) fundamental mathematical aspects and discuses some of their consequences. Based on the FC concepts, the chapter reviews the main approaches for implementing fractional operators and discusses the adoption of FC in control systems. Finally are presented some applications in the areas of modeling and control, namely fractional PID, heat diffusion systems, electromagnetism, fractional electrical impedances, evolutionary algorithms, robotics, and nonlinear system control.

Electrical Skin Phenomena: A Fractional Calculus Analysis

The internal impedance of a wire is the function of the frequency. In a conductor, where the conductivity is sufficiently high, the displacement current density can be neglected. In this case, the conduction current density is given by the product of the electric field and the conductance. One of the aspects the high-frequency effects is the skin effect (SE). The fundamental problem with SE is it attenuates the higher frequency components of a signal. The SE was first verified by Kelvin in 1887. Since then many researchers developed work on the subject and presently a comprehensive physical model, based on the Maxwell equations, is well established. The Maxwell formalism plays a fundamental role in the electromagnetic theory. These equations lead to the derivation of mathematical descriptions useful in many applications in physics and engineering. Maxwell is generally regarded as the 19th century scientist who had the greatest influence on 20th century physics, making contributions to the fundamental models of nature. The Maxwell equations involve only the integer-order calculus and, therefore, it is natural that the resulting classical models adopted in electrical engineering reflect this perspective. Recently, a closer look of some phenomas present in electrical systems and the motivation towards the development of precise models, seem to point out the requirement for a fractional calculus approach. Bearing these ideas in mind, in this study we address the SE and we re-evaluate the results demonstrating its fractional-order nature.

A Fractional Calculus Perspective in the Evolutionary Design of Combinational Circuits

This paper analyses the performance of a genetic algorithm (GA) in the synthesis of digital circuits using two novel approaches. The first concept consists in improving the static fitness function by including a discontinuity evaluation. The measure of variability in the error of the Boolean table has similarities with the function continuity issue in classical calculus. The second concept extends the static fitness by introducing a fractional-order dynamical evaluation.

Short-Circuit Impedance of Power Transformers: the Fractional Order Calculus Approach

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

On the Estimation of the Short-Circuit Impedance of Power Transformers Using Fractional Order Calculus

This paper reports investigation on the estimation of the short circuit impedance of power transformers, using fractional order calculus to analytically study the influence of the diffusion phenomena in the windings. The aim is to better characterize the medium frequency range behavior of leakage inductances of power transformer models, which include terms to represent the magnetic field diffusion process in the windings. Comparisons between calculated and measured values are shown and discussed.

Application of Fractional Calculus in Mechatronics

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses a FC perspective in the study of the dynamics and control of mechanical systems.

Wii-Workouts on Chronic Pain, Physical Capabilities and Mood of Older Women: A Randomized Controlled Double Blind Trial

Chronic Low Back Pain (CLBP) is a public health problem and older women have higher incidence of this symptom, which affect body balance, functional capacity and behavior. The purpose of this study was to verifying the effect of exercises with Nintendo Wii on CLBP, functional capacity and mood of elderly. Thirty older women (68 ± 4 years; 68 ± 12 kg; 154 ± 5 cm) with CLBP participated in this study. Elderly individuals were divided into a Control Exercise Group (n = 14) and an Experimental Wii Group (n = 16). Control Exercise Group did strength exercises and core training, while Experimental Wii Group did ones additionally to exercises with Wii. CLBP, balance, functional capacity and mood were assessed pre and post training by the numeric pain scale, Wii Balance Board, sit to stand test and Profile of Mood States, respectively. Training lasted eight weeks and sessions were performed three times weekly. MANOVA 2 x 2 showed no interaction on pain, siting, stand-up and mood (P = 0.53). However, there was significant difference within groups (P = 0.0001). ANOVA 2 x 2 showed no interaction for each variable (P > 0.05). However, there were significant differences within groups in these variables (P < 0.05). Tukey's post-hoc test showed significant difference in pain on both groups (P = 0.0001). Wilcoxon and Mann-Whitney tests identified no significant differences on balance (P > 0.01). Capacity to Sit improved only in Experimental Wii Group (P = 0.04). In conclusion, physical exercises with Nintendo Wii Fit Plus additional to strength and core training were effective only for sitting capacity, but effect size was small.

Ankle antagonist coactivation in the double-support phase of walking: Stroke vs. healthy subjects

Introduction: Lesions at ipsilateral systems related to postural control at ipsilesional side, may justify the lower performance of stroke subjects during walking. Purpose: To analyse bilateral ankle antagonist coactivation during double-support in stroke subjects. Methods: Sixteen (8 females; 8 males) subjects with a first isquemic stroke, and twenty two controls (12 females; 10 males) participated in this study. The double support phase was assessed through ground reaction forces and electromyography of ankle muscles was assessed in both limbs. Results: Ipsilesional limb presented statistical significant differences from control when assuming specific roles during double support, being the tibialis anterior and soleus pair the one in which this atypical behavior was more pronounced. Conclusion: The ipsilesional limb presents a dysfunctional behavior when a higher postural control activity was demanded.

Cross-Layer Admission Control to Enhance the Support of Real-Time Applications in WSN

Real-time monitoring applications may be used in a wireless sensor network (WSN) and may generate packet flows with strict quality of service requirements in terms of delay, jitter, or packet loss. When strict delays are imposed from source to destination, the packets must be delivered at the destination within an end-to-end delay (EED) hard limit in order to be considered useful. Since the WSN nodes are scarce both in processing and energy resources, it is desirable that they only transport useful data, as this contributes to enhance the overall network performance and to improve energy efficiency. In this paper, we propose a novel cross-layer admission control (CLAC) mechanism to enhance the network performance and increase energy efficiency of a WSN, by avoiding the transmission of potentially useless packets. The CLAC mechanism uses an estimation technique to preview packets EED, and decides to forward a packet only if it is expected to meet the EED deadline defined by the application, dropping it otherwise. The results obtained show that CLAC enhances the network performance by increasing the useful packet delivery ratio in high network loads and improves the energy efficiency in every network load.

Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus

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.