Volume 22, Issue 1

This paper analyses the citation profiles (CP) of 130 researchers in fractional calculus. In a first phase, the Canberra distance is used to measure the similarities between the researchers’ CP, and the multidimensional scaling technique (MDS) is adopted for processing and visualizing the information. In a second phase, the gamma probability distribution is used to fit the normalized CP and the gamma parameters are used to characterize the researchers. The MDS results and the gamma distribution parameters are represented graphically in 2- and 3-dimensional locus depicting the relative positions of the researchers.

Variable-order (VO) fractional differential equations (FDEs) with a time ( t ), space ( x ) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

In this paper, a wide view on the theory of the continuous time random walk ( CTRW ) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW . These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

Motivated essentially by the success of the applications of the discrete Mittag-Leffler functions (DMLF) in many areas of science and engineering, the authors present, in a unified manner, a detailed numerical implementation method of the Mittag-Leffler function. With the proposed method, the overflow problem can be well solved. To further improve the practicability, the state transition matrix described by discrete Mittag-Leffler functions are investigated. Some illustrative examples are provided to verify the effectiveness of the proposed theoretical results.

Trajectory planning for autonomous vehicles is a research topical subject. In previous studies, optimal intermediate targets have been used in the Potential Fields (PFs). PFs are only a path planning method, or a reactive obstacle avoidance method and not a trajectory tracking method. In this article, the PFs are interpreted as an on-line control method to follow an optimal trajectory. An analysis and methodological approach to design the attractive potential as a robust controller are proposed, and a new definition of a fractional repulsive potential to characterize the dangerousness of obstacles is developed. Simulation results on autonomous vehicles are given.

Stable distributions are a class of distributions that have important uses in probability theory. They also have a applications in the theory of fractional diffusions: symmetric stable density functions are the Green’s functions of the fractional heat equation. We describe efficient numerical representations for these Green’s functions, enabling their use in numerical solutions of fractional heat equations. We also describe a new connection between stable laws and the Weyl fractional derivative.

This paper presents models of economic growth for all the countries of the Group of Seven (G7) in the 1973–2016 period. The models consist of differential equations, of both integer and fractional order, where the gross domestic product (GDP) is a function of the country’s land area, arable land, population, school attendance, gross capital formation (GCF), exports of goods and services, general government final consumption expenditure (GGFCE), and broad money (M3). Results show that fractional models have a better performance, measured by several summary statistics, without increasing the number of parameters, or sacrificing the ability to predict GDP evolution in the short term. A standard validation procedure for economic growth models is presented for the assessment of future models.

The controllability and observability of fractional different orders discrete-time linear systems are analyzed. The relationship between controllability and observability of standard and fractional discrete-time linear systems are investigated. The influence of the fractional order on the controllability and observability is shown. Investigations are illustrated on numerical examples.

Optimal control problem for linear systems of arbitrary fractional order in the sense of Riemann–Liouville is treated in the paper. The technique of attainability sets and their support functions is used to obtain sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle. Theoretical results are supported by example.

Fractional impulsive differential equations are revisited first. Some fundamental solutions of linear cases are given in this study. One straightforward technique without using integral equation is adopted to obtain exact solutions which are given by use of piecewise functions. Furthermore, a class of short memory fractional differential equations is proposed and the variable case is discussed. Mittag–Leffler solutions with impulses are derived which both satisfy the equations and impulsive conditions, respectively.

The accurate modelling of an electrical coil over a wide range of frequency is the keystone for a precise modelling of an electrical machine. As a consequence of copper losses, eddy-current losses and hysteresis losses; electrical coils with conductive ferromagnetic core show different behaviour from that of an ideal coil. Throughout this paper, dynamic modelling and performance analysis of conventional as well as fractional-order models of an electrical coil with an interchangeable core are achieved. Measurement results are acquired through an integration between Matlab and the high-speed measurement system LTT24. In order to assess the accuracy of these models, simulation results are compared with experimental results whereas unknown parameters are identified through an optimization process that is based on the method of least squares. It is known that the parameters of fractional-order model ( L α , α , C β , β ) can not be measured directly. Therefore, the paper proposes a possibility based on system analysis to derive these parameters (indirect measurement) from the parameters of the classical model. A frequency band beyond the self-resonant frequency of the electrical coil is explored, thus the parasitic capacitance between coils windings must be considered as an important part of the equivalent circuit. The dependency of model parameters on frequency due to skin-effect is also examined.

In the paper we investigate the fractional-order evaluation of the fractional-order discrete integration element. We assume that the input and output signals are known. The main problem is to calculate fractional-order value. From a theoretical point of view there is no mathematical problem of the solution. One should solve linear algebraic equation or find roots of a polynomial in a variable ν . The problem arises when the measured output signal contains a noise. Then, the solution is unsettled because the polynomial roots are very sensitive to coefficients variability. In the paper we propose a method of evaluating of the discrete integrator fractional-order. The investigations are supported by numerical examples.