Volume 16, Issue 3

In this paper we investigate a fractional order logistic map and its discrete time dynamics. After a brief introduction to the discrete-time dynamical systems and fractional dynamics we show some basic properties of the fractional logistic map. We then move on to prove that the special case α = 1/2 exhibits a period doubling route to chaos. A bifurcation diagram for the special case of α = 1/2 is also included. Finally a discussion concerning the results and open problems is given.

This paper presents the effects of the temperature variation on the fractional order system of first kind realized in the electrical domain. The temperature variation, which is one of the parametric uncertainties which encounters the system’s behaviour, alters the values of the resistive and capacitive (RC) components used for the realization of this fractional system. The experimental results show that the system behaviour remains unchanged. Hence, the main objective of this work is to study analytically if the recursive factors, used to calculate the resistances and capacitance on one hand, and used to determine the poles and zeros on the other hand, are still valid and thus, the recursion is conserved. The results found show that the system is not affected by the temperature variation and the system is still robust to such uncertainties.

We consider a class of boundary value problem in a separable Banach space E, involving a nonlinear differential inclusion of fractional order with integral boundary conditions, of the form (*)$\left\{ \begin{gathered} D^\alpha u(t) \in F(t,u(t),D^{\alpha - 1} u(t)),a.e.,t \in [0,1], \hfill \\ I^\beta u(t)|_{t = 0} = 0,u(1) = \int\limits_0^1 {u(t)dt,} \hfill \\ \end{gathered} \right. $ where D α is the standard Riemann-Liouville fractional derivative, F is a closed valued mapping. Under suitable conditions we prove that the solutions set of (*) is nonempty and is a retract in W Eα,1(I). An application in control theory is also provided by using the Young measures.

In this note we propose a fractional generalization of the classical modified Bessel equation. Instead of the integer-order derivatives we use the Riemann-Liouville version. Next, we solve the fractional modified Bessel equation in terms of the power series and provide an asymptotic analysis of its solution for large arguments. We find a leading-order term of the asymptotic formula for the solution to the considered equation. This behavior is verified numerically and shows high accuracy and fast convergence. Our results reduce to the classical formulas when the order of the fractional derivative goes to integer values.

In this paper, we use waveform relaxation method to solve fractional functional differential equations. Under suitable conditions imposed on the so-called splitting functions the convergence results of the waveform relaxation method are given. Delay dependent error estimates for the method are derived. Error bounds for some special cases are considered. Numerical examples illustrate the feasibility and efficiency of the method. It is the first time for applying the method in the fractional functional differential equations.

The purpose of the paper is to investigate the global existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis. More precisely, it deals with the initial value problem (*)$\left\{ \begin{gathered} D_{0 + }^\alpha x(t) = f(t,x(t)),t \in [0,\infty ], \hfill \\ \lim _{t \to 0 + } t^{1 - \alpha } x(t) = x_0 , \hfill \\ \end{gathered} \right. $ where 0 < α < 1, D 0+α denotes the Riemann-Liouville fractional derivative of order α, and f: (0,∞) × ℝ → ℝ is a continuous function. Unlike all the previous papers dealing with the problem of existence of solutions to (*), this problem is solved here by constructing a special locally convex space which is metrizable and complete. Then Schauder’s fixed point theorem enables to provide sufficient conditions on f, ensuring that (*) possesses at least one solution. The growth conditions imposed to f are weaker than other similar conditions already used in the literature.

In this paper we investigate the stability of the equilibrium solution of the νth order linear system of difference equations $(\Delta _{a + \nu - 1}^\nu y)(t) = \Lambda y(t + \nu - 1);t \in \mathbb{N}_a ,a \in \mathbb{R},and\Lambda \in \mathbb{R}^{p \times p} ,$ subject to the initial condition $y(a + \nu - 1) = y - 1,$, where 0 < ν < 1 and y−1 ∈ ℝp.

We study the fractional integral (fI) and fractional derivative (fD), attained by the analytic continuation (AC) of Liouville’s fI (LfI) and Riemann-Liouville fI (RLfI). On the AC of RLfI, we give a detailed summary of Lavoie et al’s review. The ACs of RLfI are expressed by means of contour integrals. Two of them use the Cauchy contour, and one is using the Pochhammer contour. In this case, the latter is AC of all the others for the functions treated. For the AC of LfI, one can find studies in Campos’ papers and in Nishimoto’s books, where the AC is using only the Cauchy contour. Here we present also an AC using a modified Pochhammer’s contour. In this case, we see that any of these two ACs is not the AC of the other for all the functions treated. This fact leads to difficulties, if a careful study taking care of the domains of existence of each AC is not adopted. By taking account of this fact, we resolve the difficulties which occur in Nishimoto’s formalism.

The fractional reaction-diffusion equations play an important role in dynamical systems. Indeed, it is time consuming to numerically solve differential fractional diffusion equations. In this paper, we present a parallel algorithm for the Riesz space fractional diffusion equation. The parallel algorithm, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel matrix vector multiplication and vector vector addition. As to the authors’ knowledge, this is the first parallel algorithm for the Riesz space fractional reaction-diffusion equation. The experimental results show that the parallel algorithm is as accurate as the serial algorithm. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.3-3.4 times faster than the serial algorithm on single CPU core. The parallel efficiency of 64 processes is up to 79.39% compared with 8 processes on a distributed memory cluster system.

The problems formulated in the fractional calculus framework often require numerical fractional integration/differentiation of large data sets. Several existing fractional control toolboxes are capable of performing fractional calculus operations, however, none of them can efficiently perform numerical integration on multiple large data sequences. We developed a Fractional Integration Toolbox (FIT), which efficiently performs fractional numerical integration/differentiation of the Riemann-Liouville type on large data sequences. The toolbox allows parallelization and is designed to be deployed on both CPU and GPU platforms.

We study an initial value problem for a fractional differential equation using the Riemann-Liouville fractional derivative. We obtain some topological properties of the solution set: It is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser’s theorem on the structure solution set for ordinary differential equations.

The authors study the singular boundary value problem with fractional q-derivatives $\begin{gathered} - (D_q^\nu u)(t) = f(t,u),t \in (0,1), \hfill \\ (D_q^i u)(0) = 0,i = 0,...,n - 2,(D_q u)(1) = \sum\limits_{j = 1}^m {a_j (D_q u)(t_j ) + \lambda ,} \hfill \\ \end{gathered} $, where q ∈ (0, 1), m ≥ 1 and n ≥ 2 are integers, n − 1 < ν ≤ n, λ ≥ 0 is a parameter, f: (0, 1] × (0,∞) → [0,∞) is continuous, a i ≥ 0 and t i ∈ (0, 1) for i = 1, …,m, and D qν is the q-derivative of Riemann-Liouville type of order ν. Sufficient conditions are obtained for the existence of positive solutions. Their analysis is mainly based on a nonlinear alternative of Leray-Schauder.

In this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.

The aim of this article is to generalize the diffusion based adsorption model to a fractional diffusion and fractional adsorption model. The models are formulated as nonlinear fractional boundary value problems equivalent to a singular Hammerstein integral equation. The novelty is that not only the diffusion component of the model is fractionalized but also the adsorption part. The singular Hammerstein integral equation is solved by Sinc approximations. Specific numerical schemes are presented. Based on these solutions we are able to identify different regimes of adsorption diffusion processes controlled by fractional derivatives verified by experimental data. These regimes allow to classify experiments if examined with respect to their scaling behavior.