Fractal Fract., Volume 5, Issue 1 (March 2021) – 25 articles

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials. Full article

In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results. Full article

In this paper, the fractal derivative is introduced into a neural network-based discrete-time fractional-order PID controller in two areas, namely, in the controller’s structure and in the parameter optimization algorithm. The first use of the fractal derivative is to reconstruct the fractional-order PID controller by using the Hausdorff difference and Hausdorff sum derived from the Hausdorff derivative and Hausdorff integral. It can avoid the derivation of the Gamma function for the order updating to realize the parameter and order tuning based on neural networks. The other use is the optimization of order and parameters by using Borges derivative. Borges derivative is a kind of fractal derivative as a local fractional-order derivative. The chain rule of composite function is consistent with the integral-order derivative. It is suitable for updating the parameters and the order of the fractional-order PID controller based on neural networks. This paper improves the neural network-based PID controller in two aspects, which accelerates the response speed and improves the control accuracy. Two illustrative examples are given to verify the effectiveness of the proposed neural network-based discrete-time fractional-order PID control scheme with fractal derivatives. Full article

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory. Full article

The knowledge of the electrochemical processes inside a Fuel Cell (FC) is useful for improving FC diagnostics, and Electrochemical Impedance Spectroscopy (EIS) is one of the most used techniques for electrochemical characterization. This paper aims to propose the identification of a Fractional-Order Transfer Function (FOTF) able to represent the FC behavior in a set of working points. The model was identified by using a data-driven approach. Experimental data were obtained testing a Proton Exchange Membrane Fuel Cell (PEMFC) to measure the cell impedance. A genetic algorithm was firstly used to determine the sets of fractional-order impedance model parameters that best fit the input data in each analyzed working point. Then, a method was proposed to select a single set of parameters, which can represent the system behavior in all the considered working conditions. The comparison with an equivalent circuit model taken from the literature is reported, showing the advantages of the proposed approach. Full article

The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class ${\mathcal{C}}_W(K,a)$, which is defined here. It is also shown that from the continuity of a resolving family of operators at $t=0$ the boundedness of A follows. The existence of a resolving family is shown for $A\in {\mathcal{C}}_W(K,a)$ and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time. Full article

The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with $O\left(h^2+{\tau }^2\right)$ of accuracy on the grids. Here, h and $\frac{\sqrt{3}}{2}h$ are the step sizes in space variables $x_1$ and $x_2$, respectively and $\tau$ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for $r=\frac{\omega \tau }{h^2}\le \frac{3}{7}$, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order $O\left(h^2+{\tau }^2\right)$ is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures. Full article

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function $M\left(t\right)$ of complex materials, that their creep compliance follows a power law, $J\left(t\right)\sim t^q$ with $q\in {\mathbb{R}}^{+}$, is proportional to the fractional derivative of the Dirac delta function, $\frac{{\mathrm{d}}^q\delta (t-0)}{\mathrm{d}{t}^q}$ with $q\in {\mathbb{R}}^{+}$. This leads to the finding that the inverse Laplace transform of $s^q$ for any $q\in {\mathbb{R}}^{+}$ is the fractional derivative of the Dirac delta function, $\frac{{\mathrm{d}}^q\delta (t-0)}{\mathrm{d}{t}^q}$. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of $\frac{s^q}{s^{\alpha }\mp \lambda }$ where $\alpha <q\in {\mathbb{R}}^{+}$, which is the fractional derivative of order q of the Rabotnov function $\epsilon$${}_{\alpha -1}(\pm \lambda ,t)=t^{\alpha -1}{E}_{\alpha ,\alpha }(\pm \lambda t^{\alpha })$. The fractional derivative of order $q\in {\mathbb{R}}^{+}$ of the Rabotnov function, $\epsilon$${}_{\alpha -1}(\pm \lambda ,t)$ produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function. Full article

Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the solution of optimal control problems in cases in which bounded uncertainty in parameters and initial conditions are present. In contrast to system models with integer-order derivatives, fractional-order models have not yet gained the same attention if verified solution techniques are desired. In general, verified simulation techniques rely on interval methods, zonotopes, or Taylor model arithmetic and allow for computing guaranteed outer enclosures of the sets of solutions. As such, not only the influence of uncertain but bounded parameters can be accounted for in a guaranteed way. In addition, also round-off and (temporal) truncation errors that inevitably occur in numerical software implementations can be considered in a rigorous manner. This paper presents novel iterative and series-based solution approaches for the case of initial value problems to fractional-order system models, which will form the basic building block for implementing state estimation schemes in continuous-discrete settings, where the system dynamics is assumed as being continuous but measurements are only available at specific discrete sampling instants. Full article

Navier–Stokes equations describe the laminar flow of incompressible fluids. In most cases, one prefers to solve either these equations numerically, or the physical conditions of solving the problem are considered more straightforward than the real situation. In this paper, the Navier–Stokes equations are solved analytically and numerically for specific physical conditions. Using $F^{\alpha }$-calculus, the fractal form of Navier–Stokes equations, which describes the laminar flow of incompressible fluids, has been solved analytically for two groups of general solutions. In the analytical section, for just “the single-phase fluid” analytical answers are obtained in a two-dimensional situation. However, in the numerical part, we simulate two fluids’ flow (liquid–liquid) in a three-dimensional case through several fractal structures and the sides of several fractal structures. Static mixers can be used to mix two fluids. These static mixers can be fractal in shape. The Sierpinski triangle, the Sierpinski carpet, and the circular fractal pattern have the static mixer’s role in our simulations. We apply these structures just in zero, first and second iterations. Using the COMSOL software, these equations for “fractal mixing” were solved numerically. For this purpose, fractal structures act as a barrier, and one can handle different types of their corresponding simulations. In COMSOL software, after the execution, we verify the defining model. We may present speed, pressure, and concentration distributions before and after passing fluids through or out of the fractal structure. The parameter for analyzing the quality of fractal mixing is the Coefficient of Variation (CoV). Full article

This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique. Full article

Tree precise classification and identification of forest species is a core issue of forestry resource monitoring and ecological effect assessment. In this paper, an independent tree species classification method based on fractal features of terrestrial laser point cloud is proposed. Firstly, the terrestrial laser point cloud data of an independent tree is preprocessed to obtain terrestrial point clouds of independent tree canopy. Secondly, the multi-scale box-counting dimension calculation algorithm of independent tree canopy dense terrestrial laser point cloud is proposed. Furthermore, a robust box-counting algorithm is proposed to improve the stability and accuracy of fractal dimension expression of independent tree point cloud, which implementing gross error elimination based on Random Sample Consensus. Finally, the fractal dimension of a dense terrestrial laser point cloud of independent trees is used to classify different types of independent tree species. Experiments on nine independent trees of three types show that the fractal dimension can be stabilized under large density variations, proving that the fractal features of terrestrial laser point cloud can stably express tree species characteristics, and can be used for accurate classification and recognition of forest species. Full article

The passive electrical properties of a biological tissue, referred to as the tissue bioimpedance, are related to the underlying tissue physiology. These measurements are often well-represented by a fractional-order equivalent circuit model, referred to as the Cole-impedance model. Objective: Identify if there are differences in the fractional-order ($\alpha$) of the Cole-impedance parameters that represent the segmental right-body, right-arm, and right-leg of adult participants. Hypothesis: Cole-impedance model parameters often associated with tissue geometry and fluid ($R_{\infty }$, $R_1$, C) will be different between body segments, but parameters often associated with tissue type ($\alpha$) will not show any statistical differences. Approach: A secondary analysis was applied to a dataset collected for an agreement study between bioimpedance spectroscopy devices and dual-energy X-ray absoptiometry, identifying the Cole-model parameters of the right-side body segments of $N=174$ participants using a particle swarm optimization approach. Statistical testing was applied to the different groups of Cole-model parameters to evaluate group differences and correlations of parameters with tissue features. Results: All Cole-impedance model parameters showed statistically significant differences between body segments. Significance: The physiological or geometric features of biological tissues that are linked with the fractional-order ($\alpha$) of data represented by the Cole-impedance model requires further study to elucidate. Full article

This article presents the third part of a larger project whose final objective is to study and analyse the effects of viscous thermal losses in a flute wind musical instrument. After implementing the test bench in the first phase and modelling and validating the dynamic behaviour of the simulator, based on the previously implemented test bench (without considering the losses in the system) in the second phase, this third phase deals with the study of the viscous thermal losses that will be generated within the resonator of the flute. These losses are mainly due to the friction of the air inside the resonator with its boundaries and the changes of the temperature within this medium. They are mainly affected by the flute geometry and the materials used in the fabrication of this instrument. After modelling these losses in the frequency domain, they will be represented using a system approach where the fractional order part is separated from the system’s transfer function. Thus, this representation allows us to study, in a precise way, the influence of the fractional order behaviour on the overall system. Effectively, the fractional behavior only appears much below the 20 Hz audible frequencies, but it explains the influence of this order on the frequency response over the range [20–20,000] Hz. Some simulations will be proposed to show the effects of the fractional order on the system response. Full article

In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality of the system is investigated; its output is an electromagnetic signal propagating in media described by the time-domain two-sided fractional derivative. It is demonstrated that, for the derivative order in the range $[1,+\infty )$, the transfer function describing attenuated signal propagation is not causal for any value of the asymmetry parameter of the derivative. On the other hand, it is shown that, for derivative orders in the range $(0,1)$, the transfer function is causal if and only if the asymmetry parameter is equal to certain specific values corresponding to the left-sided Grünwald–Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers–Krönig relations for logarithm of the transfer function are presented as well. Full article

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads. Full article

The current study is of interest when performing a useful extension of a crucial physical problem through a non-local singular fractional operator. We provide solutions that include three arbitrary parameters $\alpha$, $\rho$, and $\gamma$ for the Resistance-Capacitance (RC), Inductance-Capacitance (LC), and Resistance-Inductance-Capacitance (RLC) electric circuits utilizing a generalized type fractional operator in the sense of Caputo, called non-local M-derivative. Additionally, to keep the dimensionality of the physical parameter in the proposed model, we use an auxiliary parameter. Owing to the fact that all solutions depend on three parameters unlike the other solutions containing one or two parameters in the literature, the solutions obtained in this study have more general results. On the other hand, in order to observe the advantages of the non-local M-derivative, a comprehensive comparison is carried out in the light of experimental data. We make this comparison for the RC circuit between the non-local M-derivative and Caputo derivative. It is clearly shown on graphs that the fractional M-derivative behaves closer to the experimental data thanks to the added parameters $\alpha$, $\rho$, and $\gamma$. Full article

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-$\alpha$ integral operator ${\mathrm{M}}_{\mathrm{a},\mathsf{\delta }}^{\mathrm{c}}$ to the norm of the centered fractional maximal diamond-$\alpha$ integral operator ${\mathrm{M}}_{\mathrm{a}}^{\mathrm{c}}$ on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales. Full article

This paper applies one of the special cases of auxiliary method, which is named as the Bernoulli sub-equation function method, to the nonlinear modified alpha equation. The characteristic properties of these solutions, such as complex and soliton solutions, are extracted. Moreover, the strain conditions of solutions are also reported in detail. Observing the figures plotted by considering various values of parameters of these solutions confirms the effectiveness of the approximation method used for the governing model. Full article

In many applications, real phenomena are modeled by differential problems having a time fractional derivative that depends on the history of the unknown function. For the numerical solution of time fractional differential equations, we propose a new method that combines spline quasi-interpolatory operators and collocation methods. We show that the method is convergent and reproduces polynomials of suitable degree. The numerical tests demonstrate the validity and applicability of the proposed method when used to solve linear time fractional differential equations. Full article

Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases such as epilepsy, multiple sclerosis, and Alzheimer’s disease, which have become more prevalent in the latest years. Full article

Robust stability is a major concern for real-world control applications. Realization of optimal robust stability requires a stabilization scheme, which ensures that the control system is stable and presents robust performance for a predefined range of system perturbations. This study presented an optimal robust stabilization approach for closed-loop fractional order proportional integral derivative (FOPID) control systems with interval parametric uncertainty and uncertain time delay. This stabilization approach, which is carried out in a v-plane, relies on the placement of the minimum angle system pole to a predefined target angle within the stability region of the first Riemann sheet. For this purpose, tuning of FOPID controller coefficients was performed to minimize a root angle error that is defined as the squared difference of minimum angle root of interval characteristic polynomials and the desired target angle within the stability region of the v-plane. To solve this optimization problem, a particle swarm optimization (PSO) algorithm was implemented. Findings of the study reveal that tuning of the target angle can also be used to improve the robust control performance of interval uncertain FOPID control systems. Illustrative examples demonstrated the effectiveness of the proposed v-domain, optimal, robust stabilization of FOPID control systems. Full article

Remote farms in Africa are cultivated lands planned for 100% sustainable energy and organic agriculture in the future. This paper presents the load frequency control of a two-area power system feeding those farms. The power system is supplied by renewable technologies and storage facilities only which are photovoltaics, biogas, biodiesel, solar thermal, battery storage and flywheel storage systems. Each of those facilities has 150-kW capacity. This paper presents a model for each renewable energy technology and energy storage facility. The frequency is controlled by using a novel non-linear fractional order proportional integral derivative control scheme (NFOPID). The novel scheme is compared to a non-linear PID controller (NPID), fractional order PID controller (FOPID), and conventional PID. The effect of the different degradation factors related to the communication infrastructure, such as the time delay and packet loss, are modeled and simulated to assess the controlled system performance. A new cost function is presented in this research. The four controllers are tuned by novel poor and rich optimization (PRO) algorithm at different operating conditions. PRO controller design is compared to other state of the art techniques in this paper. The results show that the PRO design for a novel NFOPID controller has a promising future in load frequency control considering communication delays and packet loss. The simulation and optimization are applied on MATLAB/SIMULINK 2017a environment. Full article

This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and Krasnosel’skii’s fixed point theorem. To illustrate the results, an example is provided. Full article