Fractal and Fractional doi: 10.3390/fractalfract5010021

Authors: Riccardo Caponetto Fabio Matera Emanuele Murgano Emanuela Privitera Maria Gabriella Xibilia

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010020

Authors: Vladimir E. Fedorov Nikolay V. Filin

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 CW(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∈CW(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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010019

Authors: Suzan Cival Buranay Ahmed Hersi Matan Nouman Arshad

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 Oh2+τ2 of accuracy on the grids. Here, h and 32h are the step sizes in space variables x1 and x2, respectively and τ 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=ωτh2≤37, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order Oh2+τ2 is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010018

Authors: Nicos Makris

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α&lt;q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010017

Authors: Andreas Rauh Luc Jaulin

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010016

Authors: Amir Pishkoo Maslina Darus

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 Fα-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).

]]>Fractal and Fractional doi: 10.3390/fractalfract5010015

Authors: Misir J. Mardanov Yagub A. Sharifov Yusif S. Gasimov Carlo Cattani

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010014

Authors: Ju Zhang Qingwu Hu Hongyu Wu Junying Su Pengcheng Zhao

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010013

Authors: Todd J. Freeborn Shelby Critcher

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&infin;, R1, 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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010012

Authors: Fractal and Fractional Editorial Office Fractal and Fractional Editorial Office

Peer review is the driving force of journal development, and reviewers are gatekeepers who ensure that Fractal and Fractional maintains its standards for the high quality of its published papers [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract5010011

Authors: Gaby Abou Haidar Xavier Moreau Roy Abi Zeid Daou

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010010

Authors: Jacek Gulgowski Dariusz Kwiatkowski Tomasz P. Stefański

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,+&infin;), 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 Gr&uuml;nwald&ndash;Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers&ndash;Kr&ouml;nig relations for logarithm of the transfer function are presented as well.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010008

Authors: Cundi Han Yiming Chen Da-Yan Liu Driss Boutat

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010009

Authors: Bahar Acay Mustafa Inc

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;.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010007

Authors: Lütfi Akın

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 Ma,&delta;c to the norm of the centered fractional maximal diamond-&alpha; integral operator Mac 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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010006

Authors: Haci Mehmet Baskonus Muzaffer Ercan

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010005

Authors: Enza Pellegrino Laura Pezza Francesca Pitolli

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010004

Authors: Corina S. Drapaca

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&rsquo;s disease, which have become more prevalent in the latest years.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010003

Authors: Sevilay Tufenkci Bilal Senol Radek Matušů Baris Baykant Alagoz

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010002

Authors: Hady H. Fayek

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract5010001

Authors: Abdelkrim Salim Mouffak Benchohra John R. Graef Jamal Eddine Lazreg

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&rsquo;s contraction principle and Krasnosel&rsquo;skii&rsquo;s fixed point theorem. To illustrate the results, an example is provided.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040060

Authors: Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen

The authors wish to make the following corrections to this paper [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract4040059

Authors: Vsevolod Bohaienko Volodymyr Bulavatsky

To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media of fractal structure. For the case of the finite thickness domain with permeable boundaries, a finite-difference technique for numerical solution of the corresponding one-dimensional non-linear boundary value problem is developed. The paper also presents a fractional-fractal model of a filtration-consolidation process in clay soils of fractal structure saturated with salt solutions. An analytical solution is found for the corresponding one-dimensional boundary value problem in the domain of finite thickness with permeable upper and impermeable lower boundaries.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040058

Authors: Xuefeng Zhang Jia Dong

The issue of robust admissibility and control for singular fractional-order systems (FOSs) with polytopic uncertainties is investigated in this paper. Firstly, a new method based on linear matrix inequalities (LMIs) is presented to solve the admissibility problems of uncertain linear systems. Then, a solid criterion of robust admissibility and a corresponding state feedback controller are derived, which overcome the conservatism of the existing results. Finally, for the sake of demonstrating the validity of proposed results, some relevant examples are provided.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040057

Authors: Moreno Concezzi Renato Spigler

A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction&ndash;diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040056

Authors: Dimiter Prodanov

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040055

Authors: Shibendu Mahata Rajib Kar Durbadal Mandal

This paper presents the optimal rational approximation of (1+&alpha;) order Butterworth filter, where &alpha; ∊ (0,1) under the continued fraction expansion framework, by employing a new cost function. Two simple techniques based on the constrained optimization and the optimal pole-zero placements are proposed to model the magnitude-frequency response of the fractional-order lowpass Butterworth filter (FOLBF). The third-order FOLBF approximants achieve good agreement to the ideal characteristic for six decades of design bandwidth. Circuit realization using the current feedback operational amplifier is presented, and the modelling efficacy is validated in the OrCAD PSPICE platform.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040054

Authors: Stavroula Kapoulea Costas Psychalinos Ahmed S. Elwakil

The Cole&ndash;Davidson function is an efficient tool for describing the tissue behavior, but the conventional methods of approximation are not applicable due the form of this function. In order to overcome this problem, a novel scheme for approximating the Cole&ndash;Davidson function, based on the utilization of a curve fitting procedure offered by the MATLAB software, is introduced in this work. The derived rational transfer function is implemented using the conventional Cauer and Foster RC networks. As an application example, the impedance model of the membrane of mesophyll cells is realized, with simulation results verifying the validity of the introduced procedure.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040053

Authors: Mehmet Giyas Sakar Onur Saldır

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to &alpha; and &beta; are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040052

Authors: Alexander Iomin Trifce Sandev

A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set F&mu; is considered. This search-like problem is treated in the framework of a diffusion&ndash;reaction equation, when an absorbing term is included inside a Fokker&ndash;Planck equation as a reaction term. This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040051

Authors: Thomas M. Michelitsch Federico Polito Alejandro P. Riascos

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process &lsquo;space-time Mittag-Leffler process&rsquo;. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a &ldquo;well-scaled&rdquo; diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the &lsquo;state density kernel&rsquo; solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040050

Authors: Xuefeng Zhang Wenkai Huang

This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufficient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a finite time. Finally, a simulation example is proposed to verify the validity of the designed procedures.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040049

Authors: H. M. Srivastava Khaled M. Saad

In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter &rho; is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of &rho;=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of &rho; and k. All calculations in this work are accomplished by using the Mathematica package.

]]>Fractal and Fractional doi: 10.3390/fractalfract4040048

Authors: Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of F1Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F2 and F3 Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030047

Authors: Ming Li

Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030046

Authors: Harry Esmonde

An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030045

Authors: Arran Fernandez Iftikhar Husain

Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030044

Authors: Caterina Balzotti Mirko D’Ovidio Paola Loreti

In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (&alpha;-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order &alpha; converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the &alpha;-SIS models.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030043

Authors: Muhammad Farman Ali Akgül Dumitru Baleanu Sumaiyah Imtiaz Aqeel Ahmad

The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system&rsquo;s actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030042

Authors: Renat T. Sibatov HongGuang Sun

The approach based on fractional advection&ndash;diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick&rsquo;s law containing the Riemann&ndash;Liouville fractional derivative is related to the well-known fractional Fokker&ndash;Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker&ndash;Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030041

Authors: Hulya Durur Esin Ilhan Hasan Bulut

This manuscript focuses on the application of the (m+1/G&prime;)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schr&ouml;dinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030040

Authors: Jocelyn Sabatier

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030039

Authors: Rafał Brociek Agata Chmielowska Damian Słota

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm&rsquo;s accuracy.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030038

Authors: Houssine Zine Delfim F. M. Torres

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler&ndash;Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030037

Authors: Guido Maione

This paper concerns cascaded, shifted, fractional-order, lead compensators made by the serial connection of two stages introducing their respective phase leads in shifted adjacent frequency ranges. Adding up leads in these intervals gives a flat phase in a wide frequency range. Moreover, the simple elements of the cascade can be easily realized by rational transfer functions. On this basis, a method is proposed in order to design a robust controller for a class of benchmark plants that are difficult to compensate due to monotonically increasing lags. The simulation experiments show the efficiency, performance and robustness of the approach.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030036

Authors: Agneta M. Balint Stefan Balint

In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann&ndash;Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030035

Authors: Mehmet Yavuz Ndolane Sene

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann&ndash;Liouville integral was introduced and the corresponding numerical discretization of the predator&ndash;prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030034

Authors: Dina A. John Saket Sehgal Karabi Biswas

In this paper, the performance of an analog PI &lambda; D &mu; controller is done for speed regulation of a DC motor. The circuits for the fractional integrator and differentiator of PI &lambda; D &mu; controller are designed by optimal pole-zero interlacing algorithm. The performance of the controller is compared with another PI &lambda; D &mu; controller&mdash;in which the fractional integrator circuit employs a solid-state fractional capacitor. It can be verified from the results that using PI &lambda; D &mu; controllers, the speed response of the DC motor has improved with reduction in settling time ( T s ), steady state error (SS error) and % overshoot (% M p ).

]]>Fractal and Fractional doi: 10.3390/fractalfract4030033

Authors: Yudhveer Singh Vinod Gill Jagdev Singh Devendra Kumar Kottakkaran Sooppy Nisar

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030032

Authors: Emilia Bazhlekova Ivan Bazhlekov

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag&ndash;Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030031

Authors: Sotiris K. Ntouyas Bashir Ahmad Ahmed Alsaedi

We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is discussed for convex valued as well as non-convex valued multi-valued map. Examples are also constructed to illustrate the main results. The results presented in this paper are not only new in the given configuration but also provide some interesting special cases.

]]>Fractal and Fractional doi: 10.3390/fractalfract4030030

Authors: Esra Karatas Akgül Ali Akgül Dumitru Baleanu

In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag&ndash;Leffler functions.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020029

Authors: Ricardo Almeida

This paper is devoted to the study of existence and uniqueness of solutions for fractional functional differential equations, whose derivative operator depends on an arbitrary function. The introduction of such function allows generalization of some known results, and others can be also obtained.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020028

Authors: Maike Antonio Faustino dos Santos

Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a non-static stochastic resetting model in the context of comb structure that consists of a structure formed by backbone in x axis and branches in y axis. Then, we find the exact analytical solutions for marginal distribution concerning x and y axis. Moreover, we show the time evolution behavior to mean square displacements (MSD) in both directions. As a consequence, the model revels that until the system reaches the equilibrium, i.e., constant MSD, there is a Brownian diffusion in y direction, i.e., &lang; ( &Delta; y ) 2 &rang; &prop; t , and a crossover between sub and ballistic diffusion behaviors in x direction, i.e., &lang; ( &Delta; x ) 2 &rang; &prop; t 1 2 and &lang; ( &Delta; x ) 2 &rang; &prop; t 2 respectively. For static stochastic resetting, the ballistic regime vanishes. Also, we consider the idealized model according to the memory kernels to investigate the exponential and tempered power-law memory kernels effects on diffusive behaviors. In this way, we expose a rich class of anomalous diffusion process with crossovers among them. The proposal and the techniques applied in this work are useful to describe random walkers with non-static stochastic resetting on comb structure.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020027

Authors: Onur Saldır Mehmet Giyas Sakar Fevzi Erdogan

In this research, obtaining of approximate solution for fractional-order Burgers&rsquo; equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020026

Authors: Lütfi Akın

Time scales have been the target of work of many mathematicians for more than a quarter century. Some of these studies are of inequalities and dynamic integrals. Inequalities and fractional maximal integrals have an important place in these studies. For example, inequalities and integrals contributed to the solution of many problems in various branches of science. In this paper, we will use fractional maximal integrals to establish integral inequalities on time scales. Moreover, our findings show that inequality is valid for discrete and continuous conditions.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020025

Authors: Dmitrii Tumakov Dmitry Chikrin Petr Kokunin

Koch-type wire dipole antennas are considered herein. In the case of a first-order prefractal, such antennas differ from a Koch-type dipole by the position of the central vertex of the dipole arm. Earlier, we investigated the dependence of the base frequency for different antenna scales for an arm in the form of a first-order prefractal. In this paper, dipoles for second-order prefractals are considered. The dependence of the base frequency and the reflection coefficient on the dipole wire length and scale is analyzed. It is shown that it is possible to distinguish a family of antennas operating at a given (identical) base frequency. The same length of a Koch-type curve can be obtained with different coordinates of the central vertex. This allows for obtaining numerous antennas with various scales and geometries of the arm. An algorithm for obtaining small antennas for Wi-Fi applications is proposed. Two antennas were obtained: an antenna with the smallest linear dimensions and a minimum antenna for a given reflection coefficient.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020024

Authors: L. K. Mork Keith Sullivan Trenton Vogt Darin J. Ulness

This work builds upon previous studies of centered polygonal lacunary functions by presenting proofs of theorems showing how rotational and dihedral mirror symmetry manifest in these lacunary functions at the modulus level. These theorems then provide a general framework for constructing other lacunary functions that exhibit the same symmetries. These investigations enable one to better explore the effects of the gap behavior on the qualitative features of the associated lacunary functions. Further, two renormalized products of centered polygonal lacunary functions are defined and a connection to Ramanunjan&rsquo;s triangular lacunary series is made via several theorems.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020023

Authors: Jocelyn Sabatier

To tackle several limitations recently highlighted in the field of fractional differentiation and fractional models, some authors have proposed new kernels for the definition of fractional integration/differentiation operators. Some limitations still remain, however, with these kernels, whereas solutions prior to the introduction of fractional models exist in the literature. This paper shows that the fractional pseudo state space description, a fractional model widely used in the literature, is a special case of the Volterra equations, equations introduced nearly a century ago. Volterra equations can thus be viewed as a serious alternative to fractional pseudo state space descriptions for modelling power law type long memory behaviours. This paper thus presents a new class of model involving a Volterra equation and several kernels associated with this equation capable of generating power law behaviours of various kinds. One is particularly interesting as it permits a power law behaviour in a given frequency band and, thus, a limited memory effect on a given time range (as the memory length is finite, the description does not exhibit infinitely slow and infinitely fast time constants as for pseudo state space descriptions).

]]>Fractal and Fractional doi: 10.3390/fractalfract4020022

Authors: Alexandre Marques de Almeida Marcelo Kaminski Lenzi Ervin Kaminski Lenzi

Multiple-input multiple-output (MIMO) systems are usually present in process systems engineering. Due to the interaction among the variables and loops in the MIMO system, designing efficient control systems for both servo and regulatory scenarios remains a challenging task. The literature reports the use of several techniques mainly based on classical approaches, such as the proportional-integral-derivative (PID) controller, for single-input single-output (SISO) systems control. Furthermore, control system design approaches based on derivatives and integrals of non-integer order, also known as fractional control or fractional order (FO) control, are frequently used for SISO systems control. A natural consequence, already reported in the literature, is the application of these techniques to MIMO systems to address some inherent issues. Therefore, this work discusses the state-of-the-art of fractional control applied to MIMO systems. It outlines different types of applications, fractional controllers, controller tuning rules, experimental validation, software, and appropriate loop decoupling techniques, leading to literature gaps and research opportunities. The span of publications explored in this survey ranged from the years 1997 to 2019.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020021

Authors: Dumitru Baleanu Hassan Kamil Jassim

In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020020

Authors: Vsevolod Bohaienko Volodymyr Bulavatsky

Since the use of the fractional-differential mathematical model of anomalous geomigration process based on the MIM (mobile&ndash;immoble media) approach in engineering practice significantly complicates simulations, a corresponding simplified mathematical model is constructed. For this model, we state a two-dimensional initial-boundary value problem of convective diffusion of soluble substances under the conditions of vertical steady-state filtration of groundwater with free surface from a reservoir to a coastal drain. To simplify the domain of simulation, we use the technique of transition into the domain of complex flow potential through a conformal mapping. In the case of averaging filtration velocity over the domain of complex flow potential, an analytical solution of the considered problem is obtained. In the general case of a variable filtration velocity, an algorithm has been developed to obtain numerical solutions. The results of process simulation using the presented algorithm shows that the constructed mathematical model can be efficiently used to simplify and accelerate modeling process.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020019

Authors: Manish Kumar Bansal Devendra Kumar Priyanka Harjule Jagdev Singh

In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms of IIF. Next, we present some interesting corollaries by specializing the parameters of IIF in the form of simpler special functions and also mention a few known results, which are very useful in solving physical or real-life problems. Finally, some graphical results are presented to demonstrate the influence of the order of the fractional integral operator on the reaction rate.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020018

Authors: Ahmed Salem Balqees Alghamdi

In the present paper, we discuss a new boundary value problem for the nonlinear Langevin equation involving two distinct fractional derivative orders with multi-point and multi-nonlocal integral conditions. The fixed point theorems for Schauder and Krasnoselskii&ndash;Zabreiko are applied to study the existence results. The uniqueness of the solution is given by implementing the Banach fixed point theorem. Some examples showing our basic results are provided.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020017

Authors: Ramazan Ozarslan Erdal Bas

In this article, the Lewis model was considered for the soybean drying process by new fractional differential operators to analyze the estimated time in 50 ∘ C , 60 ∘ C , 70 ∘ C , and 80 ∘ C . Moreover, we used dimension parameters for the physical meaning of these fractional models within generalized and Caputo fractional derivatives. Results obtained with generalized fractional derivatives were analyzed comparatively with the Caputo fractional, integer order derivatives and Page model for the soybean drying process. All results for fractional derivatives are discussed and compared in detail.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020016

Authors: Jean-Philippe Aguilar

We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs &hellip;) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020015

Authors: Ndolane Sene

This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald&ndash;Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald&ndash;Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different values of the used fractional-order derivative.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020014

Authors: Peter B. Béda

Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020013

Authors: Shorog Aljoudi Bashir Ahmad Ahmed Alsaedi

In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use of the Leray-Schauder alternative and contraction mapping principle to derive the desired results. Illustrative examples for the main results are also presented.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020012

Authors: Eva-H. Dulf Dan C. Vodnar Alex Danku Cristina-I. Muresan Ovidiu Crisan

Biochemical processes present complex mechanisms and can be described by various computational models. Complex systems present a variety of problems, especially the loss of intuitive understanding. The present work uses fractional-order calculus to obtain mathematical models for erythritol and mannitol synthesis. The obtained models are useful for both prediction and process optimization. The models present the complex behavior of the process due to the fractional order, without losing the physical meaning of gain and time constants. To validate each obtained model, the simulation results were compared with experimental data. In order to highlight the advantages of fractional-order models, comparisons with the corresponding integer-order models are presented.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020011

Authors: Komal Saxena Pushpendra Singh Pathik Sahoo Satyajit Sahu Subrata Ghosh Kanad Ray Daisuke Fujita Anirban Bandyopadhyay

Biomaterials are primarily insulators. For nearly a century, electromagnetic resonance and antenna&ndash;receiver properties have been measured and extensively theoretically modeled. The dielectric constituents of biomaterials&mdash;if arranged in distinct symmetries, then each vibrational symmetry&mdash;would lead to a distinct resonance frequency. While the literature is rich with data on the dielectric resonance of proteins, scale-free relationships of vibrational modes are scarce. Here, we report a self-similar triplet of triplet resonance frequency pattern for the four-4 nm-wide tubulin protein, for the 25-nm-wide microtubule nanowire and 1-&mu;m-wide axon initial segment of a neuron. Thus, preserving the symmetry of vibrations was a fundamental integration feature of the three materials. There was no self-similarity in the physical appearance: the size varied by 106 orders, yet, when they vibrated, the ratios of the frequencies changed in such a way that each of the three resonance frequency bands held three more bands inside (triplet of triplet). This suggests that instead of symmetry, self-similarity lies in the principles of symmetry-breaking. This is why three elements, a protein, it&rsquo;s complex and neuron resonated in 106 orders of different time domains, yet their vibrational frequencies grouped similarly. Our work supports already-existing hypotheses for the scale-free information integration in the brain from molecular scale to the cognition.

]]>Fractal and Fractional doi: 10.3390/fractalfract4020010

Authors: Paulo M. Guzmán Péter Kórus Juan E. Nápoles Valdés

In this paper, we present a number of Chebyshev type inequalities involving generalized integral operators, essentially motivated by the earlier works and their applications in diverse research subjects.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010009

Authors: Atanaska Georgieva Snezhana Hristova

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010008

Authors: Xuefeng Zhang Yuqing Yan

This paper is devoted to the admissibility issue of singular fractional order systems with order &alpha; &isin; ( 0 , 1 ) based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. Then, an observer-based controller is designed to ensure system admissible. Finally, numerical examples are given to reveal the validity of the theoretical conclusions.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010007

Authors: Omar Bazighifan

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form r 3 t r 2 t r 1 t y &prime; t &prime; &prime; &prime; + q t f y &sigma; t = 0 , where t &ge; t 0 . The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010006

Authors: Michael Ioelovich

In this research, fractal properties of a cell wall in growing cotton fibers were studied. It was found that dependences of specific pore volume (P) and apparent density (&rho;) on the scale factor, F = H/h, can be expressed by power-law equations: P = Po F(Dv&minus;E) and &rho; = &rho;o F(E&minus;D&rho;), where h is minimum thickness of the microfibrilar network in the primary cell wall, H is total thickness of cell wall in growing cotton, Dv = 2.556 and D&rho; = 2.988 are fractal dimensions. From the obtained results it follows that microfibrilar network of the primary cell wall in immature fibers is loose and disordered, and therefore it has an increased pore volume (Po = 0.037 cm3/g) and low density (&rho;o = 1.47 g/cm3). With enhance days post anthesis of growing cotton fibers, the wall thickness and density increase, while the pore volume decreases, until dense structure of completely mature fibers is formed with maximum density (1.54 g/cm3) and minimum pore volume (0.006 cm3/g). The fractal dimension for specific pore volume, Dv = 2.556, evidences the mixed surface-volume sorption mechanism of sorbate vapor in the pores. On the other hand, the fractal dimension for apparent density, D&rho; = 2.988, is very close to Euclidean volume dimension, E = 3, for the three-dimensional space.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010005

Authors: Djelloul Ziane Mountassir Hamdi Cherif Dumitru Baleanu Kacem Belghaba

The main objective of this study is to apply the local fractional homotopy analysis method (LFHAM) to obtain the non-differentiable solution of two nonlinear partial differential equations of the biological population model on Cantor sets. The derivative operator are taken in the local fractional sense. Two examples have been presented showing the effectiveness of this method in solving this model on Cantor sets.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010004

Authors: Fractal and Fractional Editorial Office

The editorial team greatly appreciates the reviewers who have dedicated their considerable time and expertise to the journal&rsquo;s rigorous editorial process over the past 12 months, regardless of whether the papers are finally published or not [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract4010003

Authors: Anguera Andújar Jayasinghe Chakravarthy Chowdary Pijoan Ali Cattani

Fractal geometry has been proven to be useful in several disciplines. In the field of antenna engineering, fractal geometry is useful to design small and multiband antenna and arrays, and high-directive elements. A historic overview of the most significant fractal mathematic pioneers is presented, at the same time showing how the fractal patterns inspired engineers to design antennas.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010002

Authors: Vineet Prasad Kajal Kothari Utkal Mehta

In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational matrix of block pulse functions to reduce computational complexity. A special test signal is developed to isolate the identification of the nonlinear static function from that of the fractional-order linear dynamic system. The merit of the proposed technique is indicated by concurrent identification of the fractional order with linear system coefficients, algebraic representation of the immeasurable nonlinear static function output, and permitting use of non-iterative procedures for identification of the nonlinearity. The efficacy of the proposed method is exhibited through simulation at various signal-to-noise ratios.

]]>Fractal and Fractional doi: 10.3390/fractalfract4010001

Authors: Jocelyn Sabatier

This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples in which these behaviors are modeled by means of fractional models. However, several limitations of fractional models have recently been reported and other solutions must be found. In the literature, the analysis of distributed delay models and integro-differential equations in general is older than that of fractional models. In this paper, it is shown that particular delay distributions and conditions on the model coefficients make it possible to obtain power laws. The class of systems considered is then used to model the input-output behavior of a lithium-ion cell.

]]>Fractal and Fractional doi: 10.3390/fractalfract3040054

Authors: Alexander Iomin Vicenç Méndez Werner Horsthemke

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport properties. The first is a continuous time random walk on a many dimensional m + n comb, where m and n are the dimensions of the backbone and branches, respectively. We observe subdiffusion, ultra-slow diffusion and random localization as a function of n. The second deals with a quantum particle in the 1 + 1 comb. It turns out that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schr&ouml;dinger equation.

]]>Fractal and Fractional doi: 10.3390/fractalfract3040053

Authors: Ricardo Almeida Sania Qureshi

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number R 0 &lt; 1 , which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model.

]]>Fractal and Fractional doi: 10.3390/fractalfract3040052

Authors: Yuanhui Wang Yiming Chen

Viscoelastic pipeline conveying fluid is analyzed with an improved variable fractional order model for researching its dynamic properties accurately in this study. After introducing the improved model, an involuted variable fractional order, which is an unknown piecewise nonlinear function for analytical solution, an equation is established as the governing equation for the dynamic displacement of the viscoelastic pipeline. In order to solve this class of equations, a numerical method based on shifted Legendre polynomials is presented for the first time. The method is effective and accurate after the numerical example verifying. Numerical results show that how dynamic properties are influenced by internal fluid velocity, force excitation, and variable fractional order through the proposed method. More importantly, the numerical method has shown great potentials for dynamic problems with the high precision model.

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Authors: Ahmed Salem Balqees Alghamdi

With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii&ndash;Zabreiko&rsquo;s and the Leray&ndash;Schauder fixed point theorems. The Banach contraction mapping principle is used to investigate the uniqueness. Illustrative examples are provided to apply of the fundamental investigations.

]]>Fractal and Fractional doi: 10.3390/fractalfract3040050

Authors: Gani Stamov Anatoliy Martynyuk Ivanka Stamova

In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka&ndash;Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated.

]]>Fractal and Fractional doi: 10.3390/fractalfract3040049

Authors: Fang Fang Raymond Aschheim Klee Irwin

In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L / S = ϕ . The corresponding pointwise dimension is 1.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L / S ratio.

]]>Fractal and Fractional doi: 10.3390/fractalfract3040048

Authors: Fang Fang Klee Irwin Julio Kovacs Garrett Sadler

In this paper, we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps of the former case are &ldquo;closed&rdquo; (in the sense that faces of adjacent tetrahedra are brought into contact to form a &ldquo;face junction&rdquo;), while translational and rotational symmetries are obtained in the latter case. In all cases, an angular displacement of &beta; = arccos 3 ϕ &minus; 1 / 4 (or a closely related angle), where ϕ = 1 + 5 / 2 is the golden ratio, is observed between faces of a junction. Additionally, the overall number of plane classes, defined as the number of distinct facial orientations in the collection of tetrahedra, is reduced following the transformation. Finally, we present several &ldquo;curiosities&rdquo; involving the structures discussed here with the goal of inspiring the reader&rsquo;s interest in constructions of this nature and their attending, interesting properties.

]]>Fractal and Fractional doi: 10.3390/fractalfract3040047

Authors: Renat T. Sibatov HongGuang Sun

New aspects of electron transport in quantum wires with L&eacute;vy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered L&eacute;vy stable distribution of spacing between scatterers or tunneling barriers. The generalized Dorokhov&ndash;Mello&ndash;Pereyra&ndash;Kumar equation contains the tempered fractional derivative on wire length. The solution describes the evolution from the anomalous conductance distribution to the Dorokhov function for a long wire. For sequential tunneling, average values and relative fluctuations of conductance and resistance are calculated for different parameters of spatial distributions. A tempered L&eacute;vy stable distribution of spacing between barriers leads to a transition in conductance scaling.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030046

Authors: Iman Malmir

Fractional integration operational matrix of Chebyshev wavelets based on the Riemann&ndash;Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030045

Authors: Dimiter Prodanov

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor&rsquo;s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith&ndash;Volterra&ndash;Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030044

Authors: Bashir Ahmad Madeaha Alghanmi Ahmed Alsaedi Sotiris K. Ntouyas

We discuss the existence of solutions for a Caputo type multi-term nonlinear fractional differential equation supplemented with generalized integral boundary conditions. The modern tools of functional analysis are applied to achieve the desired results. Examples are constructed for illustrating the obtained work. Some new results follow as spacial cases of the ones reported in this paper.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030043

Authors: Dumitru Baleanu Hassan Kamil Jassim Maysaa Al Qurashi

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030042

Authors: L.K. Mork Trenton Vogt Keith Sullivan Drew Rutherford Darin J. Ulness

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030041

Authors: Alireza Khalili Golmankhaneh Carlo Cattani

In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030040

Authors: Trenton Vogt Darin J. Ulness

This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030039

Authors: Ndolane Sene José Francisco Gómez Aguilar

This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville&ndash;Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville&ndash;Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030038

Authors: Muhammad Uzair Awan Muhammad Aslam Noor Marcela V. Mihai Khalida Inayat Noor

The main objective of this paper is to obtain certain new k-fractional estimates of Hermite&ndash;Hadamard type inequalities via s-convex functions of Breckner type essentially involving k-Appell&rsquo;s hypergeometric functions. We also present applications of the obtained results by considering particular examples.

]]>Fractal and Fractional doi: 10.3390/fractalfract3030037

Authors: Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor

In this article, certain Hermite-Hadamard-type inequalities are proven for an exponentially-convex function via Riemann-Liouville fractional integrals that generalize Hermite-Hadamard-type inequalities. These results have some relationships with the Hermite-Hadamard-type inequalities and related inequalities via Riemann-Liouville fractional integrals.

]]>Fractal and Fractional doi: 10.3390/fractalfract3020036

Authors: Ifan Johnston Vassili Kolokoltsov

We look at estimates for the Green&rsquo;s function of time-fractional evolution equations of the form D 0 + &lowast; &nu; u = L u , where D 0 + &lowast; &nu; is a Caputo-type time-fractional derivative, depending on a L&eacute;vy kernel &nu; with variable coefficients, which is comparable to y &minus; 1 &minus; &beta; for &beta; &isin; ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green&rsquo;s function of D 0 &beta; u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green&rsquo;s function of D 0 &beta; u = &Psi; ( &minus; i &nabla; ) u where &Psi; is a pseudo-differential operator with constant coefficients that is homogeneous of order &alpha; . Thirdly, we obtain local two-sided estimates for the Green&rsquo;s function of D 0 &beta; u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green&rsquo;s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green&rsquo;s functions associated with L and &Psi; , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( &nu; , t ) u = L u , where D ( &nu; , t ) is a Caputo-type operator with variable coefficients.

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