Fractal and Fractional doi: 10.3390/fractalfract7060466

Authors: Luciano Telesca Nicodemo Abate Farid Faridani Michele Lovallo Rosa Lasaponara

Xylella fastidiosa is a phytobacterium able to provoke severe diseases in many species. When it infects olive trees, it induces the olive quick decline syndrome that leads the tree to a rapid desiccation and then to the death. This phytobacterium has been recently detected in olive groves in southern Italy, representing an important threat to the olive growing of the area. In this paper, in order to identify patterns revealing the presence of Xylella fastidiosa, several hundreds pixels of MODIS satellite evapostranspiration covering infected and healthy olive groves in southern Italy were analyzed by means of the Fisher&ndash;Shannon method and the multifractal detrended fluctuation analysis. The analysis of the receiver operating characteric curve indicates that the two informational quantities (the Fisher information measure and the Shannon entropy) and the three multifractal parameters (the range of generalized Hurst exponents and the width and the maximum of the multifractal spectrum) are well suited to discriminate between infected and healthy sites, although the maximum of the multifractal spectrum performs better than the others. These results could suggest the use of both the methods as an operational tool for early detection of plant diseases.

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Authors: Xiaohong Zhang Mengyuan Li Hui Liu

As special aggregation functions, overlap functions have been widely used in the soft computing field. In this work, with the aid of overlap functions, two new groups of fuzzy mathematical morphology (FMM) operators were proposed and applied to image processing, and they obtained better results than existing algorithms. First, based on overlap functions and structuring elements, the first group of new FMM operators (called OSFMM operators) was proposed, and their properties were systematically analyzed. With the implementation of OSFMM operators and the fuzzy C-means (FCM) algorithm, a new image edge extraction algorithm (called the OS-FCM algorithm) was proposed. Then, the second group of new FMM operators (called ORFMM operators) was proposed based on overlap functions and fuzzy relations. Another new image edge extraction algorithm (called OR-FCM algorithm) was proposed by using ORFMM operators and FCM algorithm. Finally, through the edge segmentation experiments of multiple standard images, the actual segmentation effects of the above-mentioned two algorithms and relevant algorithms were compared. The acquired results demonstrate that the image edge extraction algorithms proposed in this work can extract the complete edge of foreground objects on the basis of introducing the least noise.

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Authors: Vladimir E. Fedorov Marina V. Plekhanova Daria V. Melekhina

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan&ndash;Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan&ndash;Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan&ndash;Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan&ndash;Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin&ndash;Voigt viscoelastic media.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060463

Authors: Loredana Ciurdariu Eugenia Grecu

The aim of this study was to present several improved quantum Hermite&ndash;Hadamard-type integral inequalities for convex functions using a parameter. Thus, a new quantum identity is proven to be used as the main tool in the proof of our results. Consequently, in some special cases several new quantum estimations for q-midpoints and q-trapezoidal-type inequalities are derived with an example. The results obtained could be applied in the optimization of several economic geology problems.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060462

Authors: Kinda Abuasbeh Azmat Ullah Khan Niazi Hafiza Maria Arshad Muath Awadalla Salma Trabelsi

This article investigates the approximate controllability of non-linear fractional stochastic differential inclusions with non-local conditions. We establish a set of sufficient conditions for their approximate controllability and provide results in terms of controllability for the fractional stochastic control system. Our approach relies on using fractional calculus and the fixed-point theorem for multiple-valued operators. Finally, we present an illustrative example to support our findings.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060461

Authors: H. G. Abdelwahed A. F. Alsarhana E. K. El-Shewy Mahmoud A. E. Abdelrahman

The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schr&ouml;dinger equation (CNLSE). This model outlines the energy characteristics of quantum physics&rsquo; fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the CNLSE through multiplicative noise effects. This method accurately described a variety of solitary behaviors, including bright solitons, dark periodic envelopes, solitonic forms, and dissipative and dissipative&ndash;soliton-like waves, showing how the solutions changed as the values of the studied system&rsquo;s physical parameters were changed. The stochastic parameter was shown to affect the damping, growth, and conversion effects on the bright (dark) envelope and shock-forced oscillatory wave energy, amplitudes, and frequencies. In addition, the intensity of noise resulted in enormous periodic envelope stochastic structures and shock-forced oscillatory behaviors. The proposed technique is applicable to various energy equations in the nonlinear applied sciences.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060460

Authors: Ali Yüce

System identification is an important methodology used in control theory and constitutes the first step of control design. It is known that many real systems can be better characterized by fractional-order models. However, it is often quite complex and difficult to apply classical control theory methods analytically for fractional-order models. For this reason, integer-order models are generally considered in classical control theory. In this study, an alternative approximation method is proposed for fractional-order models. The proposed method converts a fractional-order transfer function directly into an integer-order transfer function. The proposed method is based on curve fitting that uses a quadratic system model and Equilibrium Optimizer (EO) algorithm. The curve fitting is implemented based on the unit step response signal. The EO algorithm aims to determine the optimal coefficients of integer-order transfer functions by minimizing the error between general parametric quadratic model and objective data. The objective data are unit step response of fractional-order transfer functions and obtained by using the Gr&uuml;nwald-Letnikov (GL) method in the Fractional-Order Modeling and Control (FOMCON) toolbox. Thus, the coefficients of an integer-order transfer function most properly can be determined. Some examples are provided based on different fractional-order transfer functions to evaluate the performance of the proposed method. The proposed method is compared with studies from the literature in terms of time and frequency responses. It is seen that the proposed method exhibits better model approximation performance and provides a lower order model.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060459

Authors: Alicia Cordero Renso V. Rojas-Hiciano Juan R. Torregrosa Maria P. Vassileva

In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov&rsquo;s Hyperfamily. It is a substantial improvement of the classical Newton&rsquo;s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King&rsquo;s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060458

Authors: Rodica Luca Alexandru Tudorache

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann&ndash;Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo&ndash;Krasnosel&rsquo;skii fixed point theorem and the Leggett&ndash;Williams fixed point theorem.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060457

Authors: Xinxin Jiang Lianzhong Li

The current study presents a comprehensive Lie symmetry analysis for the time-fractional Mikhailov&ndash;Novikov&ndash;Wang (MNW) system with the Riemann&ndash;Liouville fractional derivative. The corresponding simplified equations with the Erd&eacute;lyi&ndash;Kober fractional derivative are constructed by group invariant solutions. Furthermore, we obtain explicit solutions with the help of the power series method and show the dynamical behavior via evolutional figures. Finally, by means of Ibragimov&rsquo;s new conservation theorem, the conservation laws are derived for the system.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060456

Authors: Meng Liu Dong Chen Hong-Guang Sun Feng Zhang

The Rouse formula and its variants have been widely used to describe the vertical distribution of the sediment concentration in sediment-laden flows in equilibrium. Han&rsquo;s formula extends the Rouse formula to non-equilibrium regimes, where the diffusive flux is still assumed to be Fickian. The turbulent flow and suspension regimes downstream of a mega-reservoir, e.g., the Three Gorges Reservoir, usually exhibit fractal and unsaturated properties, respectively. To characterize the non-Fickian dynamics of suspended sediment and the non-equilibrium regime in natural dammed rivers, this study proposes a new formula for the concentration profile of unsaturated sediment based on the Hausdorff fractal derivative advection&ndash;dispersion equation. In addition, we find that the order of the Hausdorff fractal derivative is related to the sizes of the sediment and the degrees of non-equilibrium. Compared to Rouse and Han&rsquo;s formulae, the new formula performs better in describing the sediment concentration profiles in the Jingjiang Reach, approximately 100 km below the Three Gorges Dam.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060455

Authors: María Estrella Sousa-Vieira Manuel Fernández-Veiga

In the last years of the past century, complex correlation structures were empirically observed, both in aggregated and individual traffic traces, including long-range dependence, large-timescale self-similarity and multi-fractality. The use of stochastic processes consistent with these properties has opened new research fields in network performance analysis and in simulation studies, where the efficient synthetic generation of samples is one of the main topics. Nowadays, networks have to support data services for traffic sources that are poorly understood or still insufficiently observed, for which simple, reproducible, and good traffic models are yet to be identified, and it is reasonable to expect that previous generators could be useful. For this reason, as a continuation of our previous work, in this paper, we describe efficient and online generators of the correlation structures of the generalized fractional noise process (gfGn) and the generalized Cauchy (gC) process, proposed recently. Moreover, we explain how we can use the Whittle estimator in order to choose the parameters of each process that give rise to a better adjustment of the empirical traces.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060454

Authors: Yumei Zou Yujun Cui

We consider a system of Riemann&ndash;Liouville fractional differential equations with multi-point coupled boundary conditions. Using some techniques from matrix analysis and the properties of the integral operator defined on two Banach spaces, we establish some Lyapunov-type inequalities for the problem considered. Moreover, the comparison between two Lyapunov-type inequalities is given under certain special conditions. The inequalities obtained compliment the existing results in the literature.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060453

Authors: Jian Hou Xiangyun Meng Jingjia Wang Yongshng Han Yongguang Yu

In this paper, the numerical method for a multiterm time-fractional reaction&ndash;diffusion equation with classical Robin boundary conditions is considered. The full discrete scheme is constructed with the L1-finite difference method, which entails using the L1 scheme on graded meshes for the temporal discretisation of each Caputo fractional derivative and using the finite difference method on uniform meshes for spatial discretisation. By dealing with the discretisation of Robin boundary conditions carefully, sharp error analysis at each time level is proven. Additionally, numerical results that can confirm the sharpness of the error estimates are presented.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060452

Authors: R. Perumal M. Hymavathi M. Syed Ali Batul A. A. Mahmoud Waleed M. Osman Tarek F. Ibrahim

This research investigates the synchronization of distributed delayed discrete-time fractional-order complex-valued neural networks. The necessary conditions have been established for the stability of the proposed networks using the theory of discrete fractional calculus, the discrete Laplace transform, and the theory of fractional-order discrete Mittag&ndash;Leffler functions. In order to guarantee the global asymptotic stability, adequate criteria are determined using Lyapunov&rsquo;s direct technique, the Lyapunov approach, and some novel analysis techniques of fractional calculation. Thus, some sufficient conditions are obtained to guarantee the global stability. The validity of the theoretical results are finally shown using numerical examples.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060451

Authors: Ihsan Ullah Khan Amjid Hussain Shuo Li Ali Shokri

A nonlinear mathematical model of COVID-19 containing asymptomatic as well as symptomatic classes of infected individuals is considered and examined in the current paper. The largest eigenvalue of the next-generation matrix known as the reproductive number is obtained for the model, and serves as an epidemic indicator. To better understand the dynamic behavior of the continuous model, the unconditionally stable nonstandard finite difference (NSFD) scheme is constructed. The aim of developing the NSFD scheme for differential equations is its dynamic reliability, which means discretizing the continuous model that retains important dynamic properties such as positivity of solutions and its convergence to equilibria of the continuous model for all finite step sizes. The Schur&ndash;Cohn criterion is used to address the local stability of disease-free and endemic equilibria for the NSFD scheme; however, global stability is determined by using Lyapunov function theory. We perform numerical simulations using various values of some key parameters to see more characteristics of the state variables and to support our theoretical findings. The numerical simulations confirm that the discrete NSFD scheme maintains all the dynamic features of the continuous model.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060450

Authors: Yankai Li Dongping Li Yi Jiang Xiaozhou Feng

In this article, by combining a recent critical point theorem and several theories of the &psi;-Caputo fractional operator, the multiplicity results of at least three distinct weak solutions are obtained for a new &psi;-Caputo-type fractional differential system including the generalized p-Laplacian operator. It is noted that the nonlinear functions do not need to adapt certain asymptotic conditions in the paper, but, instead, are replaced by some simple algebraic conditions. Moreover, an evaluation criterion of the equation without solutions is also provided. Finally, two examples are given to demonstrate that the &psi;-Caputo fractional operator is more accurate and can adapt to deal with complex system modeling problems by changing different weight functions.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060449

Authors: Hind H. G. Hashem Ahmed M. A. El-Sayed Shorouk M. Al-Issa

In this paper, we discuss the existence of solutions for a hybrid cubic delayed integral inclusion with fractal feedback control. We are seeking solutions for these hybrid cubic delayed integral inclusions that are defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in BC(R+). In addition, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions of that integral inclusion. Finally, some cases analyzed are in the presence and absence of the control variable, and two examples are provided in order to indicate the validity of the assumptions.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060448

Authors: Jian Sun Ling Wang Dianxuan Gong

The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060447

Authors: K. Kaliraj P. K. Lakshmi Priya Juan J. Nieto

Stability analysis over a finite time interval is a well-formulated technique to study the dynamical behaviour of a system. This article provides a novel analysis on the finite-time stability of a fractional-order system using the approach of the delayed-type matrix Mittag-Leffler function. At first, we discuss the solution&rsquo;s existence and uniqueness for our considered fractional model. Then standard form of integral inequality of Gronwall&rsquo;s type is used along with the application of the delayed Mittag-Leffler argument to derive the sufficient bounds for the stability of the dynamical system. The analysis of the system is extended and studied with impulsive perturbations. Further, we illustrate the numerical simulations of our analytical study using relevant examples.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060446

Authors: Kamel Djeddi Tahar Bouali Ahmed H. Msmali Abdullah Ali H. Ahmadini Ali N. A. Koam

The novel coronavirus disease (SARS-CoV-2) has caused many infections and deaths throughout the world; the spread of the coronavirus pandemic is still ongoing and continues to affect healthcare systems and economies of countries worldwide. Mathematical models are used in many applications for infectious diseases, including forecasting outbreaks and designing containment strategies. In this paper, we study two types of SIR and SEIR models for the coronavirus. This study focuses on the discrete-time and fractional-order of these models; we study the stability of the fixed points and orbits using the Jacobian matrix and the eigenvalues and eigenvectors of each case; moreover, we estimate the parameters of the two systems in fractional order. We present a statistical study of the coronavirus model in two countries: Saudi Arabia, which has successfully recovered from the SARS-CoV-2 pandemic, and China, where the number of infections remains significantly high.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060445

Authors: Da Xie Xinjian Wang Chunyang Wang Kai Yuan Xuyang Wei Xuelian Liu Tingsheng Huang

High-quality image restoration is typically challenging due to low signal&ndash;to&ndash;background ratios (SBRs) and limited statistics frames. To address these challenges, this paper devised a method based on fractional-order total variation (FOTV) regularization for recovering Geiger-mode avalanche photodiode (GM-APD) light detection and ranging (lidar) depth images. First, the spatial differential peak-picking method was used to extract the target depth image from low SBR and limited frames. FOTV regularization was introduced based on the total variation regularization recovery model, which incorporates the fractional-order differential operator, in order to realize FOTV-regularization-based depth image recovery. These frameworks were used to establish an algorithm for GM-APD depth image recovery based on FOTV. The simulation and experimental results demonstrate that the devised FOTV-recovery algorithm improved the target reduction degree, peak signal&ndash;to&ndash;noise ratio, and structural similarity index measurement by 76.6%, 3.5%, and 6.9% more than the TV, respectively, in the same SBR and statistic frame conditions. Thus, the devised approach is able to effectively recover GM-APD lidar depth images in low SBR and limited statistic frame conditions.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060439

Authors: Xuexi Chen Ruiyue Ma Jinsui Wu Jihong Sun

As a non-homogeneous porous medium, the structural complexity of coal directly affects pore structure parameters and gas percolation characteristics, which in turn determine the fractal dimension of coal samples. Among them, the specific surface area of coal largely determines the complexity of the pore structure and is closely related to coal and gas protrusion hazards. To investigate the relationship between the fractal dimension of coal and its specific surface area, we analyzed the pore structure of coal samples using low-temperature nitrogen adsorption, the mercury pressure method, and X-ray micro-computed tomography (CT) experiments. By calculating the fractal dimension of coal and reconstructing it in three dimensions, the morphological characteristics and distribution of pores can be described qualitatively and quantitatively. The fractal dimension of coal samples was found to increase exponentially with the specific surface area based on measurements of large pores and mesopores via the mercury pressure method and those of small pores and micropores using the nitrogen adsorption method. X-ray micro-CT experiments revealed that the fractal dimension of large pores (i.e., &gt;2 &mu;m) conformed to this pattern.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060444

Authors: Lakhlifa Sadek

In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann&ndash;Liouville D&sigma;,&gamma; and Caputo cotangent fractional derivatives CD&sigma;,&gamma;, respectively, and their corresponding integral I&sigma;,&gamma;. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if &gamma;=1 we obtain the Riemann&ndash;Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann&ndash;Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the D&sigma;,&gamma;, CD&sigma;,&gamma; and I&sigma;,&gamma;. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060443

Authors: Miglena N. Koleva Lubin G. Vulkov

The simultaneous estimation of coefficients and the initial conditions for model fractional parabolic systems of porous media is reduced to the minimization of a least-squares cost functional. This inverse problem uses information about the pressures at a finite number of space time points. The Frechet gradient of the cost functional is derived. The application of the conjugate gradient method for numerical parameter estimation is also discussed. Computational results with noise-free and noisy data illustrate the efficiency and accuracy of the proposed algorithm.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060441

Authors: Xiaoxiao Zeng Kexin Fu Xiaofei Li Junjie Du Weiran Fan

Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060442

Authors: Abd-Allah Hyder Hüseyin Budak Mohamed A. Barakat

This study aims to prove some midpoint-type inequalities for fractional extended Riemann&ndash;Liouville integrals. Crucial equality is proven to build new results. Using this equality, several midpoint-type inequalities are established via differentiable convex functions and the proposed extended fractional operators. To be more specific, the well-known H&ouml;lder, Jensen, and power mean integral inequalities are employed in the demonstrated inequalities. Additionally, many remarks based on specific selections of the main results are presented. Moreover, to illustrate the key conclusions, a few instances are provided.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060440

Authors: Vincent-Ademola Adeyemi Esteban Tlelo-Cuautle Yuma Sandoval-Ibarra Jose-Cruz Nuñez-Perez

The main objective of this work was to implement the parameter-switching chaos control scheme for fractional-order spherical systems and develop a chaos-based image encryption and transmission system. The novelty in the developed secure communication system is the application of the parameter-switching scheme in the decryption of RGB and grayscale images, which undergo one round of encryption using the chaotic states of the fractional system and a diffusion process. The secure communication system has a synchronized master and slave topology, resulting in transmitter and receiver systems for encrypting and decrypting images, respectively. This work was demonstrated numerically and also implemented on two FPGAs, namely Artix-7 AC701 and Cyclone V. The results show that the parameter-switching scheme controls chaos in the fractional-order spherical systems effectively. Furthermore, the performance analysis of the image encryption and transmission system shows that there is no similarity between the original and encrypted images, while the decryption of the encrypted images is without a loss of quality. The best result in terms of the encryption was obtained from the chaotic state x of the fractional-order system, with correlation coefficients of 0.0511 and 0.0392 for the RGB and grayscale images, respectively. Finally, the utilization of the FPGA logical resources shows that the implementation on Artix-7 AC701 is more logic-efficient than on Cyclone V.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060438

Authors: Isra Al-Shbeil Jianhua Gong Samrat Ray Shahid Khan Nazar Khan Hala Alaqad

Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the S&#259;l&#259;gean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the S&#259;l&#259;gean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coefficient estimates.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060437

Authors: Kothandapani Muthuvel Panumart Sawangtong Kalimuthu Kaliraj

The aim of this work is to analyze the relative controllability and Ulamn&ndash;Hyers stability of the&nbsp;&psi;-Caputo fractional neutral delay differential system. We use neutral&nbsp;&psi;-delayed perturbation of the Mitttag&ndash;Leffler matrix function and Banach contraction principle to examine the Ulam&ndash;Hyers stability of our considered system. We formulate the Grammian matrix to establish the controllability results of the linear fractonal differential system. Further, we employ the fixed-point technique of Krasnoselskii&rsquo;s type to establish the sufficient conditions for the relative controllability of a semilinear&nbsp;&psi;-Caputo neutral fractional system. Finally, the theoretical study is validated by providing an application.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060436

Authors: Arshad Ali Khursheed J. Ansari Hussam Alrabaiah Ahmad Aloqaily Nabil Mlaiki

In this research paper, we study a coupled system of piecewise-order differential equations (DEs) with variable kernel and impulsive conditions. DEs with variable kernel have high flexibility due to the freedom of changing the kernel. We study existence and stability theory and derive sufficient conditions for main results of the proposed problem. We apply Scheafer&rsquo;s fixed point theorem and Banach fixed point theorem for the result of at least one and unique solution, respectively. In addition, stability results based on the Ulam&ndash;Hyers concept are derived. Being a coupled system of piecewise fractional-order DEs with variable kernel and impulsive effects, the obtained results have multi-dimension applications. To demonstrate the applications, we apply the derived results to a numerical problem.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060435

Authors: Fahad M. Almasoudi Gaber Magdy Abualkasim Bakeer Khaled Saleem S. Alatawi Mahmoud Rihan

This paper proposes an efficient load frequency control (LFC) technique based on a fractional-order proportional&ndash;integral&ndash;derivative&ndash;accelerator with a low-pass filter compensator (FOPIDA-LPF) controller, which can also be accurately referred to as the PI&lambda;DND2N2 controller. A trustworthy metaheuristic optimization algorithm, known as the gray wolf optimizer (GWO), is used to fine-tune the suggested PI&lambda;DND2N2 controller parameters. Moreover, the proposed PI&lambda;DND2N2 controller is designed for the LFC of a self-contained hybrid maritime microgrid system (HM&mu;GS) containing solid oxide fuel cell energy units, a marine biodiesel generator, renewable energy sources (RESs), non-sensitive loads, and sensitive loads. The proposed controller enables the power system to deal with random variations in load and intermittent renewable energy sources. Comparisons with various controllers used in the literature demonstrate the excellence of the proposed PI&lambda;DND2N2 controller. Additionally, the proficiency of GWO optimization is checked against other powerful optimization techniques that have been extensively researched: particle swarm optimization and ant lion optimization. Finally, the simulation results performed by the MATLAB software prove the effectiveness and reliability of the suggested PI&lambda;DND2N2 controller built on the GWO under several contingencies of different load perturbations and random generation of RESs. The proposed controller can maintain stability within the system, while also greatly decreasing overshooting and minimizing the system&rsquo;s settling time and rise time.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060434

Authors: Yuqin Song Peijiang Liu

This research article investigates the application of L&eacute;vy noise to understand the dynamic aspects of measles epidemic modeling and seeks to explain the impact of vaccines on the spread of the disease. After model formulation, the study utilises uniqueness and existence techniques to derive a positive solution to the underlying stochastic model. The Lyapunov function is used to investigate the stability results associated with the proposed stochastic model. The model&rsquo;s dynamic characteristics are analyzed in the vicinity of the infection-free and endemic states of the associated ODEs model. The stochastic threshold Rs that ensures disease&rsquo;s extinction whenever Rs&lt;1 is calculated. We utilized data from Pakistan in 2019 to estimate the parameters of the model and conducted simulations to forecast the future behavior of the disease. The results were compared to actual data using standard curve fitting tools.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060433

Authors: Hani Albalawi Abualkasim Bakeer Sherif A. Zaid El-Hadi Aggoune Muhammad Ayaz Ahmed Bensenouci Amir Eisa

Currently, a two-level voltage source inverter (2L-VSI) is regarded as the cornerstone of modern industrial applications. However, the control of VSIs is a challenging task due to their nonlinear and time-varying nature. This paper proposes employing the fractional-order controller (FOC) to improve the performance of model-free predictive control (MFPC) of the 2L-VSI voltage control in uninterruptible power supply (UPS) applications. In the conventional MFPC based on the ultra-local model (ULM), the unknown variable that includes all the system disturbances is estimated using algebraic identification, which is insufficient to improve the prediction accuracy in the predictive control. The proposed FO-MFPC uses fractional-order proportional-integral control (FOPI) to estimate the unknown function associated with the MFPC. To get the best performance from the FOPI, its parameters are optimally designed using the grey wolf optimization (GWO) approach. The number of iterations of the GWO is 100, while the grey wolf&rsquo;s number is 20. The proposed GWO algorithm achieves a small fitness function value of approximately 0.156. In addition, the GWO algorithm nearly finds the optimal parameters after 80 iterations for the defined objective function. The performance of the proposed FO-MFPC controller is compared to that of conventional MFPC for the three loading cases and conditions. Using MATLAB simulations, the simulation results indicated the superiority of the proposed FO-MFPC controller over the conventional MFPC in steady state and transient responses. Moreover, the total harmonic distortion (THD) of the output voltage at different sampling times proves the excellent quality of the output voltage with the proposed FO-MFPC controller over the conventional MFPC controller. The results confirm the robustness of the two control systems against parameter mismatches. Additionally, using the TMS320F28379D kit, the experimental verification of the proposed FO-MFPC control strategy is implemented for 2L-VSI on the basis of the Hardware-in-the-Loop (HIL) simulator, demonstrating the applicability and effective performance of our proposed control strategy under realistic circumstances.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060432

Authors: Aatef Hobiny Ibrahim Abbas

This article studies the effects of fractional time derivatives on thermo-mechanical interaction in living tissue during hyperthermia treatment by using the eigenvalues approach. A comprehensive understanding of the heat transfer mechanism and the related thermo-mechanical interactions with the patient&rsquo;s living tissues is crucial for the effective implementation of thermal treatment procedures. The surface of living tissues is traction-free and is exposed to a pulse boundary heat flux that decays exponentially. The Laplace transforms and their associated techniques are applied to the generalized bio-thermo-elastic model, and analytical procedures are then implemented. The eigenvalue approach is utilized to obtain the solution of governing equations. Graphical representations are given for the temperature, the displacement, and the thermal stress results. Afterward, a parametric study was carried out to determine the best method for selecting crucial design parameters that can improve the precision of hyperthermia therapies.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060431

Authors: Alexander Iomin Ralf Metzler Trifce Sandev

An example of non-Markovian quantum dynamics is considered in the framework of a geometrical (topological) subordination approach. The specific property of the model is that it coincides exactly with the fractional diffusion equation, which describes the geometric Brownian motion on combs. Both classical diffusion and quantum dynamics are described using the dilatation operator xddx. Two examples of geometrical subordinators are considered. The first one is the Gaussian function, which is due to the comb geometry. For the quantum consideration with a specific choice of the initial conditions, it corresponds to the integral representation of the Mittag&ndash;Leffler function by means of the subordination integral. The second subordinator is the Dirac delta function, which results from the memory kernels that define the fractional time derivatives in the fractional diffusion equation.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060430

Authors: Ruiqing Shi Yang Li Cuihong Wang

In this paper, a fractional-order model for African swine fever with limited medical resources is proposed and analyzed. First, the existence and uniqueness of a positive solution is proven. Second, the basic reproduction number and the conditions sufficient for the existence of two equilibriums are obtained. Third, the local stability of the two equilibriums is studied. Next, some numerical simulations are performed to verify the theoretical results. The mathematical and simulation results show that the values of some parameters, such as fractional order and medical resources, are critical for the stability of the equilibriums.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060429

Authors: Ahmad Shafee Yousuf Alkhezi Rasool Shah

In this paper, we present an efficient solution method for solving fractional system partial differential equations (FSPDEs) using the Laplace residual power series (LRPS) method. The LRPS method is a powerful technique for solving FSPDEs, as it allows for the efficient computation of the solution in the form of a power series expansion. The method is based on the Laplace transform and the residual power series, and is applied to a system of coupled FSPDEs. The method is validated using several test problems, and the results show that the LRPS method is a reliable and efficient method for solving FSPDEs.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060427

Authors: Tengfei Shen

The purpose of this paper is to investigate the initial value problem of Hadamard-type fractional relativistic oscillator equation with p-Laplacian operator. By overcoming the perturbation of singularity to fractional relativistic oscillator equation, the multiplicity of positive solutions to the problem were proved via the methods of reducing and topological degree in cone, which extend and enrich some previous results.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060428

Authors: Jianhong Zhu Junbin Chen Xiaoliang Duanmu Xiaoming Wang Diguang Gong Xiangrong Nie

After multi-stage volume hydraulic fracturing in a shale oil reservoir, massive amounts of water can be imbibed into the matrix pores. One of the key imbibition characteristics of a shale reservoir is the imbibition water and its height distribution. Based on high pressure mercury injection (HPMI) experiments and nuclear magnetic resonance (NMR) analyses, this study quantitatively evaluated the pore-size distribution of Chang 7 continental shale oil reservoirs in Yanchang Formation, Ordos Basin. The pores could be divided into three types as micropores (&le;0.1 &mu;m), mesopores (0.1&ndash;1.0 &mu;m), and macropores (&gt;1.0 &mu;m), while the volume of micropores and mesopores accounted for more than 90%. This demonstrated that there were strong heterogeneity and micro&ndash;nano characteristics. According to the spontaneous imbibition (SI) experiments, the cumulative proportion of imbibition water content was the largest in micropores, exceeding 43%, followed by mesopores around 30%, and that of macropores was the lowest, and basically less than 20%. The negative values of stage water content in the macropore or mesopore indicated that these pores became a water supply channel for other dominant imbibition pores. Additionally, combining the fractal theory with the NMR T2 spectrum, the relative imbibition water and actual height were calculated in different pores, while the height distribution varied with cores and shale oil. The shorter the core, the higher was the relative height, while the radius of macropores filled with imbibition water was reduced. This indicates that the height distribution was affected by the pore structure, oil viscosity, and core length.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060426

Authors: Farhana Tasnim Md Ali Akbar Mohamed S. Osman

In the field of nonlinear optics, quantum mechanics, condensed matter physics, and wave propagation in rigid and other nonlinear instability phenomena, the nonlinear Schr&ouml;dinger equation has significant applications. In this study, the soliton solutions of the space-time fractional cubic nonlinear Schr&ouml;dinger equation with Kerr law nonlinearity are investigated using an extended direct algebraic method. The solutions are found in the form of hyperbolic, trigonometric, and rational functions. Among the established solutions, some exhibit wide spectral and typical characteristics, while others are standard. Various types of well-known solitons, including kink-shape, periodic, V-shape, and singular kink-shape solitons, have been extracted here. To gain insight into the internal formation of these phenomena, the obtained solutions have been depicted in two- and three-dimensional graphs with different parameter values. The obtained solitons can be employed to explain many complicated phenomena associated with this model.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060424

Authors: Kuo-Chen Lu Kuo-Shing Chen

This study aimed to uncover the impact of COVID-19 on the leading cryptocurrency (Bitcoin) and on sustainable finance with specific attention to their potential long memory properties. In this article, the application of the selected methodologies is based on a fractal and entropy analysis of the econometric model in the financial market. To detect the regularity/irregularity property of a time series, approximate entropy is introduced to measure deterministic chaos. Using daily data for Bitcoin and sustainable finance, namely DJSW, Green Bond, Carbon, and Clean Energy, we examine long memory behaviour by employing a rescaled range statistic (R/S) methodology. The results of the research present that the returns of Bitcoin, the Dow Jones Sustainability World Index (DJSW), Green Bond, Carbon, and Clean Energy have a significant long memory. Contrastingly, an interdisciplinary approach, namely wavelet analysis, is also used to obtain complementary results. Wavelet analysis can provide warning information about turmoil phenomena and offer insights into co-movements in the time&ndash;frequency space. Our findings reveal that approximate entropy shows crisis (turmoil) conditions in the Bitcoin market, despite the nature of the pandemic&rsquo;s origin. Crucially, compared to Bitcoin assets, sustainable financial assets may play a better safe haven role during a pandemic turmoil period. The policy implications of this study could improve trading strategies for the sake of portfolio managers and investors during crisis and non-crisis periods.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060425

Authors: Ahmed Salem Kholoud N. Alharbi

This study demonstrates the total control of a class of hybrid neutral fractional evolution equations with non-instantaneous impulses and non-local conditions. The boundary value problem with non-local conditions is created using the Caputo fractional derivative of order 1&lt;&alpha;&le;2. In order to create novel, strongly continuous associated operators, the infinitesimal generator of the sine and cosine families is examined. Additionally, two approaches are used to discuss the solution&rsquo;s total controllability. A compact strategy based on the non-linear Leray&ndash;Schauder alternative theorem is one of them. In contrast, a measure of a non-compactness technique is implemented using the Sadovskii fixed point theorem with the Kuratowski measure of non-compactness. These conclusions are applied using simulation findings for the non-homogeneous fractional wave equation.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060423

Authors: Shuang Song Mingkun Pang Yi Guo Lei Zhang Tianjun Zhang Hongyu Pan

The permeability of crushed coal bodies plays a bottom neck role in seepage processes, which significantly limits the coal resource utilisation. To study the permeability of crushed coal bodies under pressure, the particle size distribution of crushed coal body grains is quantitatively considered by fractal theory. In addition, the parameters of the percolation characteristics of crushed coal body grains are calculated. Moreover, the permeability of the crushed coal body during recrushing is determined by the fractal dimension and porosity. A lateral limit compression test with the crushed coal bodies was carried out to illustrate the effect of the porosity on the permeability, In addition, a compressive crushed coal body size fractal&ndash;permeability model was proposed by combination of the fractal dimension and the non-Darcy equivalent permeability. The results show (1) the migration and loss of fine particles lead to a rapid increase in the porosity of the crushed coal body. (2) Increases in the effective stress cause the porosity and permeability to decrease. When the porosity decreases to approximately 0.375, its effect is undermined. (3) The migration and loss of fine particles change the pore structure and enhance the permeability properties of the skeleton, causing sudden seepage changes. (4) At low porosity, the permeability k is slightly larger than the non-Darcy equivalent permeability ke. Thus, the experimental data show an acceptable agreement with the present model. A particle size fractal&ndash;percolation model for crushed coal bodies under pressure provides a solution for effectively determining the grain permeability of the crushed coal bodies. The research results can contribute to the formation of more fractal-seepage theoretical models in fractured lithosphere, karst column pillars and coal goaf, and provide theoretical guidance for mine water disaster prevention.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060422

Authors: Yuzhen Chen Hui Wang Suzhen Li Rui Dong

To accurately predict the time series of energy data, an optimized Hausdorff fractional grey seasonal model was proposed based on the complex characteristics of seasonal fluctuations and local random oscillations of seasonal energy data. This paper used a new seasonal index to eliminate the seasonal variation of the data and weaken the local random fluctuations. Furthermore, the Hausdorff fractional accumulation operator was introduced into the traditional grey prediction model to improve the weight of new information, and the particle swarm optimization algorithm was used to find the nonlinear parameters of the model. In order to verify the reliability of the new model in energy forecasting, the new model was applied to two different energy types, hydropower and wind power. The experimental results indicated that the model can effectively predict quarterly time series of energy data. Based on this, we used China&rsquo;s quarterly natural gas production data from 2015 to 2021 as samples to forecast those for 2022&ndash;2024. In addition, we also compared the proposed model with the traditional statistical models and the grey seasonal models. The comparison results showed that the new model had obvious advantages in predicting quarterly data of natural gas production, and the accurate prediction results can provide a reference for natural gas resource allocation.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060421

Authors: Chuanjing Song

In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060417

Authors: Ibtehal Alazman Mohamed Jleli Bessem Samet

A Schr&ouml;dinger equation with a time-fractional derivative, posed in (0,&infin;)&times;I, where I=]a,b], is investigated in this paper. The equation involves a singular Hardy potential of the form &lambda;(x&minus;a)2, where the parameter &lambda; belongs to a certain range, and a nonlinearity of the form &mu;(x&minus;a)&minus;&rho;|u|p, where &rho;&ge;0. Using some a priori estimates, necessary conditions for the existence of weak solutions are obtained.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060420

Authors: Xinyu Liu Yi-Fei Pu

Edge detection is a highly researched topic in the field of image processing, with numerous methods proposed by previous scholars. Among these, ant colony algorithms have emerged as a promising approach for detecting image edges. These algorithms have demonstrated high efficacy in accurately identifying edges within images. For this paper, due to the long-term memory, nonlocality, and weak singularity of fractional calculus, fractional-order ant colony algorithm combined with fractional differential mask and coefficient of variation (FACAFCV) for image edge detection is proposed. If we set the order of the fractional-order ant colony algorithm and fractional differential mask to v=0, the edge detection method we propose becomes an integer-order edge detection method. We conduct experiments on images that are corrupted by multiplicative noise, as well as on an edge detection dataset. Our experimental results demonstrate that our method is able to detect image edges, while also mitigating the impact of multiplicative noise. These results indicate that our method has the potential to be a valuable tool for edge detection in practical applications.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060419

Authors: Jose Vanterler da C. Sousa

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in &psi;-fractional space Hp&alpha;,&beta;;&psi;(&Omega;). The results obtained in this paper are the first to make use of the theory of &psi;-Hilfer fractional operators with p-Laplacian.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060418

Authors: Liliana Guran Khurram Shabbir Khushdil Ahmad Monica-Felicia Bota

We have developed a Jungck version of the DK iterative scheme called the Jungck&ndash;DK iterative scheme. Our analysis focuses on the convergence and stability of the Jungck&ndash;DK scheme for a pair of non-self-mappings using the more general contractive condition. We demonstrate that this iterative scheme converges faster than all other leading Jungck-type iterative schemes. To further illustrate its effectiveness, we provide an example to verify the rate of convergence and prove the data dependence result for the Jungck&ndash;DK iterative scheme. Finally, we calculate the escape criteria for generating Mandelbrot and Julia sets for polynomial functions and present visually appealing images of these sets by our modified iteration.

]]>Fractal and Fractional doi: 10.3390/fractalfract7060416

Authors: Mohamed Naji Muftah Ahmad Athif Mohd Faudzi Shafishuhaza Sahlan Shahrol Mohamaddan

This study aims to improve the performance of a pneumatic positioning system by designing a control system based on Fuzzy Fractional Order Proportional Integral Derivative (Fuzzy FOPID) controllers. The pneumatic system&rsquo;s mathematical model was obtained using a system identification approach, and the Fuzzy FOPID controller was optimized using a PSO algorithm to achieve a balance between performance and robustness. The control system&rsquo;s performance was compared to that of a Fuzzy PID controller through real-time experimental results, which showed that the former provided better rapidity, stability, and precision. The proposed control system was applied to a pneumatically actuated ball and beam (PABB) system, where a Fuzzy FOPID controller was used for the inner loop and another Fuzzy FOPID controller was used for the outer loop. The results demonstrated that the intelligent pneumatic actuator, when coupled with a Fuzzy FOPID controller, can accurately and robustly control the positioning of the ball and beam system.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050415

Authors: Hari Mohan Srivastava

This Special Issue of the MDPI journal, Fractal and Fractional, on the subject area of &ldquo;Operators of Fractional Calculus and Their Multidisciplinary Applications&rdquo; consists of 19 peer-reviewed papers, including some invited feature articles, originating from all over the world [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract7050414

Authors: Danila Cherkashin Yana Teplitskaya

We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that S&cup;A is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties).

]]>Fractal and Fractional doi: 10.3390/fractalfract7050413

Authors: Ved Prakash Dubey Jagdev Singh Sarvesh Dubey Devendra Kumar

In this paper, the Kharrat&ndash;Toma transforms of the Prabhakar integral, a Hilfer&ndash;Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP fractional derivative via Kharrat&ndash;Toma transform. The solutions of Cauchy problems and the diffusion equations modeled with the HP fractional derivative are computed in the form of the generalized Mittag&ndash;Leffler function.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050412

Authors: Viorel-Puiu Paun

Advances in our knowledge of nonlinear dynamical networks, systems and processes (as well as their unified repercussions) currently allow us to study many typical complex phenomena taking place in nature, from the nanoscale to the extra-galactic scale, in an comprehensive manner [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract7050411

Authors: Suha B. Al-Shaikh Ahmad A. Abubaker Khaled Matarneh Mohammad Faisal Khan

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator (DR&lambda;,qm,n) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator (DR&lambda;,qm,n), we establish the q-analogues of two new integral operators (F&lambda;,&gamma;1,&gamma;2,&hellip;&gamma;lm,n,q and G&lambda;,&gamma;1,&gamma;2,&hellip;&gamma;lm,n,q), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator (DR&lambda;,qm,n) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050410

Authors: Marko Kostić Belkacem Chaouchi Wei-Shih Du Daniel Velinov

In this paper, we analyze the Bohr &rho;-almost periodic type sequences and the generalized &rho;-almost periodic type sequences of the form F:I&times;X&rarr;Y, where &empty;&ne;I&sube;Zn, X and Y are complex Banach spaces and &rho; is a general binary relation on Y. We provide many structural results, observations and open problems about the introduced classes of &rho;-almost periodic sequences. Certain applications of the established theoretical results to the abstract Volterra integro-difference equations are also given.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050409

Authors: Asad Khan Azmat Ullah Khan Niazi Waseem Abbasi Airish Jamil Jaleel Ahsan Malik

In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weighted digraph. A disturbance term is introduced into the delayed and non-delayed controller part to make it more practical. Our method involves proposing algebraic criteria by utilizing non-delayed and delayed protocols, applying the Razumikhin technique and graph theory respectively. The presented method can well overcome the difficulty resulting from fractional calculus, time delays and fractional derivatives. To demonstrate the validity and effectiveness of our findings, we provide an example at the end of our study.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050408

Authors: Yujie Tang Yun Peng Guitian He Wenjie Liang Weiting Zhang

This work aims to address the P-bifurcation of a stochastic nonlinear system with fractional damping driven by Gaussian white noise. Based on a stochastic averaging method, a fractional damping stochastic nonlinear equation has been studied. Furthermore, the expressions of drift and diffusion coefficients of the Fokker&ndash;Planck equation (FPKE) have been obtained. The probability density function (PDF), the steady solution of FPKE, has also been derived. Then, PDFs of two fractional damping Morse oscillators have been obtained. One can note that the analytical results coincide with the results of numerical simulation. Importantly, stochastic P-bifurcation of a fractional damping stochastic nonlinear Morse oscillator has been further addressed and analyzed.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050407

Authors: Prasantha Bharathi Dhandapani Víctor Leiva Carlos Martin-Barreiro Maheswari Rangasamy

In this paper, we introduce a SIVR model using the Laplace Adomian decomposition. This model focuses on a new trend in mathematical epidemiology dedicated to studying the characteristics of vaccination of infected communities. We analyze the epidemiological parameters using equilibrium stability and numerical analysis techniques. New mathematical strategies are also applied to establish our epidemic model, which is a pandemic model as well. In addition, we mathematically establish the chance for the next wave of any pandemic disease and show that a consistent vaccination strategy could control it. Our proposal is the first model introducing a vaccination strategy to actively infected cases. We are sure this work will serve as the basis for future research on COVID-19 and pandemic diseases since our study also considers the vaccinated population.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050406

Authors: Rodica Luca

Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemistry, biophysics, biology, medical sciences, financial economics, ecology, bioengineering, control theory, signal and image processing, aerodynamics, transport dynamics, thermodynamics, viscoelasticity, hydrology, statistical mechanics, electromagnetics, astrophysics, cosmology, and rheology [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract7050405

Authors: Muhammad Tariq Sotiris K. Ntouyas Asif Ali Shaikh

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m,tgs)-type s-preinvex function, Godunova&ndash;Levin (s,m)-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite&ndash;Hadamard (H&ndash;H), Fej&eacute;r, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H&ndash;H and Fej&eacute;r-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050404

Authors: Moldir Muratbekova Bakhtiyar Kadirkulov Maira Koshanova Batirkhan Turmetov

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050403

Authors: Vladimir Kulish Navid Aslfattahi Michal Schmirler

Fractional differ-integral operators are used to obtain the equation of state for a substance that can be seen as fractal. Two equations of state have been obtained, the first of which depends on two parameters that characterize the fractal dimension of the material and the thermal energy of the particles, respectively. The second equation involves three parameters, and expressions for the Helmholtz free energy and the bulk modulus have also been obtained for this equation. The model presented in this study has been validated using experimental data available in literature, and fractional exponent have been determined for various metals.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050402

Authors: Tao Chen

A new fractional accumulation technique based on discrete sequence convolution transform was developed. The accumulation system, whose unit impulse response is the accumulation convolution sequence, was constructed; then, the order was extended to fractional orders. The fractional accumulative convolution grey forecasting model GMr*(1,1) was established on the sequence convolution. From the viewpoint of sequence convolution, we can better understand the mechanism of accumulative generation. Real cases were used to verify the validity and effectiveness of the fractional accumulative convolution method.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050401

Authors: Bao Lv Kai Han Yongzhen Wang Xiaolong Li

Aiming to accurately predict the leakage rate of the sealing interface, this work proposes a two-dimensional finite element model of a proton exchange membrane fuel cell, which includes the microscopic surface morphology and the asperity contact process of the components. First of all, we constructed the surface morphology of the seal by the two-dimensional W-M (Weierstrass&ndash;Mandelbrot) fractal function and explored the influence of fractal dimension (D) and scale parameter (G) on the surface profile. Furthermore, the finite element method and Poiseuille fluid theory were adopted to obtain the deformation variables of the asperity under different clamping pressures and leakage rates. Moreover, we quantitatively analyzed the impact of surface roughness on the clamping pressure and leakage rate. It was found that both the surface amplitude and surface roughness are positively correlated with G and negatively correlated with D. Surface morphology is proportional to D but has no relationship with G. Additionally, the deformation asperity decreases exponentially with growing clamping pressure, and the leakage rate is consistent with the experimental values at a clamping pressure of 0.54 MPa. With the same leakage rate, when the seal surface roughness value is less than 1 &mu;m, a doubled roughness value leads to an increase of 31% in the clamping pressure. In contrast, when the surface roughness of the seal is greater than 1 &mu;m, a doubled roughness value induces an increase of 50% in the corresponding clamping pressure.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050400

Authors: Sultan Alyobi Rashid Jan

Infectious diseases can have a significant economic impact, both in terms of healthcare costs and lost productivity. This can be particularly significant in developing countries, where infectious diseases are more prevalent, and healthcare systems may be less equipped to handle them. It is recognized that the hepatitis B virus (HBV) infection remains a critical global public health issue. In this study, we develop a comprehensive model for HBV infection that includes vaccination and hospitalization through a fractional framework. It has been shown that the solutions of the recommended system of HBV infection are positive and bounded. We examine the steady states of the model and determine the basic reproduction number; denoted by R0. The qualitative and quantitative behavior of the model is demonstrated using mathematical skills and numerical techniques. It has been proved that the infection-free steady state of the system is locally asymptotically stable if R0&lt;1 and unstable otherwise. Furthermore, the Ulam&ndash;Hyers stability (UHS) of the recommended fractional models is investigated and the significant conditions are provided. We present an iterative technique to visualize the dynamical behavior of the system. We perform different simulations to illustrate the effect of different input factors on the solution pathways of the system of HBV infection to conceptualize the role of parameters in the control and prevention of the infection.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050399

Authors: Najmeddine Attia Rim Amami Rimah Amami

Let &part;T be a super-critical Galton&ndash;Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of &part;T along which the sequence SnX(t)/SnX&tilde;(t) has a given set of limit points, where SnX(t) and SnX&tilde;(t) are two branching random walks defined on &part;T. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence s=(sn), we consider E&alpha;,s=t&isin;&part;T:SnX(t)&minus;&alpha;SnX&tilde;(t)&sim;snasn&rarr;+&infin;. We will give a sufficient condition on (sn) so that E&alpha;,s has a maximal Hausdorff and packing dimension.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050397

Authors: Jian Cao Bai-Ni Guo Wei-Shih Du Feng Qi

In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1x&alpha;x to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions 1+1x&alpha;x and (1+x)&alpha;/x, and present a closed-form formula of the partial Bell polynomials Bn,k(H0(x),H1(x),&#8943;,Hn&minus;k(x)) for n&ge;k&ge;0, where Hk(x)=&int;0&infin;eu&minus;1&minus;ueuuk&minus;1e&minus;xudu for k&ge;0 are completely monotonic on (0,&infin;).

]]>Fractal and Fractional doi: 10.3390/fractalfract7050398

Authors: Mohamed Adel Mohamed M. Khader Salman Algelany

Presenting and simulating the numerical treatment of the nine-dimensional fractional chaotic Lorenz system is the goal of this work. The spectral collocation method (SCM), which makes use of Changhee polynomials of the Appell type, is the suggested approximation technique to achieve this goal. A rough formula for the Caputo fractional derivative is first derived, and it is used to build the numerical strategy for the suggested model&rsquo;s solution. This procedure creates a system of algebraic equations from the model that was provided. We validate the effectiveness and precision of the provided approach by evaluating the residual error function (REF). We compare the results obtained with the fourth-order Runge&ndash;Kutta technique and other existing published work. The outcomes demonstrate that the technique used is a simple and effective tool for simulating such models.

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Authors: Ravi P. Agarwal Snezhana Hristova

A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050395

Authors: Khalid Hattaf

The present paper introduces a new class of generalized differential and integral operators. This class includes and generalizes a large number of definitions of fractal-fractional derivatives and integral operators used to model the complex dynamics of many natural and physical phenomena found in diverse fields of science and engineering. Some properties of the newly introduced class are rigorously established. As applications of this new class, two illustrative examples are presented, one for a simple problem and the other for a nonlinear problem modeling the dynamical behavior of a chaotic system.

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Authors: Tingsheng Huang Xinjian Wang Da Xie Chunyang Wang Xuelian Liu

Depth image enhancement techniques can help to improve image quality and facilitate computer vision tasks. Traditional image-enhancement methods, which are typically based on integer-order calculus, cannot exploit the textural information of an image, and their enhancement effect is limited. To solve this problem, fractional differentiation has been introduced as an innovative image-processing tool. It enables the flexible use of local and non-local information by taking into account the continuous changes between orders, thereby improving the enhancement effect. In this study, a fractional differential is applied in depth image enhancement and used to establish a novel algorithm, named the fractional differential-inverse-distance-weighted depth image enhancement method. Experiments are performed to verify the effectiveness and universality of the algorithm, revealing that it can effectively solve edge and hole interference and significantly enhance textural details. The effects of the order of fractional differentiation and number of iterations on the enhancement performance are examined, and the optimal parameters are obtained. The process data of depth image enhancement associated with the optimal number of iterations and fractional order are expected to facilitate depth image enhancement in actual scenarios.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050393

Authors: Qisong Sun Minggang Zhu Jiaming Bai Qiang Wang

Remanence is an important parameter of magnetic property for Nd-Fe-B magnets, and high remanent magnetization is a prerequisite for high-performance magnets. In this paper, the surface morphology perpendicular to the texture orientation direction and parallel to the texture orientation direction is analyzed by Nd-Fe-B permanent magnets with different compositions. For the first time, the relationship between the remanence of a magnet and the degree of texture orientation is explained in depth using the fractal dimension. The fractal dimension of surface morphology combined with the remanence equation yields the degree of texture orientation of the magnet, which is in agreement with the trend of the squareness factor of the demagnetization curves. Among the three samples, the Nd-Fe-B sample has the highest degree of texture orientation, the Pr-Nd-Fe-B sample has the lowest degree of texture orientation, and the Nd-Ce-Fe-B sample is in between the first two. The multiples of uniform (pole) density obtained by EBSD further prove the correctness of the degree of texture orientation calculated by the fractal dimension. The combination of EBSD morphology and fractal dimension to obtain novel insights into the correlation between remanence and the degree of texture orientation will contribute to the development of high-performance Nd-Fe-B with high remanence.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050392

Authors: Zareen A. Khan Sajjad Ahmad Salman Zeb Hussam Alrabaiah

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration and differentiation are developed. With the aid of these operational matrices, the considered problems are converted to algebraic-type equations, which can be easily solved using computational software. Various examples are solved by applying the method described above, and their graphical presentation and accuracy performance are provided.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050391

Authors: Masoud Alilou Hatef Azami Arman Oshnoei Behnam Mohammadi-Ivatloo Remus Teodorescu

The worldwide energy revolution has accelerated the utilization of demand-side manageable energy systems such as wind turbines, photovoltaic panels, electric vehicles, and energy storage systems in order to deal with the growing energy crisis and greenhouse emissions. The control system of renewable energy units and energy storage systems has a high effect on their performance and absolutely on the efficiency of the total power network. Classical controllers are based on integer-order differentiation and integration, while the fractional-order controller has tremendous potential to change the order for better modeling and controlling the system. This paper presents a comprehensive review of the energy system of renewable energy units and energy storage devices. Various papers are evaluated, and their methods and results are presented. Moreover, the mathematical fundamentals of the fractional-order method are mentioned, and the various studies are categorized based on different parameters. Various definitions for fractional-order calculus are also explained using their mathematical formula. Different studies and numerical evaluations present appropriate efficiency and accuracy of the fractional-order techniques for estimating, controlling, and improving the performance of energy systems in various operational conditions so that the average error of the fractional-order methods is considerably lower than other ones.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050390

Authors: Badr Saad T. Alkahtani Ibtehal Alazman Shahid Ahmad Wani

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro&ndash;differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also discovered.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050389

Authors: Samuel Megameno Nuugulu Frednard Gideon Kailash C. Patidar

After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2+k2).

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Authors: Evgeniya Gospodinova Penio Lebamovski Galya Georgieva-Tsaneva Mariya Negreva

The dynamics of cardiac signals can be studied using methods for nonlinear analysis of heart rate variability (HRV). The methods that are used in the article to investigate the fractal, multifractal and informational characteristics of the intervals between heartbeats (RR time intervals) are: Rescaled Range, Detrended Fluctuation Analysis, Multifractal Detrended Fluctuation Analysis, Poincar&eacute; plot, Approximate Entropy and Sample Entropy. Two groups of people were studied: 25 healthy subjects (15 men, 10 women, mean age: 56.3 years) and 25 patients with arrhythmia (13 men, 12 women, mean age: 58.7 years). The results of the application of the methods for nonlinear analysis of HRV in the two groups of people studied are shown as mean &plusmn; std. The effectiveness of the methods was evaluated by t-test and the parameter Area Under the Curve (AUC) from the Receiver Operator Curve (ROC) characteristics. The studied 11 parameters have statistical significance (p &lt; 0.05); therefore, they can be used to distinguish between healthy and unhealthy subjects. It was established by applying the ROC analysis that the parameters Hq=2(MFDFA), F(&alpha;)(MFDFA) and SD2(Poincar&eacute; plot) have a good diagnostic value; H(R/S), &alpha;1(DFA), SD1/SD2(Poincar&eacute; plot), ApEn and SampEn have a very good score; &alpha;2(DFA), &alpha;all(DFA) and SD1(Poincar&eacute; plot) have an excellent diagnostic score. In conclusion, the methods used for nonlinear analysis of HRV have been evaluated as effective, and with their help, new perspectives are opened in the diagnosis of cardiovascular diseases.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050387

Authors: Antonio Iodice Gerardo Di Martino Alessio Di Simone Daniele Riccio Giuseppe Ruello

Marine and terrestrial natural surfaces exhibit statistical scale invariance properties that are well modelled by fractional Brownian motion (fBm), two-dimensional random processes. Accordingly, for microwave remote sensing applications it is useful to evaluate the normalized radar cross section (NRCS) of fBm surfaces. This task has been accomplished in the past by using either the Kirchhoff approximation (KA) or the small perturbation method (SPM). However, KA and SPM have rather limited ranges of application in terms of surface roughness and incidence angle: a wider range of application is achieved by the small slope approximation (SSA), more recently developed, but the latter has not been applied yet to fBm surfaces. In this paper, the first-order SSA (SSA-1) is applied to the evaluation of scattering from fBm surfaces obtaining an analytical formulation of their NRCS. It is then shown that the obtained SSA-1 expression reduces to the KA and SPM ones at near-specular and far-from-specular scattering directions, respectively. Finally, the results of the proposed method are compared to experimental data available in the literature.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050386

Authors: Mustafa Ç. Korkmaz Víctor Leiva Carlos Martin-Barreiro

The continuous Bernoulli distribution is defined on the unit interval and has a unique property related to fractiles. A fractile is a position on a probability density function where the corresponding surface is a fixed proportion. This article presents the derivation of properties of the continuous Bernoulli distribution and formulates a fractile or quantile regression model for a unit response using the exponentiated continuous Bernoulli distribution. Monte Carlo simulation studies evaluate the performance of point and interval estimators for both the continuous Bernoulli distribution and the fractile regression model. Real-world datasets from science and education are analyzed to illustrate the modeling abilities of the continuous Bernoulli distribution and the exponentiated continuous Bernoulli quantile regression model.

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Authors: Vladimir E. Fedorov Marko Kostić Tatyana A. Zakharova

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers A&gamma; for an operator A such that &minus;A generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of A&gamma; for some &gamma;&isin;(0,1) of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov&ndash;Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.

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Authors: Özge Dinç Göğüş Elif Avşar Kayhan Develi Ayten Çalık

Since each rock type represents different deformation characteristics, prediction of the damage beforehand is one of the most fundamental problems of industrial activities and rock engineering studies. Previous studies have predicted the stress&ndash;strain behaviors preceding rock failure; however, quantitative analyses of the progressive damage in different rocks under stress have not been accurately presented. This study aims to quantify pre-failure rock damage by investigating the stress-induced microscale cracking process in three different rock types, including diabase, ignimbrite, and marble, representing strong, medium-hard, and weak rock types, respectively. We demonstrate crack intensity at critical stress levels where cracking initiates (&sigma;ci), propagates (&sigma;cd), and where failure occurs (&sigma;peak) based on scanning electron microscope (SEM) images. Furthermore, the progression of rock damage was quantified for each rock type through the fractal analyses of crack patterns on these images. Our results show that the patterns in diabase have the highest fractal dimensions (DB) for all three stress levels. While marble produces the lowest DB value up to &sigma;ci stress level, it presents greater DB values than those of ignimbrite, starting from the &sigma;cd level. This is because rock damage in ignimbrite is controlled by the groundmass, proceeding from such stress level. Rock texture controls the rock stiffness and, hence, the DB values of cracking. The mineral composition is effective on the rock strength, but the textural pattern of the minerals has a first-order control on the rock deformation behavior. Overall, our results provide a better understanding of progressive damage in different rock types, which is crucial in the design of engineering structures.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050384

Authors: Trayan Stamov Gani Stamov Ivanka Stamova

The paper is oriented on the existence of almost periodic solutions of factional-order impulsive delayed reaction-diffusion gene regulatory networks. Caputo type fractional-order derivatives and impulsive disturbances at not fixed instants of time are considered. New almost periodic and perfect Mittag&ndash;Leffler stability criteria are proposed. Lyapunov&rsquo;s like impulsive functions, the properties of the fractional derivatives and comparison principle are the main tools in the investigation. Illustrative examples are also presented to demonstrate the proposed criteria. Our results contribute to the development of qualitative the theory of fractional-order gene regulatory networks.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050382

Authors: Mengchen Zhang Ming Shen Hui Chen

This paper investigates a two-dimensional Riemann&ndash;Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of O(&sigma;2+&rho;2+&tau;+hx+hy). After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050381

Authors: Raheel Kamal Kamran Saleh M. Alzahrani Talal Alzahrani

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0&lt;&alpha;&lt;1, and 1&lt;&beta;&lt;2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050380

Authors: Bo Yu Xiang Li Ning Dong

The implicit difference approach is used to discretize a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and of nonsingular M-matrix structure, respectively. A low-ranked doubling Smith method with determined optimally iterative parameters is presented for solving the corresponding matrix equation. In comparison to the existing Krylov solver with Fast Fourier Transform (FFT) for the sequence Toeplitz linear system, numerical examples demonstrate that the proposed method is more effective on CPU time for solving large-scale problems.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050379

Authors: Vladimir Kulish Navid Aslfattahi Michal Schmirler

Based on the derivation of the equation of state for systems with a fractional power spectrum, the relationship between the van der Waals constant and the fractional derivative order has been established. The fractional model of electron&ndash;phonon interaction has received additional consideration, which may be pertinent when interpreting the experimental results. This model is valuable for describing superconductivity at high temperatures because it predicts relatively large values for the electron&ndash;phonon interaction constant.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050377

Authors: M. Shehata M. Shokry R. A. Abd-Elmonem I. L. El-Kalla

In this article, we solve the second type of nonlinear Volterra picture fuzzy integral equation (NVPFIE) using an accelerated form of the Adomian decomposition method (ADM). Based on (&alpha;,&delta;,&beta;)-cut, we convert the NVPFIE to the nonlinear Volterra integral equations in a crisp form. An accelerated version of the ADM is used to solve this transformed system, which is based on a new formula for the Adomian polynomial. The sufficient condition that guarantees a unique solution is obtained using this new Adomian polynomial, error estimates are given, and the convergence of the series solution is proven. Numerical cases are discussed to illustrate the effectiveness of this approach.

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Authors: Haleemah Ghazwani Muhammad Faisal Nadeem Faiza Ishfaq Ali N. A. Koam

Shannon entropy, also known as information entropy or entropy, measures the uncertainty or randomness of probability distribution. Entropy is measured in bits, quantifying the average amount of information required to identify an event from the distribution. Shannon&rsquo;s entropy theory initiates graph entropies and develops information-theoretic magnitudes for structural computational evidence of organic graphs and complex networks. Graph entropy measurements are valuable in several scientific fields, such as computing, chemistry, biology, and discrete mathematics. In this study, we investigate the entropy of fractal-type networks by considering cycle, complete, and star networks as base graphs using degree-based topological indices.

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Authors: Khalil Ullah Muhammad Arif Ibtisam Mohammed Aldawish Sheza M. El-Deeb

Determining the sharp bounds for coefficient-related problems that appear in the Taylor&ndash;Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of problems, such as the first three initial coefficient problems, the Zalcman inequalities, the Fekete&ndash;Szeg&ouml; type results, and the second-order Hankel determinant for families of Sakaguchi-type functions related to the cardioid-shaped domain. Further, we study the logarithmic coefficients for both of these classes.

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Authors: Vasile Preda Răzvan-Cornel Sfetcu

We introduce fractal Tsallis entropy and show that it satisfies Shannon&ndash;Khinchin axioms. Analogously to Tsallis divergence (or Tsallis relative entropy, according to some authors), fractal Tsallis divergence is defined and some properties of it are studied. Within this framework, Lesche stability is verified and an example concerning the microcanonical ensemble is given. We generalize the LMC complexity measure (LMC is Lopez-Ruiz, Mancini and Calbert), apply it to a two-level system and define the statistical complexity by using the Euclidean and Wootters&rsquo; distance measures in order to analyze it for two-level systems.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050374

Authors: Zhongshu Wu Xinxia Zhang Jihan Wang Xiaoyan Zeng

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi&ndash;Gauss&ndash;Lobatto or Jacobi&ndash;Gauss&ndash;Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, the fractional differential equations were transformed into linear systems, which are easier to solve. Two types of fractional differential equations were used for the numerical simulations, and the numerical results demonstrated the fast convergence and high accuracy of the proposed methods.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050373

Authors: Timur Gamilov Ruslan Yanbarisov

Recent studies have demonstrated the benefits of using fractional derivatives to simulate a blood pressure profile. In this work we propose to combine a one-dimensional model of coronary blood flow with fractional-order Windkessel boundary conditions. This allows us to obtain a greater variety of blood pressure profiles for better model personalization An algorithm of parameter identification is described, which is used to fit the measured mean value of arterial pressure and estimate the fractional flow reserve (FFR) for a given patient. The proposed framework is used to investigate sensitivity of mean blood pressure and fractional flow reserve to fractional order. We demonstrate that the fractional derivative order significantly affects the fractional flow reserve (FFR), which is used as an indicator of stenosis significance.

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Authors: Munirah A. Almulhim Muneerah Al Nuwairan

The current work is devoted to studying the dynamical behavior of the Sakovich equation with beta derivatives. We announce the conditions of problem parameters leading to the existence of periodic, solitary, and kink solutions by applying the qualitative theory of planar dynamical systems. Based on these conditions, we construct some new solutions by integrating the conserved quantity along the possible interval of real wave propagation in order to obtain real solutions that are significant and desirable in real-world applications. We illustrate the dependence of the solutions on the initial conditions by examining the phase plane orbit. We graphically show the fractional order beta effects on the width of the solutions and keep their amplitude approximately unchanged. The graphical representations of some 3D and 2D solutions are introduced.

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Authors: Mustafa Zeki Ramazan Tinaztepe Salih Tatar Suleyman Ulusoy Rami Al-Hajj

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent.

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Authors: Emmanuel Addai Adejimi Adeniji Olumuyiwa J. Peter Janet O. Agbaje Kayode Oshinubi

The rising tide of smoking-related diseases has irreparably damaged the health of both young and old people, according to the World Health Organization. This study explores the dynamics of the age-structure smoking model under fractal-fractional (F-F) derivatives with government intervention coverage. We present a new fractal-fractional model for two-age structure smokers in the Caputo&ndash;Fabrizio framework to emphasize the potential of this operator. For the existence-uniqueness criterion of the given model, successive iterative sequences are defined with limit points that are the solutions of our proposed age-structure smoking model. We also use the functional technique to demonstrate the proposed model stability under the Ulam&ndash;Hyers condition. The two age-structure smoking models are numerically characterized using the Newton polynomial. We observe that in Groups 1 and 2, a change in the fractal-fractional orders has a direct effect on the dynamics of the smoking epidemic. Moreover, testing the inherent effectiveness of government interventions shows a considerable impact on potential, occasional, and temporary smokers when the fractal-fractional order is 0.95. It is the view that this study will contribute to the applicability of the schemes, the rich dynamics of the fractal, and the fractional perspective of future predictions.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050368

Authors: Esteban González Genly Leon Guillermo Fernandez-Anaya

This paper investigates exact solutions of cosmological interest in fractional cosmology. Given &mu;, the order of Caputo&rsquo;s fractional derivative, and w, the matter equation of state, we present specific exact power-law solutions. We discuss the exact general solution of the Riccati Equation, where the solution for the scale factor is a combination of power laws. Using cosmological data, we estimate the free parameters. An analysis of type Ia supernovae (SNe Ia) data and the observational Hubble parameter data (OHD), also known as cosmic chronometers, and a joint analysis with data from SNe Ia + OHD leads to best-fit values for the free parameters calculated at 1&sigma;, 2&sigma; and 3&sigma; confidence levels (CLs). On the other hand, these best-fit values are used to calculate the age of the Universe, the current deceleration parameter (both at 3&sigma; CL) and the current matter density parameter at 1&sigma; CL. Finding a Universe roughly twice as old as the one of &Lambda;CDM is a distinction of fractional cosmology. Focusing our analysis on these results, we can conclude that the region in which &mu;&gt;2 is not ruled out by observations. This parameter region is relevant because fractional cosmology gives a power-law solution without matter, which is accelerated for &mu;&gt;2. We present a fractional origin model that leads to an accelerated state without appealing to &Lambda; or dark energy.

]]>Fractal and Fractional doi: 10.3390/fractalfract7050369

Authors: Lei Wang Shengwen Tang

Fractals is a new branch of nonlinear science that was established in the 1970s, focusing on irregularities, haphazard phenomena and self-similarities in nature [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract7050367

Authors: Banan Al-Homidan Nasser-eddine Tatar

In this paper, we consider a fractional equation of order between one and two which may be looked at as an interpolation between the heat and wave equations. The problem is non-linear as it involves a power-type non-linearity. We investigate the possibilities of stabilizing the system by a lower-order fractional term and/or a memory term involving the Laplacian. We prove a global Mittag&ndash;Leffler stability result in case a fractional frictional damping is active and a local Mittag&ndash;Leffler stability result when the material is viscoelastic in case of small relaxation functions. Unlike the integer-order problems, additional serious difficulties arise in the present case. These difficulties are highlighted clearly in the introduction. They are mainly due to the memory dependence of the fractional derivatives which is the cause of the invalidity of the product rule in particular. We utilize several properties in fractional calculus. Moreover, we introduce new Lyapunov-type functionals in the context of the multiplier technique.

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