Fractal and Fractional doi: 10.3390/fractalfract6100562

Authors: Poongjin Cho Kyungwon Kim

The efficient market hypothesis (EMH) assumes that all available information in an efficient financial market is ideally fully reflected in the price of an asset. However, whether the reality that asset prices are not informational efficient is an opportunity for profit or a systemic risk of the financial system that needs to be corrected is still a ubiquitous concept, so many economic participants and research scholars have conducted related studies in order to understand the phenomenon of the financial market. This research employed attention entropy of the log-returns of 27 global assets to analyze the time-varying informational efficiency. International markets could be classified hierarchically into groups with similar long-term efficiency trends; however, at the same time, the ranks and clusters were found to remain stable only for a short period of time in terms of short-term efficiency. Therefore, a complex network representation analysis was performed to express whether the short-term efficiency patterns have interacted with each other over time as a coherent picture. It was confirmed that the network of 27 international markets was fully connected, strongly globalized and entangled. In addition, the complex network was composed of two modular structures grouped together with similar efficiency dynamics. As a result, although the informational efficiency of financial markets may be globalized to a high-efficiency state, it shows a collective dynamics pattern in which the global system may fall into risk due to the spread of systemic risk.

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Authors: Xu Tang Ying Luo Bin Han

In this paper, a fractional-order model of the gas film is proposed for the dynamic characteristics of an air bearing. Based on the dynamic characteristics common between gas film and viscoelastic body, the idea of the fractional-order equivalent modeling of the dynamic characteristics of the gas film is presented to improve the modeling accuracy. Four fractional-order gas film (FOGF) models are introduced based on generalization of traditional viscoelastic models. The analysis of the characteristics of the FOGF models shows that the FOGF model can capture more complex dynamic characteristics and fit the real dynamic data of the gas film better than traditional models. A genetic algorithm particle swarm optimization (GA-PSO) method is used for parameter identification of the proposed models. The experimental results tested on the air bearing motion platform show that the FOGF models are superior in accuracy to the traditional equivalent models for the gas film. In particular, the fractional-order Maxwell gas film (FOMGF) model has the best capture accuracy compared to the other FOGF models and traditional models.

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Authors: Hongwen Hu Chunna Zhao Jing Li Yaqun Huang

As one of the main areas of value investing, the stock market attracts the attention of many investors. Among investors, market index movements are a focus of attention. In this paper, combining the efficient market hypothesis and the fractal market hypothesis, a stock prediction model based on mixed fractional Brownian motion (MFBM) and an improved fractional-order particle swarm optimization algorithm is proposed. First, the MFBM model is constructed by adjusting the parameters to mix geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM). After that, an improved fractional-order particle swarm optimization algorithm is proposed. The position and velocity formulas of the fractional-order particle swarm optimization algorithm are improved using new fractional-order update formulas. The inertia weight in the update formula is set to be linearly decreasing. The improved fractional-order particle swarm optimization algorithm is used to optimize the coefficients of the MFBM model. Through experiments, the accuracy and validity of the prediction model are proven by combining the error analysis. The model with the improved fractional-order particle swarm optimization algorithm and MFBM is superior to GBM, GFBM, and MFBM models in stock price prediction.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100559

Authors: Hassan M. Serag Abd-Allah Hyder Mahmoud El-Badawy Areej A. Almoneef

This paper discusses the optimal control issue for elliptic k&times;k cooperative fractional systems. The fractional operators are proposed in the Laplace sense. Because of the nonlocality of the Laplace fractional operators, we reformulate the issue as an extended issue on a semi-infinite cylinder in Rk+1. The weak solution for these fractional systems is then proven to exist and be unique. Moreover, the existence and optimality conditions can be inferred as a consequence.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100557

Authors: Jing Yang Xiaorong Hou Yajun Li

Based on the generalized Routh&ndash;Hurwitz criterion, we propose a sufficient and necessary criterion for testing the stability of fractional-order linear systems with order &alpha;&isin;1,2, called the fractional-order Routh&ndash;Hurwitz criterion. Compared with the existing criterion, ours involves fewer and simpler expressions, which is significant for analyzing the robust stability of high-dimensional uncertain systems. All these expressions are explicit ones about the coefficients of the characteristic polynomial of the system matrix, so the stable parameter region of fractional-order systems can be described directly. Some examples show the effectiveness of our method.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100558

Authors: Xiaoyu Xia Yinmeng Chen Litan Yan

In this paper, we study a class of time-fractal-fractional stochastic differential equations with the fractal&ndash;fractional differential operator of Atangana under the meaning of Caputo and with a kernel of the power law type. We first establish the Ho&uml;lder continuity of the solution of the equation. Then, under certain averaging conditions, we show that the solutions of original equations can be approximated by the solutions of the associated averaged equations in the sense of the mean square convergence. As an application, we provide an example with numerical simulations to explore the established averaging principle.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100556

Authors: Aqeel Shahzad Abdullah Shoaib Nabil Mlaiki Suhad Subhi Aiadi

The purpose of this paper is to develop some fuzzy fixed point results for the sequence of locally fuzzy mappings satisfying rational type almost contractions in complete dislocated metric spaces. We apply our results to obtain new results for set-valued and single-valued mappings. We also study the stability of fuzzy fixed point &gamma;-level sets. An example is presented in favor of these results.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100555

Authors: Bilgi Görkem Yazgaç Mürvet Kırcı

In this paper, we propose two fractional-order calculus-based data augmentation methods for audio signals. The first approach is based on fractional differentiation of the Mel scale. By using a randomly selected fractional derivation order, we are warping the Mel scale, therefore, we aim to augment Mel-scale-based time-frequency representations of audio data. The second approach is based on previous fractional-order image edge enhancement methods. Since multiple deep learning approaches treat Mel spectrogram representations like images, a fractional-order differential-based mask is employed. The mask parameters are produced with respect to randomly selected fractional-order derivative parameters. The proposed data augmentation methods are applied to the UrbanSound8k environmental sound dataset. For the classification of the dataset and testing the methods, an arbitrary convolutional neural network is implemented. Our results show that fractional-order calculus-based methods can be employed as data augmentation methods. Increasing the dataset size to six times the original size, the classification accuracy result increased by around 8.5%. Additional tests on more complex networks also produced better accuracy results compared to a non-augmented dataset. To our knowledge, this paper is the first example of employing fractional-order calculus as an audio data augmentation tool.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100554

Authors: Changlong Yu Si Wang Jufang Wang Jing Li

Due to the great application potential of fractional q-difference system in physics, mechanics and aerodynamics, it is very necessary to study fractional q-difference system. The main purpose of this paper is to investigate the solvability of nonlinear fractional q-integro-difference system with the nonlocal boundary conditions involving diverse fractional q-derivatives and Riemann-Stieltjes q-integrals. We acquire the existence results of solutions for the systems by applying Schauder fixed point theorem, Krasnoselskii&rsquo;s fixed point theorem, Schaefer&rsquo;s fixed point theorem and nonlinear alternative for single-valued maps, and a uniqueness result is obtained through the Banach contraction mapping principle. Finally, we give some examples to illustrate the main results.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100553

Authors: Basma Souayeh Zulqurnain Sabir Najib Hdhiri Wael Al-Kouz Mir Waqas Alam Tarfa Alsheddi

This motive of current research is to provide a stochastic platform based on the artificial neural networks (ANNs) along with the Bayesian regularization approach for the fractional food chain supply system (FFSCS) with Allee effects. The investigations based on the fractional derivatives are applied to achieve the accurate and precise results of FFSCS. The dynamical FFSCS is divided into special predator category P(&eta;), top-predator class Q(&eta;), and prey population dynamics R(&eta;). The computing numerical performances for three different variations of the dynamical FFSCS are provided by using the ANNs along with the Bayesian regularization approach. The data selection for the dynamical FFSCS is selected for train as 78% and 11% for both test and endorsement. The accuracy of the proposed ANNs along with the Bayesian regularization method is approved using the comparison performances. For the rationality, ability, reliability, and exactness are authenticated by using the ANNs procedure enhanced by the Bayesian regularization method through the regression measures, correlation values, error histograms, and transition of state performances.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100552

Authors: Didier Samayoa Andriy Kryvko Gelasio Velázquez Helvio Mollinedo

A new approach for solving the fractal Euler-Bernoulli beam equation is proposed. The mapping of fractal problems in non-differentiable fractals into the corresponding problems for the fractal continuum applying the fractal continuum calculus (FdH3-CC) is carried out. The fractal Euler-Bernoulli beam equation is derived as a generalization using FdH3-CC under analogous assumptions as in the ordinary calculus and then it is solved analytically. To validate the spatial distribution of self-similar beam response, three different classical beams with several fractal parameters are analysed. Some mechanical implications are discussed.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100551

Authors: Yeonju Kim Duc Anh Pham Ratanak Phon Sungjoon Lim

This paper proposes a millimeter-wave lens antenna using 3-dimensional (3D) printing technology to reduce weight and provide stable gain performance. The antenna consists of a four-layer cylindrical gradient-index (GRIN) lens fed by a wideband Yagi antenna. We designed a fractal cell geometry to achieve the desired effective permittivity for a GRIN lens. Among different candidates, the honeycomb structure is chosen to provide high mechanical strength with light weight, low dielectric loss, and lens dispersion for a lens antenna. Therefore, the measured peak gain was relatively flat at 16.86 &plusmn; 0.5 dBi within 25&minus;31.5 GHz, corresponding to 1 dB gain bandwidth = 23%. The proposed 3D-printed GRIN lens is cost-effective, with rapid and easy manufacturing.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100550

Authors: Nezha Maamri Jean-Claude Trigeassou

The usual approach to the integration of fractional order initial value problems is based on the Caputo derivative, whose initial conditions are used to formulate the classical integral equation. Thanks to an elementary counter example, we demonstrate that this technique leads to wrong free-response transients. The solution of this fundamental problem is to use the frequency-distributed model of the fractional integrator and its distributed initial conditions. Using this model, we solve the previous counter example and propose a methodology which is the generalization of the integer order approach. Finally, this technique is applied to the modeling of Fractional Differential Systems (FDS) and the formulation of their transients in the linear case. Two expressions are derived, one using the Mittag&ndash;Leffler function and a new one based on the definition of a distributed exponential function.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100549

Authors: Jairo Viola YangQuan Chen

This paper presents a design and evaluation of a fractional-order self optimizing control (FOSOC) architecture for process control. It is based on a real-time derivative-free optimization layer that adjusts the parameters of a discrete-time fractional-order proportional integral (FOPI) controller according to an economic cost function. A simulation benchmark is designed to assess the performance of the FOSOC controller based on a first order plus dead time system. Similarly, an acceleration mechanism is proposed for the fractional-order self optimizing control framework employing fractional-order Gaussian noise with long-range dependence given by the Hurst exponent. The obtained results show that the FOSOC controller can improve the system closed-loop response under different operating conditions and reduce the convergence time of the real-time derivative-free optimization algorithm by using fractional-order stochasticity.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100548

Authors: Amira Hassan Mokhtar Aly Ahmed Elmelegi Loai Nasrat Masayuki Watanabe Emad A. Mohamed

Modern structures of electrical power systems are expected to have more domination of renewable energy sources. However, renewable energy-based generation systems suffer from their lack of or reduced rotating masses, which is the main source of power system inertia. Therefore, the frequency of modern power systems represents an important indicator of their proper and safe operation. In addition, the uncertainties and randomness of the renewable energy sources and the load variations can result in frequency undulation problems. In this context, this paper presents an improved cascaded fractional order-based frequency regulation controller for a two-area interconnected power system. The proposed controller uses the cascade structure of the tilt integral derivative (TID) with the fractional order proportional integral derivative with a filter (FOPIDN or PI&lambda;D&mu;N) controller (namely the cascaded TID-FOPIDN or TID-PI&lambda;D&mu;N controller). Moreover, an optimized TID control method is presented for the electric vehicles (EVs) to maximize their benefits and contribution to the frequency regulation of power systems. The recent widely employed marine predators optimization algorithm (MPA) is utilized to design the new proposed controllers. The proposed controller and design method are tested and validated at various load and renewable source variations, as is their robustness against parameter uncertainties of power systems. Performance comparisons of the proposed controller with featured frequency regulation controllers in the literature are provided to verify the superiority of the new proposed controller. The obtained results confirm the stable operation and the frequency regulation performance of the new proposed controller with optimized controller parameters and without the need for complex design methods.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100547

Authors: Chengwei Dong

To further understand the dynamical characteristics of chaotic systems with a hidden attractor and coexisting attractors, we investigated the fundamental dynamics of a novel three-dimensional (3D) chaotic system derived by adding a simple constant term to the Yang&ndash;Chen system, which includes the bifurcation diagram, Lyapunov exponents spectrum, and basin of attraction, under different parameters. In addition, an offset boosting control method is presented to the state variable, and a numerical simulation of the system is also presented. Furthermore, the unstable cycles embedded in the hidden chaotic attractors are extracted in detail, which shows the effectiveness of the variational method and 1D symbolic dynamics. Finally, the adaptive synchronization of the novel system is successfully designed, and a circuit simulation is implemented to illustrate the flexibility and validity of the numerical results. Theoretical analysis and simulation results indicate that the new system has complex dynamical properties and can be used to facilitate engineering applications.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100546

Authors: Yabin Shao Gauhar Rahman Yasser Elmasry Muhammad Samraiz Artion Kashuri Kamsing Nonlaopon

In the recent era of research, the field of integral inequalities has earned more recognition due to its wide applications in diverse domains. The researchers have widely studied the integral inequalities by utilizing different approaches. In this present article, we aim to develop a variety of certain new inequalities using the generalized fractional integral in the sense of multivariate Mittag-Leffler (M-L) functions, including Gr&uuml;ss-type and some other related inequalities. Also, we use the relationship between the Riemann-Liouville integral, the Prabhakar integral, and the generalized fractional integral to deduce specific findings. Moreover, we support our findings by presenting examples and corollaries.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100545

Authors: Najla M. Alarifi Rabha W. Ibrahim

A special function is a function that is typically entitled after an early scientist who studied its features and has a specific application in mathematical physics or another area of mathematics. There are a few significant examples, including the hypergeometric function and its unique species. These types of special functions are generalized by fractional calculus, fractal, q-calculus, (q,p)-calculus and k-calculus. By engaging the notion of q-fractional calculus (QFC), we investigate the geometric properties of the generalized Prabhakar fractional differential operator in the open unit disk &nabla;:={&xi;&isin;C:|&xi;|&lt;1}. Consequently, we insert the generalized operator in a special class of analytic functions. Our methodology is indicated by the usage of differential subordination and superordination theory. Accordingly, numerous fractional differential inequalities are organized. Additionally, as an application, we study the solution of special kinds of q&ndash;fractional differential equation.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100543

Authors: Yuan Ma Dehong Ji

A general system of fractional differential equations with coupled fractional Stieltjes integrals and a Riemann&ndash;Liouville fractional integral in boundary conditions is studied in the context of pattern formation. We need to transform the fractional differential system into the corresponding integral operator to obtain the existence and uniqueness of solutions for the system. The contraction mapping principle in Banach space and the alternative theorem of Leray&ndash;Schauder are applied. Finally, we give two applications to illustrate our theoretical results.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100544

Authors: Yuxing Li Bingzhao Tang Bo Geng Shangbin Jiao

Fuzzy dispersion entropy (FuzzDE) is a very recently proposed non-linear dynamical indicator, which combines the advantages of both dispersion entropy (DE) and fuzzy entropy (FuzzEn) to detect dynamic changes in a time series. However, FuzzDE only reflects the information of the original signal and is not very sensitive to dynamic changes. To address these drawbacks, we introduce fractional order calculation on the basis of FuzzDE, propose FuzzDE&alpha;, and use it as a feature for the signal analysis and fault diagnosis of bearings. In addition, we also introduce other fractional order entropies, including fractional order DE (DE&alpha;), fractional order permutation entropy (PE&alpha;) and fractional order fluctuation-based DE (FDE&alpha;), and propose a mixed features extraction diagnosis method. Both simulated as well as real-world experimental results demonstrate that the FuzzDE&alpha; at different fractional orders is more sensitive to changes in the dynamics of the time series, and the proposed mixed features bearing fault diagnosis method achieves 100% recognition rate at just triple features, among which, the mixed feature combinations with the highest recognition rates all have FuzzDE&alpha;, and FuzzDE&alpha; also appears most frequently.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100542

Authors: Feng Feng Kexin Zhang Xinghui Li Yousheng Xia Meng Yuan Pingfa Feng

Fractal dimension (D) is widely utilized in various fields to quantify the complexity of signals and other features. However, the fractal nature is limited to a certain scope of concerned scales, i.e., scaling region, even for a theoretically fractal profile generated through the Weierstrass-Mandelbrot (W-M) function. In this study, the scaling characteristics curves of profiles were calculated by using the roughness scaling extraction (RSE) algorithm, and an interception method was proposed to locate the two ends of the scaling region, which were named corner and drop phenomena, respectively. The results indicated that two factors, sampling length and flattening order, in the RSE algorithm could influence the scaling region length significantly. Based on the scaling region interception method and the above findings, the RSE algorithm was optimized to improve the accuracy of the D calculation, and the influence of sampling length was discussed by comparing the lower critical condition of the W-M function. To improve the ideality of fractal curves generated through the W-M function, the strategy of reducing the fundamental frequency was proposed to enlarge the scaling region. Moreover, the strategy of opposite operation was also proposed to improve the consistency of generated curves with actual signals, which could be conducive to practical simulations.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100541

Authors: Vladimir E. Fedorov Marina V. Plekhanova Elizaveta M. Izhberdeeva

In this paper, a criterion for generating an analytic family of operators, which resolves a linear equation solved with respect to the Dzhrbashyan&ndash;Nersesyan fractional derivative, via a linear closed operator is obtained. The properties of the resolving families are investigated and applied to prove the existence of a unique solution for the corresponding initial value problem of the inhomogeneous equation with the Dzhrbashyan&ndash;Nersesyan fractional derivative. A solution is presented explicitly using resolving families of operators. A theorem on perturbations of operators from the found class of generators of resolving families is proved. The obtained results are used for a study of an initial-boundary value problem to a model of the viscoelastic Oldroyd fluid dynamics. Thus, the Dzhrbashyan&ndash;Nersesyan initial value problem is investigated in the essentially infinite-dimensional case. The use of the proved abstract results to study initial-boundary value problems for a system of partial differential equations is demonstrated.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100540

Authors: Asena Çetinkaya Luminita-Ioana Cotîrlă

We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the Briot&ndash;Bouquet differential subordinations for the linear operator involving the normalized form of the generalized Struve function. We also obtain univalent solutions to the Briot&ndash;Bouquet differential equations and observe that these solutions are the best possible solutions to the Briot&ndash;Bouquet differential subordinations for the exponential starlike function class. Moreover, we give an application of fractional integral calculus for a complex-valued function associated with the generalized Struve function. The significance of this paper is due to the technique employed in proving the results and novelty of these results for the Struve functions. The approach used in this paper can lead to several new problems in geometric function theory associated with special functions.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100539

Authors: Changxu Shao Hao Guo Songhe Meng Yingfeng Shao Shanxiang Wang Shangjian Xie Fei Qi

Ceramics are commonly used as high-temperature structural materials which are easy to fracture because of the propagation of thermal shock cracks. Characterizing and controlling crack propagation are significant for the improvement of the thermal shock resistance of ceramics. However, observing crack morphology, based on macro and SEM images, costs much time and potentially includes subjective factors. In addition, complex cracks cannot be counted and will be simplified or omitted. Fractals are suitable to describe complex and inhomogeneous structures, and the multifractal spectrum describes this complexity and heterogeneity in more detail. This paper proposes a crack characterization method based on the multifractal spectrum. After thermal shocks, the multifractal spectrum of alumina ceramics was obtained, and the crack fractal features were extracted. Then, a deep learning method was employed to extract features and automatically classify ceramic crack materials with different strengths, with a recognition accuracy of 87.5%.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100538

Authors: Yijun Zhu Huilin Shang

The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method is applied to derive the critical conditions for global bifurcations. By engaging in the fractal erosion of safe basin to depict the phenomenon pull-in instability intuitively, the point-mapping approach is used to present numerical simulations which are in close agreement with the analytical prediction, showing the validity of the analysis. It is found that chaos and pull-in instability, two initial-sensitive phenomena of MEMS resonators, can be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. On this basis, two types of delayed feedback are proposed to control the complex dynamics successively. Their control mechanisms and effect are then studied. It follows that under a positive gain coefficient, delayed position feedback and delayed velocity feedback can both reduce pull-in instability; nevertheless, to suppress chaos, only the former can be effective. The results may have some potential value in broadening the application fields of global bifurcation theory and improving the performance reliability of capacitive MEMS devices.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100537

Authors: Yongqing Wu Xiao Zhang

This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and hybrid directed networks (Networks-C), and rigorously derive the analytical expressions of the eigenvalue spectrum on the basis of their supra-Laplacian matrix. It is found that network structure parameters (such as the number of layers and nodes, the intra-layer and the inter-layer coupling strengths) have a significant impact on the synchronizability in the case of the two typical synchronized regions. Finally, in order to confirm that the theoretical conclusions are correct, simulation experiments of multilayer directed network are delivered.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100536

Authors: Jinping Zhang Keming Zhang

Risk management is very important for individual investors or companies. There are several ways to measure the risk of investment. Prices of risky assets vary rapidly and randomly due to the complexity of finance market. Random interval is a good tool to describe uncertainty including both randomness and imprecision. Considering the uncertainty of financial market, we employ random intervals to describe returns of a risk asset and define an interval-valued risk measurement, which considers the tail risk. It is called the interval-valued conditional value-at-risk (ICVaR, for short). Similar to the classical conditional value-at-risk, ICVaR satisfies the sub-additivity. Under the new risk measure ICVaR, as a manner similar to the classical Mean-CVaR portfolio model, two optimal interval-valued portfolio selection models are built. The sub-additivity of ICVaR guarantees the global optimal solution to the Mean-ICVaR portfolio model. Based on the real data from mainland Chinese stock markets and international stock markets, the case study shows that our models are interpretable and consistent with the practical scenarios.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100535

Authors: Jingmin Pi Tianxiu Lu Yuanlin Chen

The concepts of collectively accessible, collectively sensitive, collectively infinitely sensitive, and collectively Li&ndash;Yorke sensitive are defined in non-autonomous discrete systems. It is proved that, if the mapping sequence h1,&infin;=(h1,h2,&#8943;) is W-chaotic, then hn,&infin;=(hn,hn+1,&#8943;)(&forall;n&isin;N={1,2,&#8943;}) would also be W-chaotic. W-chaos represents one of the following five properties: collectively accessible, sensitive, collectively sensitive, collectively infinitely sensitive, and collectively Li&ndash;Yorke sensitive. Then, the relationship of chaotic properties between the product system (H1&times;H2,f1,&infin;&times;g1,&infin;) and factor systems (H1,f1,&infin;) and (H2,g1,&infin;) was presented. Furthermore, in this paper, it is also proved that, if the autonomous discrete system (X,h^) induced by the p-periodic discrete system (H,h1,&infin;) is W-chaotic, then the p-periodic discrete system (H,f1,&infin;) would also be W-chaotic.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100534

Authors: Filippo Beretta Jesse Dimino Weike Fang Thomas C. Martinez Steven J. Miller Daniel Stoll

We investigate Benford&rsquo;s law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational numbers, and through statistical testing, we demonstrate that the numerators and denominators are a good fit for Benford&rsquo;s law. We offer additional conjectures and observations about these coefficients. In particular, we highlight certain arithmetic subsequences related to the coefficients&rsquo; denominators, provide an estimate for their slope, and describe efficient methods to compute them.

]]>Fractal and Fractional doi: 10.3390/fractalfract6100533

Authors: Mohammed Subhi Hadi Bülent Bilgehan

A fractional-order coronavirus disease of 2019 (COVID-19) model is constructed of five compartments in the Caputo-Fabrizio sense. The main aim of the paper is to study the effects of successive optimal control policies in different susceptible classes; a susceptible unaware class where awareness control is observed, a susceptible aware class where vaccine control is observed, and a susceptible vaccinated class where optimal vaccination control is observed. These control policies are considered awareness and actions toward vaccination and non-pharmaceuticals to control infection. Equilibrium points are calculated, which subsequently leads to the computation of the basic reproduction ratio. The existence and uniqueness properties of the model are established. The optimal control problem is constructed and subsequently analyzed. Numerical simulations are carried out and the significance of the fractional-order from the biological point of view is established. The results showed that applying various control functions will lead to a decrease in the infected population, and it is evident that introducing the three control measures together causes a drastic decrease in the infected population.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090532

Authors: Chandra Bose Sindhu Varun Bose Ramalingam Udhayakumar

This manuscript focuses on the existence of a mild solution Hilfer fractional neutral integro-differential inclusion with almost sectorial operator. By applying the facts related to fractional calculus, semigroup, and Martelli&rsquo;s fixed point theorem, we prove the primary results. In addition, the application is provided to demonstrate how the major results might be applied.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090531

Authors: Mostafijur Rahaman Reda M. S. Abdulaal Omer A. Bafail Manojit Das Shariful Alam Sankar Prasad Mondal

The present paper aims to demonstrate the combined impact of memory, selling price, and exhibited stock on a retailer&rsquo;s decision to maximizing the profit. Exhibited stock endorses demand and low selling prices are also helpful for creating demand. The proposed mathematical model considers demand as a linear function of selling price and displayed inventory. This work utilized fractional calculus to design a memory-based decision-making environment. Following the analytical theory, an algorithm was designed, and by using the Mathematica software, we produced the numerical optimization results. Firstly, the work shows that memory negatively influences the retailer&rsquo;s goal of maximum profit, which is the most important consequence of the numerical result. Secondly, raising the selling price will maximize the profit though the selling price, and demand will be negatively correlated. Finally, compared to the selling price, the influence of the visible stock is slightly lessened. The theoretical and numerical results ultimately imply that there can be no shortage and memory restrictions, leading to the highest average profit. The recommended approach may be used in retailing scenarios for small start-up businesses when a warehouse is required for continuous supply, but a showroom is not a top concern.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090530

Authors: Nguyen Hoang Luc Donal O’Regan Anh Tuan Nguyen

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton&ndash;Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag&ndash;Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090529

Authors: José Morán

The ability of the Langevin equation to predict coagulation kernels in the transition regime (ranging from ballistic to diffusive) is not commonly discussed in the literature, and previous numerical works are lacking a theoretical justification. This work contributes to the conversation to gain better understanding on how the trajectories of suspended particles determine their collision frequency. The fundamental link between the Langevin equation and coagulation kernels based on a simple approximation of the former is discussed. The proposed approximation is compared to a fractal model from the literature. In addition, a new, simple expression for determining the coagulation kernels in the transition regime is proposed. The new expression is in good agreement with existing methods such as the flux-matching approach proposed by Fuchs. The new model predicts an asymptotic limit for the kinetics of coagulation in the transition regime.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090528

Authors: Muhammad Abbas Afreen Bibi Ahmed S. M. Alzaidi Tahir Nazir Abdul Majeed Ghazala Akram

Numerous fields, including the physical sciences, social sciences, and earth sciences, benefit greatly from the application of fractional calculus (FC). The fractional-order derivative is developed from the integer-order derivative, and in recent years, real-world modeling has performed better using the fractional-order derivative. Due to the flexibility of B-spline functions and their capability for very accurate estimation of fractional equations, they have been employed as a solution interpolating polynomials for the solution of fractional partial differential equations (FPDEs). In this study, cubic B-spline (CBS) basis functions with new approximations are utilized for numerical solution of third-order fractional differential equation. Initially, the CBS functions and finite difference scheme are applied to discretize the spatial and Caputo time fractional derivatives, respectively. The scheme is convergent numerically and theoretically as well as being unconditionally stable. On a variety of problems, the validity of the proposed technique is assessed, and the numerical results are contrasted with those reported in the literature.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090527

Authors: Cheng Li Yao Xu Zhouting Jiang Boming Yu Peng Xu

The mapping relationships between the conductivity properties are not only of great importance for understanding the transport phenomenon in porous material, but also benefit the prediction of transport parameters. Therefore, a fractal pore-scale model with capillary bundle is applied to study the fluid flow and heat conduction as well as gas diffusion through saturated porous material, and calculate the conductivity properties including effective permeability, thermal conductivity and diffusion coefficient. The results clearly show that the correlations between the conductivity properties of saturated porous material are prominent and depend on the way the pore structure changes. By comparing with available experimental results and 2D numerical simulation on Sierpinski carpet models, the proposed mapping relationships among transport properties are validated. The present mapping method provides a new window for understanding the transport processes through porous material, and sheds light on oil and gas resources, energy storage, carbon dioxide sequestration and storage as well as fuel cell etc.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090526

Authors: Juan J. Gude Pablo García Bringas

This paper aims to present a general identification procedure for fractional first-order plus dead-time (FFOPDT) models. This identification method is general for processes having S-shaped step responses, where process information is collected from an open-loop step-test experiment, and has been conducted by fitting three arbitrary points on the process reaction curve. In order to validate this procedure and check its effectiveness for the identification of fractional-order models from the process reaction curve, analytical expressions of the FFOPDT model parameters have been obtained for both situations: as a function of any three points and three points symmetrically located on the reaction curve, respectively. Some numerical examples are provided to show the simplicity and effectiveness of the proposed procedure. Good results have been obtained in comparison with other well-recognized identification methods, especially when simplicity is emphasized. This identification procedure has also been applied to a thermal-based experimental setup in order to test its applicability and to obtain insight into the practical issues related to its implementation in a microprocessor-based control hardware. Finally, some comments and reflections about practical issues relating to industrial practice are offered in this context.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090525

Authors: Shijing Cheng Ning Du Hong Wang Zhiwei Yang

A finite element scheme for solving a two-timescale Hadamard time-fractional equation is discussed. We prove the error estimate without assuming the smoothness of the solution. In order to invert the fractional order, a finite-element Levenberg&ndash;Marquardt method is designed. Finally, we give corresponding numerical experiments to support the correctness of our analysis.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090524

Authors: Salemah A. Almutlak Rasool Shah Wajaree Weera Samir A. El-Tantawy Lamiaa S. El-Sherif

This study investigates the fractional-order Swift&ndash;Hohenberg equations using the natural decomposition method with non-singular kernel derivatives. The fractional derivative in the sense of Caputo&ndash;Fabrizio is considered. The Adomian decomposition technique (ADT) is a great deal to the overall natural transformation to create closed-form results of the given models. This technique provides a closed-form result for the suggested models. In addition, this technique is attractive, simple, and preferred over other techniques. The graphs of the solution in fractional and integer-order show that the achieved solutions are very close to the actual result of the examples. It is also investigated that the result of fractional-order models converges to the integer-order model&rsquo;s solution. Furthermore, the proposed method validity is examined using numerical examples. The obtained results for the given problems fully support the theory of the proposed method. The present method is a straightforward and accurate analytical method to analyze other fractional-order partial differential equations, such as many evolution equations that govern the dynamics of nonlinear waves in plasma physics.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090522

Authors: Sivajiganesan Sivasankar Ramalingam Udhayakumar

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We start by confirming the existence of the mild solution by using Dhage&rsquo;s fixed-point theorem. Finally, an example is provided to demonstrate the considered Hilferr fractional stochastic differential systems theory.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090523

Authors: Tongxin Wang Ziwen Jiang Ailing Zhu Zhe Yin

In this paper, the transverse vibration of a fractional viscoelastic beam is studied based on the fractional calculus, and the corresponding scheme of a viscoelastic beam is established by using the mixed finite volume element method. The stability and convergence of the algorithm are analyzed. Numerical examples demonstrate the effectiveness of the algorithm. Finally, the values of different parameter sets are tested, and the test results show that both the damping coefficient and the fractional derivative have significant effects on the model. The results of this paper can be used for the damping modeling of viscoelastic structures.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090521

Authors: Laura Pezza Luca Tallini

In recent years, we found that some multiscale methods applied to fractional differential problems, are easy and efficient to implement, when we use some fractional refinable functions introduced in the literature. In fact, these functions not only generate a multiresolution on R, but also have fractional (non-integer) derivative satisfying a very convenient recursive relation. For this reason, in this paper, we describe this class of refinable functions and focus our attention on their approximating properties.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090520

Authors: Lihua Zhang Bo Shen Hongbing Jiao Gangwei Wang Zhenli Wang

Fractional calculus is useful in studying physical phenomena with memory effects. In this paper, the fractional KMM (FKMM) system with beta-derivative in (2+1)-dimensions was studied for the first time. It can model short-wave propagation in saturated ferromagnetic materials, which has many applications in the high-tech world, especially in microwave devices. Using the properties of beta-derivatives and a proper transformation, the FKMM system was initially changed into the KMM system, which is a (2+1)-dimensional generalization of the sine-Gordon equation. Lie symmetry analysis and the optimal system for the KMM system were investigated. Using the optimal system, we obtained eight (1+1)-dimensional reduction equations. Based on the reduction equations, new soliton solutions, oblique analytical solutions, rational function solutions and power series solutions for the KMM system and FKMM system were derived. Using the properties of beta-derivatives and another transformation, the FKMM system was changed into a system of ordinary differential equations. Based on the obtained system of ordinary differential equations, Jacobi elliptic function solutions and solitary wave solutions for the FKMM system were derived. For the KMM system, the results about Lie symmetries, optimal system, reduction equations, and oblique traveling wave solutions are new, since Lie symmetry analysis method has not been applied to such a system before. For the FKMM system, all of the exact solutions are new. The main novelty of the paper lies in the fact that beta-derivatives have been used to change fractional differential equations into classical differential equations. The technique can also be extended to other fractional differential equations.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090518

Authors: Waqar Afzal Mujahid Abbas Jorge E. Macías-Díaz Savin Treanţă

Interval analysis distinguishes between different types of order relations. As a result of these order relations, convexity and nonconvexity contribute to different kinds of inequalities. Despite this, convex theory is commonly known to rely on Godunova&ndash;Levin functions because their properties make it more efficient for determining inequality terms than convex ones. The purpose of this study is to introduce the notion of cr-h-Godunova&ndash;Levin functions by using total order relation between two intervals. Considering their properties and widespread use, center-radius order relation appears to be ideally suited for the study of inequalities. In this paper, various types of inequalities are introduced using center-radius order (cr) relation. The cr-order relation enables us firstly to derive some Hermite&ndash;Hadamard (H.H) inequalities, and then to present Jensen-type inequality for h-Godunova&ndash;Levin interval-valued functions (GL-IVFS) using a Riemann integral operator. This kind of convexity unifies several new and well-known convex functions. Additionally, the study includes useful examples to support its findings. These results confirm that this new concept is useful for addressing a wide range of inequalities. We hope that our results will encourage future research into fractional versions of these inequalities and optimization problems associated with them.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090519

Authors: Zhiyu Cui Lu Liu Boyu Zhu Lichuan Zhang Yang Yu Zhexuan Zhao Shiyuan Li Mingwei Liu

Autonomous underwater vehicles (AUVs) have broad applications owing to their ability to undertake long voyages, strong concealment, high level of intelligence and ability to replace humans in dangerous operations. AUV motion control systems can ensure stable operation in the complex ocean environment and have attracted significant research attention. In this paper, we propose a single-input fractional-order fuzzy logic controller (SIFOFLC) as an AUV motion control system. First, a single-input fuzzy logic controller (SIFLC) was proposed based on the signed distance method, whose control input is the linear combination of the error signal and its derivative. The SIFLC offers a significant reduction in the controller design and calculation process. Then, a SIFOFLC was obtained with the derivative of the error signal extending to a fractional order and offering greater flexibility and adaptability. Finally, to verify the superiority of the proposed control algorithm, comparative numerical simulations in terms of spiral dive motion control were conducted. Meanwhile, the parameters of different controllers were optimized according to the hybrid particle swarm optimization (HPSO) algorithm. The simulation results illustrate the superior stability and transient performance of the proposed control algorithm.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090517

Authors: Kholoud Saad Albalawi Ibtehal Alazman

In this paper, we analyze the novel type of COVID-19 caused by the Omicron virus under a new operator of fractional order modified by Caputo&ndash;Fabrizio. The whole compartment is chosen in the sense of the said operator. For simplicity, the model is distributed into six agents along with the inclusion of the Omicron virus infection agent. The proposed fractional order model is checked for fixed points with the help of fixed point theory. The series solution is carried out by the technique of the Laplace Adomian decomposition technique. The compartments of the proposed problem are simulated for graphical presentation in view of the said technique. The numerical simulation results are established at different fractional orders along with the comparison of integer orders. This consideration will also show the behavior of the Omicron dynamics in human life and will be essential for its control and future prediction at various time durations. The sensitivity of different parameters is also checked graphically.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090513

Authors: Vladimir Uchaikin Elena Kozhemiakina

This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville differential equation of the first order in time, it is described by a fractional differential equation of a distributed equation in the interval (0, 1] order. The key characteristic of its motion is a spectral function, representing the order distribution over the interval. As a specific case of the process, a system with single-point spectrum is investigated. It follows the fractional Poisson process method evolution, obeying via a time-fractional differential equation with a unique order. The article ends with description of statistical estimation of parameters of seismic shocks imitated by Monte Carlo simulated fractional Poisson process.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090516

Authors: Xinxin Su Yongtao Zhou

In this paper, we focus on the computation of Caputo-type fractional differential equations. A high-order predictor&ndash;corrector method is derived by applying the quadratic interpolation polynomial approximation for the integral function. In order to deal with the weak singularity of the solution near the initial time of the fractional differential equations caused by the fractional derivative, graded meshes were used for time discretization. The error analysis of the predictor&ndash;corrector method is carefully investigated under suitable conditions on the data. Moreover, an efficient sum-of-exponentials (SOE) approximation to the kernel function was designed to reduce the computational cost. Lastly, several numerical examples are presented to support our theoretical analysis.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090515

Authors: Nattakan Boonsatit Santhakumari Rajendran Chee Peng Lim Anuwat Jirawattanapanit Praneesh Mohandas

The issue of adaptive finite-time cluster synchronization corresponding to neutral-type coupled complex-valued neural networks with mixed delays is examined in this research. A neutral-type coupled complex-valued neural network with mixed delays is more general than that of a traditional neural network, since it considers distributed delays, state delays and coupling delays. In this research, a new adaptive control technique is developed to synchronize neutral-type coupled complex-valued neural networks with mixed delays in finite time. To stabilize the resulting closed-loop system, the Lyapunov stability argument is leveraged to infer the necessary requirements on the control factors. The effectiveness of the proposed method is illustrated through simulation studies.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090514

Authors: Jun-Sheng Duan Yun-Yun Zhang

The impulsive response of the fractional vibration equation z&prime;&prime;(t)+bDt&alpha;z(t)+cz(t)=F(t), b&gt;0,c&gt;0,0&le;&alpha;&le;2, is investigated by using the complex path-integral formula of the inverse Laplace transform. Similar to the integer-order case, the roots of the characteristic equation s2+bs&alpha;+c=0 must be considered. It is proved that for any b&gt;0, c&gt;0 and &alpha;&isin;(0,1)&cup;(1,2), the characteristic equation always has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface. Particular attention is paid to the problem as to how the couple conjugated complex roots approach the two roots of the integer case &alpha;=1, especially to the two different real roots in the case of b2&minus;4c&gt;0. On the upper-half complex plane, the root s(&alpha;) is investigated as a function of order &alpha; and with parameters b and c, and so are the argument &theta;(&alpha;), modulus r(&alpha;), real part &lambda;(&alpha;) and imaginary part &omega;(&alpha;) of the root s(&alpha;). For the three cases of the discriminant b2&minus;4c: &gt;0, =0 and &lt;0, variations of the argument and modulus of the roots according to &alpha; are clarified, and the trajectories of the roots are simulated. For the case of b2&minus;4c&lt;0, the trajectories of the roots are further clarified according to the change rates of the argument, real part and imaginary part of root s(&alpha;) at &alpha;=1. The solution components, i.e., the residue contribution and the Hankel integral contribution to the impulsive response, are distinguished for the three cases of the discriminant.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090511

Authors: Davood Jabari Sabegh Reza Ezzati Omid Nikan António M. Lopes Alexandra M. S. F. Galhano

This paper proposes an accurate numerical approach for computing the solution of two-dimensional fractional Volterra integral equations. The operational matrices of fractional integration based on the Hybridization of block-pulse and Taylor polynomials are implemented to transform these equations into a system of linear algebraic equations. The error analysis of the proposed method is examined in detail. Numerical results highlight the robustness and accuracy of the proposed strategy.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090512

Authors: Mariam Sultana Uroosa Arshad Abdel-Haleem Abdel-Aty Ali Akgül Mona Mahmoud Hichem Eleuch

In this work, the fractional novel analytic method (FNAM) is successfully implemented on some well-known, strongly nonlinear fractional partial differential equations (NFPDEs), and the results show the approach&rsquo;s efficiency. The main purpose is to show the method&rsquo;s strength on FPDEs by minimizing the calculation effort. The novel numerical approach has shown to be the simplest technique for obtaining the numerical solution to any form of the fractional partial differential equation (FPDE).

]]>Fractal and Fractional doi: 10.3390/fractalfract6090510

Authors: Wei Gao Xin Chen Chengjie Hu

The fracture of interfacial crack is the main failure type of jointed rock mass. Therefore, it is very important to study the interfacial fracture of jointed rock mass. For the similarity of jointed rock mass and composites (all are composed by two parts, intact materials and their contact interfaces), the interface fracture mechanics widely used for analysis the interface crack of the composites (bimaterials) can be applied to study the interfacial fracture of jointed rock mass. Therefore, based on the basic theories of interface fracture mechanics, the interfacial fracture of jointed rock mass was analyzed, and one new criterion of interfacial crack initiation for jointed rock mass is proposed. Moreover, based on the proposed interfacial crack initiation criterion, the effect of main influence factors on the interfacial crack initiation of jointed rock mass was analyzed comprehensively. At last, by using the triaxial compression numerical tests on a jointed rock mass specimen with interfacial crack, the theoretical studies were verified.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090508

Authors: Yanzhu Zhang Tingting Liu Fan Yang Qi Yang

Following the traditional total variational denoising model in removing medical image noise with blurred image texture details, among other problems, an adaptive medical image fractional-order total variational denoising model with an improved sparrow search algorithm is proposed in this study. This algorithm combines the characteristics of fractional-order differential operators and total variational models. The model preserves the weak texture region of the image improvement based on the unique amplitude-frequency characteristics of the fractional-order differential operator. The order of the fractional-order differential operator is adaptively determined by the improved sparrow search algorithm using both the sine search strategy and the diversity variation processing strategy, which can greatly improve the denoising ability of the fractional-order differential operator. The experimental results reveal that the model not only achieves the adaptivity of fractional-order total variable differential order, but also can effectively remove noise, preserve the texture structure of the image to the maximum extent, and improve the peak signal-to-noise ratio of the image; it also displays favorable prospects for applications in medical image denoising.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090509

Authors: Carmelo Agnese Giorgio Baiamonte Elvira Di Nardo Stefano Ferraris Tommaso Martini

The Poisson-stopped sum of the Hurwitz&ndash;Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz&ndash;Lerch zeta distribution and the one inflated Hurwitz&ndash;Lerch zeta distribution. Within this framework, the capability of these three distributions to fit the main statistical features of rainfall time series was tested on a dataset never previously considered in the literature and chosen in order to represent very different climates from the rainfall characteristics point of view. The results address the Hurwitz&ndash;Lerch zeta distribution as a natural framework in rainfall modelling using the additional random convolution induced by the Poisson-stopped model as a further refinement. Indeed the Poisson contribution allows more flexibility and depiction in reproducing statistical features, even in the presence of very different climates.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090507

Authors: Yingjin He Jun Peng Song Zheng

This study is concerned with the dynamic investigation and fixed-time synchronization of a fractional-order financial system with the Caputo derivative. The rich dynamic behaviors of the fractional-order financial system with variations of fractional orders and parameters are discussed analytically and numerically. Through using phase portraits, bifurcation diagrams, maximum Lyapunov exponent diagrams, 0&ndash;1 testing and time series, it is found that chaos exists in the proposed fractional-order financial system. Additionally, a complexity analysis is carried out utilizing approximation entropy SE and C0 complexity to detect whether chaos exists. Furthermore, a synchronization controller and an adaptive parameter update law are designed to synchronize two fractional-order chaotic financial systems and identify the unknown parameters in fixed time simultaneously. The estimate of the setting time of synchronization depends on the parameters of the designed controller and adaptive parameter update law, rather than on the initial conditions. Numerical simulations show the effectiveness of the theoretical results obtained.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090506

Authors: Soubhagya Kumar Sahoo Muhammad Amer Latif Omar Mutab Alsalami Savin Treanţă Weerawat Sudsutad Jutarat Kongson

The objective of this manuscript is to establish a link between the concept of inequalities and Center-Radius order functions, which are intriguing due to their properties and widespread use. We introduce the notion of the CR (Center-Radius)-order interval-valued preinvex function with the help of a total order relation between two intervals. Furthermore, we discuss some properties of this new class of preinvexity and show that the new concept unifies several known concepts in the literature and also gives rise to some new definitions. By applying these new definitions, we have amassed many classical and novel special cases that serve as applications of the key findings of the manuscript. The computations of cr-order intervals depend upon the following concept B=&#10216;Bc,Br&#10217;=&#10216;B&macr;+B&#818;2,B&macr;&minus;B&#818;2&#10217;. Then, for the first time, inequalities such as Hermite&ndash;Hadamard, Pachpatte, and Fej&eacute;r type are established for CR-order in association with the concept of interval-valued preinvexity. Some numerical examples are given to validate the main results. The results confirm that this new concept is very useful in connection with various inequalities. A fractional version of the Hermite&ndash;Hadamard inequality is also established to show how the presented results can be connected to fractional calculus in future developments. Our presented results will motivate further research on inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090505

Authors: Eylem Öztürk Joseph L. Shomberg

We examine a viscous Cahn&ndash;Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the nonlinear potential allow for arbitrary polynomial growth.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090504

Authors: Lichuang Jin Shuai Zhang Yingjie Xu

Fractal analysis is an effective tool to describe real world phenomena. Water evaporation from the soil surface under extreme climatic conditions, such as drought, causes salt to accumulate in the soil, resulting in soil salinization, which aggravates soil shrinkage, deformation, and cracking. Hippophae is an alkali tolerant plant that is widely grown in Northwest China. Laboratory drying shrinkage tests of Saline-Alkali soil samples with 0%, 0.5%, 1%, and 2% concentrations of hippophae roots were carried out to study the effect of hippophae roots on the evaporation and cracking of Saline-Alkali soil and to determine variation characteristics of the soil samples&rsquo; fractal dimensions. A series of changes in the cracking parameters of Saline-Alkali soil were obtained during the cracking period. Based on fractal theory and the powerful image processing function of ImageJ software, the relationships between samples&rsquo; cracking process parameters were evaluated qualitatively and quantitatively. The experimental results show that the residual water contents of Saline-Alkali soil samples with 0%, 0.5%, 1%, and 2% concentrations of hippophae roots were 2.887%, 4.086%, 5.366%, and 6.696%, respectively. The residual water content of Saline-Alkali soil samples with 0.5% and 1% concentrations of hippophae roots increased by 41.53% and 85.87%, respectively; the residual water content of the sample with a 2% concentration of hippophae roots was 131.94% higher than that of the sample without hippophae roots. The final crack ratios of Saline-Alkali soil samples with 0%, 0.5%, 1%, and 2% concentrations of hippophae roots were 21.34%, 20.3%, 18.93%, and 17.18%, respectively. The final crack ratios of Saline-Alkali soil samples with 0.5%, 1%, and 2% concentrations of hippophae roots reduced by 4.87%, 11.29%, and 19.49%, respectively, compared with that of the sample without hippophae roots. Fractal dimensions at the end of cracking were 1.6217, 1.5656, 1.5282, and 1.4568, respectively. Fractal dimensions increased with an increase in the crack ratio and with a decrease in water content. The relationship between water content and fractal dimension can be expressed using a quadratic function. Results indicate that hippophae roots can effectively inhibit the cracking of Saline-Alkali soil and improve its water holding capacity.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090503

Authors: Omar Bazighifan

Differential equations, both fractional and ordinary, give key tools in understanding the mechanisms of physical systems and solving various problems of nonlinear phenomena [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract6090502

Authors: Tianyuan Jia Xiangyong Chen Liping He Feng Zhao Jianlong Qiu

Finite-time synchronization (FTS) of uncertain fractional-order memristive neural networks (FMNNs) with leakage and discrete delays is studied in this paper, in which the impacts of uncertain parameters as well as external disturbances are considered. First, the fractional-order adaptive terminal sliding mode control scheme (FATSMC) is designed, which can effectively estimate the upper bounds of unknown external disturbances. Second, the FTS of the master&ndash;slave FMNNs is realized and the corresponding synchronization criteria and the explicit expression of the settling time (ST) are obtained. Finally, a numerical example and a secure communication application are provided to demonstrate the validity of the obtained results.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090501

Authors: Litong Fang Chang Chen Yubin Wang

Porous alumina ceramics with different porosities were prepared via atmospheric pressure sintering using a sacrificial template method with alumina powder as the raw material and carbon fiber (CF) and graphite as pore-forming agents. The effects of the contents and ratios of the pore-forming agents and the aspect ratios of CF on the microstructure, mechanical properties, pore size, and pore-size distribution of the porous alumina samples were investigated. In addition, the surface fractal dimension (Ds) of porous alumina samples with different pore-forming agents was evaluated based on the mercury intrusion porosimetry data. The pore-size distribution of the prepared porous alumina samples showed single, double, or multiple peaks. The pore structure of the samples maintained the fibrous shape of the original CF and the flake morphology of graphite with a uniform pore-size distribution, but the pore structure and morphology were different. With the increase in the content of the pore-forming agent, the porosity of the samples gradually increased to a maximum of 63.2%, and the flexural strength decreased to a minimum of 12.36 MPa. The pore structure of the porous alumina samples showed obvious fractal characteristics. Ds was closely related to the pore structure parameters of the samples when the content of the pore-forming agent was 70 vol.%. It decreased with an increase in the sample porosity, most probable pore size and median pore size, but increased with an increase in the sample flexural strength.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090500

Authors: Jorge E. Macías-Díaz Tassos Bountis

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo&ndash;Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090499

Authors: Yong Li

It is necessary to quantitatively describe or illustrate the characteristics of abnormal stock price fluctuations in order to prevent and control financial risks. This paper studies the fractal structure of China&rsquo;s stock market by calculating the fractal dimension and scaling behavior on the timeline of its eight big slumps, the results show that the slumps have multifractal characteristics, which are correlated with the policy intervention, institutional arrangements, and investors&rsquo; rationality. The empirical findings are a perfect match with the anomalous features of the stock prices. The fractal dimensions of the eight stock collapses are between 0.84 and 0.98. The fractal dimension distribution of the slumps is sensitive to market conditions and the active degree of speculative trading. The more mature market conditions and the more risk-averse investors correspond to the higher fractal dimension and the fall which is less deep. Therefore, the fractal characteristics could reflect the evolution characteristics of the stock market and investment philosophy. The parameter set calculated in this paper could be used as an effective tool to foresee the slumps on the horizon.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090498

Authors: V. Dhanya A. Arunkumar Kantapon Chaisena

This study inspects the issue of robust reliable sampled data control (SDC) for a class of Takagi-Sugeno (TS) fuzzy CE151 Helicopter systems with time-varying delays and linear fractional uncertainties. Specifically, both the variation range and the distribution probability of the time delay are considered in the control input. The essential aspect of the suggested results in this study is that the time variable delay in the control input is dependent not only on the bound but also on the distribution probability of the time delay. The prime intent of this study is to enhance a state feedback reliable sampled-data controller. By constructing an appropriate Lyapunov-Krasovskii functional (LKF) and employing a linear matrix inequalities (LMIs) approach, a new set of delay-dependent necessary conditions is obtained to ensure the asymptotic stabilisation of a TS fuzzy CE151 Helicopter system with a prescribed mixed H&infin; and passivity (MH&infin;P) performance index. The acquired results are expressed as LMIs, which are easily addressed using standard optimization algorithms. In addition, an exemplary scenario based on the CE151 helicopter model is presented to demonstrate the less conservative nature of the obtained results as well as the application of the recommended unique design approaches.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090497

Authors: Emmanuel Adeyefa Ezekiel Omole Ali Shokri Shao-Wen Yao

A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite polynomial was interpolated at the first two successive points, while the collocation occurred at all the suitably chosen points. The major scheme and its complementary scheme were united together to form the HFBI. The analysis of the HFBI showed that it had a convergence order of eight with small error constants, was zero-stable, absolutely-stable, and satisfied the condition for convergence. In order to confirm the usefulness, accuracy, and efficiency of the HFBI, the method of lines approach was applied to discretize the second-order anisotropic elliptic partial differential equation PDE into a system of second-order ODEs and consequently used the derived HFBI to obtain the approximate solutions for the PDEs. The computed solution generated by using the HFBI was compared to the exact solutions of the problems and other existing methods in the literature. The proposed method compared favorably with other existing methods, which were validated through test problems whose solutions are presented in tabular form, and the comparisons are illustrated in the curves.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090496

Authors: Abd-Allah Hyder Hüseyin Budak Areej A. Almoneef

In this study, new midpoint-type inequalities are given through recently generalized Riemann&ndash;Liouville fractional integrals. Foremost, we present an identity for a class of differentiable functions including the proposed fractional integrals. Then, several midpoint-type inequalities containing generalized Riemann&ndash;Liouville fractional integrals are proved by employing the features of convex and concave functions. Furthermore, all obtained results in this study can be compared to previously published results.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090495

Authors: Vaijanath L. Chinchane Asha B. Nale Satish K. Panchal Christophe Chesneau

The Caputo&ndash;Fabrizio fractional integral operator is one of the important notions of fractional calculus. It is involved in numerous illustrative and practical issues. The main goal of this paper is to investigate weighted fractional integral inequalities using the Caputo&ndash;Fabrizio fractional integral operator with non-singular e&minus;1&minus;&delta;&delta;(&#1008;&minus;s), 0&lt;&delta;&lt;1. Furthermore, based on a family of n positive functions defined on [0,&infin;), we investigate some new extensions of weighted fractional integral inequalities.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090494

Authors: Xiongfeng Deng Lisheng Wei

In this paper, the adaptive finite-time control problem for fractional-order systems with uncertainties and unknown dead-zone fault was studied by combining a fractional-order command filter, radial basis function neural network, and Nussbaum gain function technique. First, the fractional-order command filter-based backstepping control method is applied to avoid the computational complexity problem existing in the conventional recursive procedure, where the fractional-order command filter is introduced to obtain the filter signals and their fractional-order derivatives. Second, the radial basis function neural network is used to handle the uncertain nonlinear functions in the recursive design step. Third, the Nussbaum gain function technique is considered to handle the unknown control gain caused by the unknown dead-zone fault. Moreover, by introducing the compensating signal into the control law design, the virtual control law, adaptive laws, and the adaptive neural network finite-time control law are constructed to ensure that all signals associated with the closed-loop system are bounded in finite time and that the tracking error can converge to a small neighborhood of origin in finite time. Finally, the validity of the proposed control law is confirmed by providing simulation cases.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090493

Authors: Minghao Wang Enli Chen Ruilan Tian Cuiyan Wang

A smooth and discontinuous (SD) oscillator is a typical multi-stable state system with strong nonlinear properties and has been widely used in many fields. The nonlinear dynamic characteristics of the system have not been thoroughly investigated because the nonlinear restoring force cannot be integrated. In this paper, the nonlinear restoring force is represented by a piecewise nonlinear function. The equivalent coefficients of fractional damping are obtained with an orthogonal function. The influence of fractional damping on the transition set, the amplitude&ndash;frequency response and the snap-through of the SD oscillator are analyzed. The conclusions are as follows: The nonlinear piecewise function accurately mimics the nonlinear restoring force and maintains a nonlinearity property. Fractional damping can significantly affect the stiffness and damping property simultaneously. The equivalent coefficients of the fractional damping are variable with regard to the fractional-order power of the excitation frequency. A hysteresis point, a bifurcation point, a frequency island, pitchfork bifurcations and transcritical bifurcations were discovered in the small-amplitude resonant region. In the non-resonant region, the increase in the fractional parameters leads to the probability of snap-through declining by increasing the symmetry of the attraction domain or reducing the number of stable states.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090492

Authors: Zhiyuan Li Qintong Chen Yulan Wang Xiaoyu Li

Fractional-order calculus has become a useful mathematical framework to describe the complex super-diffusive process; however, numerical solutions of the two-sided space-fractional super-diffusive model with variable coefficients are difficult to obtain, and almost no method can obtain an analytical solution. In this paper, a class of new fractional dimensional reproducing kernel spaces (RKS) based on Caputo fractional derivatives is given, and we give analytical and numerical solutions of the two-sided space-fractional super-diffusive model based on the class of new RKS. The analytical solution is represented in the form of series in the reproducing kernel space. Numerical experiments indicate that the piecewise reproducing kernel method is more accurate than the traditional reproducing kernel method (RKM), and these new fractional reproducing kernel spaces are efficient for the two-sided space-fractional super-diffusive model.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090491

Authors: Bhargav Panchal Soney Varghese

This paper presents the FEM modeling and simulation of a thin-film bulk acoustic resonator (FBAR) for a tetrachloroethene (PCE) gas-sensing application. A zinc oxide layer is used as a piezoelectric material; an aluminum layer is used as the electrode material in the structure of the FBAR. Polyisobutylene (PIB) is used as the sensitive layer for PCE gas detection. The study was carried out in commercially available FEM-based COMSOL software. The proposed structure was exposed to six different organic gases with concentrations ranging from 0 to 1000 ppm. The structure showed high selectivity for PCE gas. Incorporating the 3rd-order Hilbert fractal geometry in the top electrode of the FBAR increased the sensitivity of the sensor which showed high selectivity for PCE gas detection. A sensitivity enhancement of 66% was obtained using fractal geometry on the top electrode of the FBAR without alteration in size or cost. In addition, a reduction in the cross-sensitivity was achieved. Further, the PIB layer thickness and active area of the FBAR were optimized to obtain high sensitivity. The equivalent circuit was also analyzed to understand the behavior of the sensing effect and mechanism.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090490

Authors: Rania Saadeh Ahmad Qazza Kawther Amawi

The objective of this work is to investigate analytical solutions of some models of cancer tumors using the Laplace residual power series method (LRPSM). The proposed method was effective and required simple calculations to find the analytic series solution, utilizing computer software such as the Mathematica package. Figures and graphs of the attained analytical Maclaurin solutions are presented to depict the procedure. The outcomes we obtained in this research showed the applicability and strength of the proposed approach in studying numerical series solutions of differential equations of fractional orders.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090489

Authors: Fredrick M. Mwema Tien-Chien Jen Pavel Kaspar

A bibliometric analysis of publications on fractal theory and thin films is presented in this article. Bibliographic information is extracted from the Web of Science digital database and the bibliographic mapping undertaken using VOSviewer software. Based on the analysis, there is a growing trend in research on the applications of fractal theory in thin film technology. The factors driving this trend are discussed in the article. The co-citation, co-authorship and bibliographic coupling among authors, institutions and regions are presented. The applications of fractal theory in thin film technology are clarified based on the bibliometric study and the directions for future research provided.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090488

Authors: Haiyong Xu Lihong Zhang Guotao Wang

This paper studies the existence of extremal solutions for a nonlinear boundary value problem of Bagley&ndash;Torvik differential equations involving the Caputo&ndash;Fabrizio-type fractional differential operator with a non-singular kernel. With the help of a new inequality with a Caputo&ndash;Fabrizio fractional differential operator, the main result is obtained by applying a monotone iterative technique coupled with upper and lower solutions. This paper concludes with an illustrative example.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090487

Authors: Mohammad Dehghan Angelo B. Mingarelli

We continue the study of a non-self-adjoint fractional three-term Sturm&ndash;Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann&ndash;Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter &alpha;, 0&lt;&alpha;&lt;1, there is a finite set of real eigenvalues and that, for &alpha; near 1/2, there may be none at all. As &alpha;&rarr;1&minus; we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm&ndash;Liouville problem with the composition of the operators becoming the operator of second order differentiation.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090485

Authors: Chunna Zhao Murong Jiang Yaqun Huang

Fractional-order PID control is a landmark in the development of fractional-order control theory. It can improve the control precision and accuracy of systems and achieve more robust control results. As a theorem-proving formal verification method, it can be applied to an arbitrary system represented by a mathematical model. It is the ideal verification method because it is not subject to limits on state numbers. This paper presents the higher-order logic (HOL) formal verification and modeling of fractional-order PID controller systems. Firstly, a fractional-order PID controller was designed. The accuracy of fractional-order PID control can be supported by simulation, comparing integral-order PID controls. Secondly, the superior property of fractional-order PID control is validated via higher-order logic theorem proofs. An important basic property, the relationship between fractional-order differential calculus and integral-order differential calculus, was analyzed via a higher-order logic theorem proof. Then, the relations between the fractional-order PID controller and integral-order PID controller were verified based on the fractional-order Gr&uuml;nwald&ndash;Letnikov definition for higher-order logic theorem proofs. Formalization models of the fractional-order PID controller and the fractional-order closed-loop control system were established. Finally, the stability of the fractional-order control systems was verified based on established formal models and theorems. The results show that the fractional-order PID controllers can be conducive to the control performance of control systems, and the higher-order logic formal verification method can ensure the reliability and security of fractional-order control systems.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090486

Authors: Kaijun Miao Shihao Tu Hongsheng Tu Xun Liu Wenlong Li Hongbin Zhao Long Tang Jieyang Ma Yan Li

A fractal realizes the quantitative characterization of complex and disordered mining fracture networks, and it is of great significance to grasp the fractal characteristics of rock movement law to guide mine production. To prevent the water-conducting fracture (WF) under the gullies from conducting the surface water body, and to realize the purpose of safe production and surface water body protection. The evolution of overburden fissures in the working face with shallow buried gulley landform and thick bedrock conditions is studied. The development height of water-conducting fracture (DHWF) is theoretically analyzed. The evolution characteristics of overlying fissures with different mining heights were observed by similarity simulation, and the observation results were analyzed by fractal theory. The results show that the main factor that determines the height of WF is mining height. The working face is mined at different mining heights, and the corresponding indexes such as the height of the WF, the area of the caving zone and the fractal dimension are related to engineering phenomena. In particular, the appearance and disappearance of the separation space correspond to the fractal dimension fluctuation phase. The safe mining technology under a gully water body, which mainly reduces mining height, is adopted, and the fissures of the working face are not connected to the surface water body after mining.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090484

Authors: Hongming Zhan Xizhe Li Zhiming Hu Xianggang Duan Wei Wu Wei Guo Wei Lin

The occurrence and flow of shale gas are substantially impacted by nanopore structures. The fractal dimension provides a new way to explore the pore structures of shale reservoirs. In this study, eight deep shale samples from Longmaxi Formation to Wufeng Formation in Southern Sichuan were selected to perform a series of analysis tests, which consisted of small-angle neutron scattering, low-pressure nitrogen adsorption, XRD diffraction, and large-scale scanning electron microscopy splicing. The elements that influence the shale fractal dimension were discussed from two levels of mineral composition and pore structures, and the relationship between the mass fractal dimension and surface fractal dimension was focused on during a comparative analysis. The results revealed that the deep shale samples both had mass fractal characteristics and surface fractal characteristics. The mass fractal dimension ranged from 2.499 to 2.991, whereas the surface fractal dimension ranged from 2.814 to 2.831. The mass fractal dimension was negatively correlated with the surface fractal dimension. The mass fractal dimension and the surface fractal dimension are controlled by organic matter pores, and their development degree significantly affects the fractal dimension. The mass fractal dimension increases with the decrease of a specific surface area and pore volume and increases with the increase of the average pore diameter. The permeability and surface fractal dimension are negatively correlated, but no significant correlation exists between the permeability and mass fractal dimension, and the internal reason is the dual control effect of organic matter on shale pores. This study comprehensively analyses the mass fractal characteristics and surface fractal characteristics, which helps in a better understanding of the pore structure and development characteristics of shale gas reservoirs.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090482

Authors: Yunmei Fang Siyang Li Juntao Fei

A second-order sliding mode control (SOSMC) with a fractional module using adaptive fuzzy controller is developed for an active power filter (APF). A second-order sliding surface using a fractional module which can decrease the discontinuities and chattering is designed to make the system work stably and simplify the design process. In addition, a fuzzy logic control is utilized to estimate the parameter uncertainties. Simulation and experimental discussion illustrated that the designed fractional SOSMC with adaptive fuzzy controller is valid in satisfactorily eliminating harmonic, showing good robustness and stability compared with an integer order one.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090483

Authors: Jamiu Olusegun Hamzat Matthew Olanrewaju Oluwayemi Alina Alb Lupaş Abbas Kareem Wanas

In the present article, using the subordination principle, the authors employed certain generalized multiplier transform to define two new subclasses of analytic functions with respect to symmetric and conjugate points. In particular, bi-univalent conditions for function f(z) belonging to these new subclasses and their relevant connections to the famous Fekete-Szeg&ouml; inequality |a3&minus;va22| were investigated using a succinct mathematical approach.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090481

Authors: Hari M. Srivastava Jose Vanterler da Costa Sousa

In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the &chi;-fractional space H&kappa;(x)&gamma;,&beta;;&chi;(&Delta;). Using the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem, we show that under certain suitable assumptions the considered problem has at least k pairs of non-trivial solutions.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090480

Authors: Keyu Zhang Fehaid Salem Alshammari Jiafa Xu Donal O’Regan

In this paper, we use the fixed-point index to establish positive solutions for a system of Riemann&ndash;Liouville type fractional-order integral boundary value problems. Some appropriate concave and convex functions are used to characterize coupling behaviors of our nonlinearities.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090479

Authors: Chang-Hua Lien Hao-Chin Chang Ker-Wei Yu Hung-Chi Li Yi-You Hou

In this paper, we propose synchronous switching of rule and input to achieve H&infin; performance for an uncertain switched delay system with linear fractional perturbations. Our developed simple scheme utilizes the linear matrix inequality optimization problem to provide a feasible solution for the proposed results; if the optimization problem was feasible, our proposed robust H&infin; control could be designed. The feasibility of the optimization problem could be solved using the LMI toolbox of Matlab. In this paper, robust control with sampling is proposed to stabilize uncertain switching with interval time-varying delay and achieve H&infin; performance. Interval time-varying delay and sampling were considered instead of constant delay and pointwise sampling. A full-matrix formulation approach is presented to improve the conservativeness of our proposed results. Some numerical examples are demonstrated to show our main contributions.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090478

Authors: Helber Meneses Orlando Arrieta Fabrizio Padula Antonio Visioli Ramon Vilanova

This paper deals with the design of a control system based on fractional order models and fractional order proportional-integral-derivative (FOPID) controllers and fractional-order proportional-integral (FOPI) controllers. The controller design takes into account the trade-off between robustness and performance as well as the trade-off between the load disturbance rejection and set-point tracking tasks. The fractional order process model is able to represent an extensive range of dynamics, including over-damped and oscillatory behaviors and this simplifies the process modelling. The tuning of the FOPID and FOPI controllers is achieved by using an optimization, as a first step, and in a second step, several fitting functions were used to capture the behavior of the optimal parameters of the controllers. In this way, a new set of tuning rules called FOMCoRoT (Fractional Order Model and Controllers Robust Tuning) is obtained for both FOPID and FOPI controllers. Simulation examples show the effectiveness of the proposed control strategy based on fractional calculus.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090477

Authors: Puxi Li Ran Tao Shijie Yang Di Zhu Ruofu Xiao

Vortex rope is a common phenomenon in the draft tube of hydraulic turbines. It may cause strong pressure pulsation, noise, and strong vibration of the unit especially when it is helical. Therefore, the study of vortex rope is of great significance. In order to study the helical vortex rope, the embedded large eddy simulation (ELES) method in the hybrid methods is used based on the vortex rope generator case. The Liutex method can show the three-dimensional shape of the vortex rope well. In order to quantitatively describe the helical vortex rope, the three-dimensional structure is divided into multiple two-dimensional sections, and then the shape of vortex rope on each section is processed to extract the perimeter and area of the vortex. Combined with the change trend of vortex number and section area, the helical vortex rope is divided into four zones. Then, the fractal dimension on each zone and section can be obtained, and it can be used to quantitatively analyze the change trend of the vortex rope in time and space. The fractal analysis method can be applied to the analysis of the vortex rope in the draft tube to help judge the flow pattern shape and the stability of the unit operating conditions.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090476

Authors: Lei Wang Shihua Zhou Yan Shi Yajun Huang Feng Zhao Tingting Huo Shengwen Tang

Concrete-face slabs are the primary anti-permeability structures of the concrete-face rockfill dam (CFRD), and the resistance of face slab concrete to permeability is the key factor affecting the operation and safety of CFRDs. Herein, the influences of five fly ash dosages (namely 10%, 20%, 30%, 40% and 50%) on the permeability property of face slab concretes were investigated. Moreover, the difference in the permeability caused by the fly ash dosage variations is revealed in terms of the pore structure and fractal theory. The results illustrate that: (1) The inclusion of 10&ndash;50% fly ash lowered the compressive strength of face slab concretes before 28 days of hydration, whereas it contributed to the 180-day strength increment. (2) The incorporation of 10&ndash;50% fly ash raised the average water-seepage height (Dm) and the relative permeability coefficient (Kr) of the face slab concrete by about 14&ndash;81% and 30&ndash;226% at 28 days, respectively. At 180 days, the addition of fly ash improved the 180-day impermeability by less than 30%. (3) The permeability of face slab concretes is closely correlated with their pore structures and Ds. (4) The optimal fly ash dosage in terms of the long-term impermeability and pore refinement of face slab concretes is around 30%. Nevertheless, face slab concretes containing a high dosage of fly ash must be cured for a relatively long period before they can withstand high water pressure.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090475

Authors: Junying Cao Zhongqing Wang Ziqiang Wang

In this paper, the time fractional diffusion equations optimal control problem is solved by 3&minus;&alpha; order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are used to make the space variables discrete, and obtain the semidiscrete scheme of the state and adjoint state. The priori error estimates for the semidiscrete scheme for state and adjoint state equation are established. Furthermore, the 3&minus;&alpha; order uniform accuracy scheme is used to make the time variable discrete in the semidiscrete scheme and construct the full discrete scheme for the control problems based on the first optimal condition and &lsquo;first optimize, then discretize&rsquo; approach. The fully discrete scheme&rsquo;s stability and truncation error are analyzed. Finally, two numerical examples are denoted to show that the theoretical analysis are correct.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090474

Authors: Gang Zhang Hongyu Wang Jahanzaib Israr Wenguo Ma Youzhen Yang Keliang Ren

In this study, a rigorous mathematical approach used to compute an effective diameter based on particle size distribution (PSD) has been presented that can predict the hydraulic conductivity of granular soils with enhanced rigor. The PSD was discretized based on an abstract interval system of fractal entropy, while the effective diameter of soil was computed using the grading entropy theory. The comparisons between current entropy-based effective diameter (DE) and those computed using existing procedures show that the current DE can capture the particle size information of a given soil more accurately than others. Subsequently, the proposed DE was successfully implicated into Kozeny&ndash;Carman&rsquo;s formula to deduce the saturated hydraulic conductivity of soils with enhanced accuracy. The proposed model was tested using current and previously published experimental data from literature. Not surprisingly, the results of the current model and those from previous experimental studies were found to be consistent, which can sufficiently verify the proposed entropy-based effective diameter model.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090473

Authors: Kang Xu Tingli Cheng António M. Lopes Liping Chen Xiaoxuan Zhu Minwu Wang

A new control strategy is proposed to suppress earthquake-induced vibrations on uncertain building structures. The control strategy embeds fuzzy logic in a fractional-order (FO) proportional derivative (FOPD) controller. A new improved FO particle swarm optimization (IFOPSO) algorithm is derived to adjust the initial parameters of the FOPD controller. An original fuzzy logic-FOPD (FFOPD) controller is then designed by combining the advantages of the fuzzy logic and FOPD control, to deal with large displacements on structures under earthquake excitation. Simulation experiments are carried out on uncertain building structures subjected to the effects of different kinds of seismic signals, illustrating the validity and feasibility of the proposed method.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090472

Authors: Hyun Geun Lee

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the L2-gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the L2-gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical H&minus;1-gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090471

Authors: Wanqing Song Shouwu Duan Dongdong Chen Enrico Zio Wenduan Yan Fan Cai

In this paper, an efficient prediction model based on the fractional generalized Pareto motion (fGPm) with Long-Range Dependent (LRD) and infinite variance characteristics is proposed. Firstly, we discuss the meaning of each parameter of the generalized Pareto distribution (GPD), and the LRD characteristics of the generalized Pareto motion are analyzed by taking into account the heavy-tailed characteristics of its distribution. Then, the mathematical relationship H=1&frasl;&alpha; between the self-similar parameter H and the tail parameter &alpha; is obtained. Also, the generalized Pareto increment distribution is obtained using statistical methods, which offers the subsequent derivation of the iterative forecasting model based on the increment form. Secondly, the tail parameter &alpha; is introduced to generalize the integral expression of the fractional Brownian motion, and the integral expression of fGPm is obtained. Then, by discretizing the integral expression of fGPm, the statistical characteristics of infinite variance is shown. In addition, in order to study the LRD prediction characteristic of fGPm, LRD and self-similarity analysis are performed on fGPm, and the LRD prediction conditions H&gt;1&frasl;&alpha; is obtained. Compared to the fractional Brownian motion describing LRD by a self-similar parameter H, fGPm introduces the tail parameter &alpha;, which increases the flexibility of the LRD description. However, the two parameters are not independent, because of the LRD condition H&gt;1&frasl;&alpha;. An iterative prediction model is obtained from the Langevin-type stochastic differential equation driven by fGPm. The prediction model inherits the LRD condition H&gt;1&frasl;&alpha; of fGPm and the time series, simulated by the Monte Carlo method, shows the superiority of the prediction model to predict data with high jumps. Finally, this paper uses power load data in two different situations (weekdays and weekends), used to verify the validity and general applicability of the forecasting model, which is compared with the fractional Brown prediction model, highlighting the &ldquo;high jump data prediction advantage&rdquo; of the fGPm prediction model.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090470

Authors: Adel R. Hadhoud Faisal E. Abd Alaal Ayman A. Abdelaziz Taha Radwan

This article seeks to show a general framework of the cubic polynomial spline functions for developing a computational technique to solve the space-fractional Fisher&rsquo;s equation. The presented approach is demonstrated to be conditionally stable using the von Neumann technique. A numerical illustration is given to demonstrate the proposed algorithm&rsquo;s effectiveness. The novelty of the present work lies in the fact that the results suggest that the presented technique is accurate and convenient in solving such problems.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090469

Authors: Kaihong Zhao

The fractional Langevin equation is a very effective mathematical model for depicting the random motion of particles in complex viscous elastic liquids. This manuscript is mainly concerned with a class of nonlinear fractional Langevin equations involving nonsingular Mittag&ndash;Leffler (ML) kernel. We first investigate the existence and uniqueness of the solution by employing some fixed-point theorems. Then, we apply direct analysis to obtain the Ulam&ndash;Hyers (UH) type stability. Finally, the theoretical analysis and numerical simulation of some interesting examples show that there is a great difference between the fractional Langevin equation and integer Langevin equation in describing the random motion of free particles.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090468

Authors: Gangwei Wang Bo Shen Mengyue He Fei Guan Lihua Zhang

In the present paper, PT-symmetric extension of the fifth-order Korteweg-de Vries-like equation are investigated. Several special equations with PT symmetry are obtained by choosing different values, for which their symmetries are obtained simultaneously. In particular, for the particular equation, its conservation laws are obtained, including conservation of momentum and conservation of energy. Reciprocal Ba&uml;cklund transformations of conservation laws of momentum and energy are presented for the first time. The important thing is that for the special case of &#1013;=3, the corresponding time fractional case are studied by Lie group method. And what is interesting is that the symmetry of the time fractional equation is obtained, and based on the symmetry, this equation is reduced to a fractional ordinary differential equation. Finally, for the general case, the symmetry of this equation is obtained, and based on the symmetry, the reduced equation is presented. Through the results obtained in this paper, it can be found that the Lie group method is a very effective method, which can be used to deal with many models in natural phenomena.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090467

Authors: Dongping Li Yankai Li Fangqi Chen

This paper deals with a class of nonlinear fractional Sturm&ndash;Liouville boundary value problems. Each sub equation in the system is a fractional partial equation including the second kinds of Fredholm integral equation and the p-Laplacian operator, simultaneously. Infinitely many solutions are derived due to perfect involvements of fractional calculus theory and variational methods with some simpler and more easily verified assumptions.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090466

Authors: Jehad Alzabut Ravi P. Agarwal Said R. Grace Jagan M. Jonnalagadda

This survey paper is devoted to succinctly reviewing the recent progress in the field of oscillation theory for linear and nonlinear fractional differential equations. The paper provides a fundamental background for all interested researchers who would like to contribute to this topic.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090465

Authors: Li Li Yang Zhang Yuqiong Shi Zhigang Xue Mingli Cao

Destruction pattern analysis of building materials subjected to fire provide the basis for strengthening, restoring the bearing capacity, and optimizing the function of the building structure. The surface cracking and fractal characteristics of calcium carbonate whisker-reinforced cement pastes subjected to high temperatures were studied herein. The test results showed that at 400 &deg;C, the surface crack area, length, and fractal dimension of cement pastes specimen increases from 0 to 35 mm2, 100 mm, and 1.0, respectively, due to the increase of vapor pressure. When the temperature is above 900 &deg;C, the calcium carbonate whisker (CW) and other hydration products in the specimen begin to decompose, causing the surface crack area, length, and fractal dimension of the cement paste specimen to increase from 0 to 120 mm2, 310 mm, and 1.2, respectively. Compared with the length and width of cracks, the area, and fractal dimension of cracks are less affected by the size and shape of specimen. This paper uses image processing methods to analyze the cracking patterns and fractal characteristics of specimens after high-temperature treatment. The aim is to elucidate the quantitative relationship between concrete material, temperature, and cracking characteristics, providing theoretical basis for structural evaluation after exposure to high temperature.

]]>Fractal and Fractional doi: 10.3390/fractalfract6090464

Authors: Mohammed Kbiri Alaoui

The aim of this paper is to investigate the following non local p-Laplacian problem with data a bounded Radon measure &thetasym;&isin;Mb(&Omega;): (&minus;&Delta;)psu=&thetasym;in&Omega;, with vanishing conditions outside &Omega;, and where s&isin;(0,1),2&minus;sN&lt;p&le;N. An existence result is provided, and some sharp regularity has been investigated. More precisely, we prove by using some fractional isoperimetric inequalities the existence of weak solution u such that: 1. If &thetasym;&isin;Mb(&Omega;), then u&isin;W0s1,q(&Omega;) for all s1&lt;s and q&lt;N(p&minus;1)N&minus;s. 2. If &thetasym; belongs to the Zygmund space LLog&alpha;L(&Omega;),&alpha;&gt;N&minus;sN, then the limiting regularity u&isin;W0s1,N(p&minus;1)N&minus;s(&Omega;) (for all s1&lt;s). 3. If &thetasym;&isin;LLog&alpha;L(&Omega;), and &alpha;=N&minus;sN with p=N, then we reach the maximal regularity with respect to s and N,u&isin;W0s,N(&Omega;).

]]>Fractal and Fractional doi: 10.3390/fractalfract6090463

Authors: Kouqi Liu Mehdi Ostadhassan

Rock permeability, defined as the ability of fluid to flow through the rocks, is one of the most important properties of rock. Many researchers have developed models to predict the permeability of rock from the porosity and pore size based on the mercury intrusion. However, these existing models still have some limitations. In this study, based on data regarding the fractal nature of the mercury intrusion of the rocks, we built a new model to predict the permeability of the rocks. In order to verify the new model, we extracted data regarding different kinds of samples from the literature and estimated the permeability using the new model. The results showed that the model could predict various types of rocks, such as tight sandstone, carbonates, and shale. The comparison of the calculated permeability using the new model is closer to the measured value than the value estimated from the existing models, indicating that the new model is better in predicting the permeability of rock samples.

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