Fractal and Fractional doi: 10.3390/fractalfract8030176
Authors: Benjamin Ducharne Hamed Hamzehbahmani Yanhui Gao Patrick Fagan Gael Sebald
Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces a model for estimating GO FeSi laminations’ magnetic behavior under these elevated operational frequencies. The proposed model combines the Maxwell diffusion equation and a material law derived from a fractional differential equation, capturing the viscoelastic characteristics of the magnetization process. Remarkably, the model’s dynamical contribution, characterized by only two parameters, achieves a notable 4.8% Euclidean relative distance error across the frequency spectrum from 50 Hz to 1 kHz. The paper’s initial section offers an exhaustive description of the model, featuring comprehensive comparisons between simulated and measured data. Subsequently, a methodology is presented for the localized segregation of magnetic losses into three conventional categories: hysteresis, classical, and excess, delineated across various tested frequencies. Further leveraging the model’s predictive capabilities, the study extends to investigating the very high-frequency regime, elucidating the spatial distribution of loss contributions. The application of proportional–iterative learning control facilitates the model’s adaptation to standard characterization conditions, employing sinusoidal imposed flux density. The paper deliberates on the implications of GO FeSi behavior under extreme operational conditions, offering insights and reflections essential for understanding and optimizing magnetic core performance in high-frequency applications.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030175
Authors: Neda Bauman Jelena Srbljanović Ivana Čolović Čalovski Olivera Lijeskić Vladimir Ćirković Jelena Trajković Branko Bobić Andjelija Ž. Ilić Tijana Štajner
Toxoplasma gondii is an obligate intracellular parasite existing in three infectious life stages—tachyzoites, bradyzoites, and sporozoites. Rupture of tissue cysts and re-conversion of bradyzoites to tachyzoites leads to reactivated toxoplasmosis (RT) in an immunocompromised host. The aim of this study was to apply ImageJ software for analysis of T. gondii brain cysts obtained from a newly established in vivo model of RT. Mice chronically infected with T. gondii (BGD1 and BGD26 strains) were treated with cyclophosphamide and hydrocortisone (experimental group—EG) or left untreated as infection controls (ICs). RT in mice was confirmed by qPCR (PCR+); mice remaining chronically infected were PCR−. A total of 90 images of cysts were analyzed for fractal dimension (FD), lacunarity (L), diameter (D), circularity (C), and packing density (PD). Circularity was significantly higher in PCR+ compared to IC mice (p < 0.05 for BGD1, p < 0.001 for the BGD26 strain). A significant negative correlation between D and PD was observed only in IC for the BGD1 strain (ρ = −0.384, p = 0.048), while fractal parameters were stable. Significantly higher D, C, and PD and lower lacunarity, L, were noticed in the BGD1 compared to the more aggressive BGD26 strain. In conclusion, these results demonstrate the complexity of structural alterations of T. gondii cysts in an immunocompromised host and emphasize the application potential of ImageJ in the experimental models of toxoplasmosis.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030174
Authors: Shanyuan Qin Jidong Yang Ning Qin Jianping Huang Kun Tian
In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse isotropic (TTI) media. However, it involves a fractional pseudo-differential operator that depends on the anisotropy parameters, making it unsuitable for resolution using conventional solvers for fractional operators. To address this issue, we propose a novel pure quasi-P-wave equation with a generalized fractional convolution operator in TTI media. First, we decompose the conventional pure quasi-P-wave equation into an elliptical anisotropy equation and a fractional pseudo-differential correction term. Then, we use a generalized fractional convolution stencil to approximate the spatial-domain pseudo-differential term through the solution of an inverse problem. The proposed approximation method is accurate, and the wavefield modeling method based on it also accurately describes quasi-P-wave propagation in TTI media. Moreover, it only increases the computational cost for calculating mixed partial derivatives compared to those in vertical transverse isotropic (VTI) media. Finally, the proposed wavefield modeling method is utilized in RTM to correct the anisotropic effects in seismic imaging. Numerical RTM experiments demonstrate the flexibility and viability of the proposed method.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030173
Authors: Mengru Liu Lihong Zhang
This article mainly studies the double index logarithmic nonlinear fractional g-Laplacian parabolic equations with the Marchaud fractional time derivatives ∂tα. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional g-Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional g-Laplacian parabolic equations is studied.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030172
Authors: Ni Zhang Wu-Yang Zhu Peng Jin Guo Huang Yi-Fei Pu
With the rise of social media and the internet, the rapid dissemination of information has increased the likelihood of reputation infringement. This study utilizes judicial big data and AI to analyze intrinsic connections in reputation infringement cases, aiding judges in delivering consistent rulings. The challenge lies in balancing freedom of speech with the right to reputation and addressing the ambiguity and subjectivity in infringement cases. This research constructs a structured reputation infringement case dataset from Chinese Judgments Online. It introduces a Fractional Fuzzy Neural System (FFNS) to tackle the vagueness in reputation infringement acts and judicial language, enhancing prediction accuracy for case outcomes. The FFNS, integrating fractional calculus, fuzzy logic, and neural networks, excels in adaptability and nonlinear modeling. It uses fractional order fuzzy membership functions to depict the extent and severity of reputation infringement accurately, combining these outputs with neural networks for predictive analysis. The result is a more precise adjudication tool, demonstrating significant potential for judicial application.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030171
Authors: Zheng-Xin Wang Yue-Ting Li Ling-Fei Gao
The quantile regression technique is introduced into the Lotka–Volterra ecosystem analysis framework. The quantile grey Lotka–Volterra model is established to reveal the dynamic trade relationship between China and the United States. An optimisation model is constructed to solve optimum quantile parameters. The empirical results show that the quantile grey Lotka–Volterra model shows higher fitting accuracy and reveals the trade relationships at different quantiles based on quarterly data on China–US trade from 1999 to 2019. The long-term China–US trade relationship presents a prominent predator–prey relationship because exports from China to the US inhibited China’s imports from the United States. Moreover, we divide samples into five stages according to four key events, China’s accession to the WTO, the 2008 global financial crisis, the weak global economic recovery in 2015, and the 2018 China–US trade war, recognising various characteristics at different stages.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030170
Authors: Yuquan Chen Zhenlong Wu Yixiang Lu Yangquan Chen Yong Wang
In this paper, we mainly consider two kinds of perturbed accelerated gradient descents driven by Lévy perturbations, which is of great importance for enhancing the global search ability. By using Lévy representation, Lévy perturbations can be divided into two parts: small jumps and large jumps, whose properties are then carefully discussed. By introducing the concept of attraction domain for local minima, Makovian transition properties are proven for the proposed two perturbed accelerated gradient descents with different infinitesimal matrices. Finally, all the results are extended to the vector case and two simulation examples are provided to validate all the conclusions.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030169
Authors: Jianhong Guo Che-Jung Chang Yingyi Huang
Electronic commerce plays a vital role in the digital age, and the creation of a good electronic commerce ecosystem is crucial to maintaining economic growth. The electronic commerce service industry is a leading indicator of electronic commerce development, and its possible changes imply the future trends and innovation directions of the electronic commerce industry. An accurate grasp of the possible future revenue scale of the electronic commerce service industry can provide decision-making information for government policy formulation. Electronic commerce companies must formulate operational plans based on the latest information to determine strategic directions that are reasonable and consistent with the actual situation. Although there exist many prediction methods, they often fail to produce ideal results when the number of observations is insufficient. The fractional-order grey model is a common method used to deal with small data set prediction problems. This study therefore proposes a new modeling procedure for the fractional-order grey model to predict the revenue scale of China’s electronic commerce service industry. The results of experiments demonstrate that the proposed procedure can yield robust outputs under the condition of small data sets to reduce decision-making risks. Therefore, it can be regarded as a practical small data set analysis tool for managers.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030168
Authors: Xiaofeng Xu Jiangang Lu Jinshui Chen
Iterative learning control is widely applied to address the tracking problem of dynamic systems. Although this strategy can be applied to fractional order systems, most existing studies neglected the impact of the system initialization on operation repeatability, which is a critical issue since memory effect is inherent for fractional operators. In response to the above deficiencies, this paper derives robust convergence conditions for iterative learning control under non-repetitive initialization functions, where the bound of the final tracking error depends on the shift degree of the initialization function. Model nonlinearity, initial error, and channel noises are also discussed in the derivation. On this basis, a novel initialization learning strategy is proposed to obtain perfect tracking performance and desired initialization trajectory simultaneously, providing a new approach for fractional order system design. Finally, two numerical examples are presented to illustrate the theoretical results and their potential applications.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030167
Authors: Changsheng Lu Xixin Wang Shuwei Ma Shaohua Li Ting Xue Qiangqiang Li
The evaluation of shale reservoir quality is of great significance for the exploration and development of shale oil. To more effectively study the distribution characteristics of shale reservoir quality, thin-section observation, scanning electron microscopy and pressure-controlled porosimetry were used to obtain the pore structure characteristics of shale in Chang 7, including pore types, pore size distribution, etc. In addition, the fractal dimensions of the shale samples were calculated based on pressure-controlled porosimetry data. The results show that residual interparticle pores, dissolution pores and clay-dominated pores were the main pore types. The overall pore size was mainly distributed between 3 nm and 50 μm. The pore system was divided into four types using fractal features, and the shale reservoir was divided into four types based on the proportion of different types of pore system. In different types of reservoirs, the production capacity of exploration wells varies significantly, as does the production capacity of horizontal wells. The classification of shale reservoirs using mercury intrusion fractal analysis proved to be suited for the efficient development of Chang 7 shale oil reservoirs.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030166
Authors: James Abah Ugboh Joseph Oboyi Mfon Okon Udo Hossam A. Nabwey Austine Efut Ofem Ojen Kumar Narain
In this paper, we consider a faster iterative method for approximating the fixed points of generalized α-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030165
Authors: Adel Salim Tayyah Waggas Galib Atshan
This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030164
Authors: Haiyu Liu Shujuan Lü
In this paper, we numerically solve the nonlinear time-fractional diffusion equation of distributed order on an unbounded domain with a weak singularity. A fully discrete implicit scheme is developed based on the L1 formula on graded meshes in time and the Galerkin spectral method using the Laguerre function in space. We obtained an α-robust discrete Gronwall inequality and the a priori error estimation of the numerical solution. Then, the existence and uniqueness of the numerical solution are discussed. Next, we present the α-robust stability and convergence of the fully discrete scheme, where the convergence was obtained based on the regularity conditions of the exact solution. A numerical example demonstrates the validity of the theoretical results.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030163
Authors: Fengzhou Tian Yulan Wang Zhiyuan Li
The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030162
Authors: Mudassir Shams Naila Rafiq Bruno Carpentieri Nazir Ahmad Mir
In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030161
Authors: Keping Zhou Yifan Zhang Chun Yang Niange Yang Zheng Pan
To study the dynamic response of granite to different levels of microwave power, an intelligent microwave rock-breaking instrument is used to irradiate different power from three directions. The servo universal testing machine is used to carry out a uniaxial compression test on the granite after microwave damage to analyze the strength damage characteristics and the degree of pore damage. Pore fractal characteristics are analyzed based on nuclear magnetic resonance to establish the microwave damage degradation model. In parallel, the energy evolution process of granite under the influence of various power levels is analyzed using the theory of energy dissipation. Simultaneously, based on the energy dissipation theory, we analyze the energy evolution process of granite under the action of different powers. The results show that with higher microwave power, the peak strength and modulus of elasticity show a linear decreasing law. The degree of fragmentation is more obvious, showing the damage characteristics with two big ends and little in the middle. The higher the power, the greater the porosity and the more sensitive the micropore becomes to microwaves. Additionally, the damage degradation model established to evaluate the microwave damage of the rock showed that it was feasible. The higher the power, the lower the total energy, elastic energy, and dissipation energy, and the granite is gradually transformed from elastic deformation to plastic deformation. The elastic energy ratio decreases, the dissipation energy ratio increases, and the degree of damage becomes more and more serious. This study provides theoretical support for exploring the mechanical behavior and mechanism of microwave-assisted rock breaking and is of great practical significance.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030160
Authors: Changhui Wang Wencheng Li Mei Liang
In this article, an event-triggered adaptive fuzzy finite-time dynamic surface control (DSC) is presented for a class of strict-feedback nonlinear fractional-order systems (FOSs) with full-state constraints. The fuzzy logic systems (FLSs) are employed to approximate uncertain nonlinear functions in the backstepping process, the dynamic surface method is applied to overcome the inherent computational complexity from the virtual controller and its fractional-order derivative, and the barrier Lyapunov function (BLF) is used to handle the full-state constraints. By introducing the finite-time stability criteria from fractional-order Lyapunov method, it is verified that the tracking error converges to a small neighborhood near the zero and the full-state constraints are satisfied within a predetermined finite time. Moreover, reducing the communication burden can be guaranteed without the occurrence of Zeno behavior, and the example is given to demonstrate the effectiveness of the proposed controller.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030159
Authors: Ancheng Wang Lei Wang
To study the dynamic crack evolution process of loaded coal from the perspective of fractals, we carried out in situ industrial CT scanning tests of loaded coal under different confining pressures, visualizing loaded coal fracturing. Combined with fractal theory, the temporal and spatial evolution law of coal cracks is described quantitatively. The results provide two findings: (1) from the perspective of two-dimensional images and three-dimensional space, the evolution characteristics of cracks in coal under different confining pressures were basically the same in each loading stage. During the loading stages, the cracks exhibited a change rule of a slow reduction, initiation/development, rapid increase, expansion, and penetration. (2) The fractal dimension of coal was calculated by introducing fractal theory, and its change law was in good agreement with the dynamic changes of the cracks, which can explain the influence of the confining pressure on the loaded coal. The fractal dimension showed three stages: a slight decrease, a stable increase, and then a significant increase. The larger the confining pressure, the more obvious the limiting effect. Thus, our approach provides a more accurate method for evaluating the spatial and temporal evolution of cracks in loaded coal. This study can be used to predict the instability failure of loaded coal samples.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030158
Authors: Nikola Popović Soley Ersoy İbrahim İnce Ana Savić Vladimir Baltić
In this paper, we introduce a membership function used to form the fuzzy Mandelbric set and investigate the structural effects of additive and multiplicative dynamic noises on it. The newly defined membership function of this fuzzy set and its perturbations is a generalization of the indicator function for the classical Mandelbric set. We present an algorithm for detecting each complex number’s fuzzy membership degree. Through the use of the membership degrees of each complex number and experimental mathematics based on the visualizations of a variety of versions by utilizing computer-aided design, we gain a deep foresight for the structure characteristics of the additive and multiplicative perturbed fuzzy Mandelbric sets. Our novel approach allows us to identify the symmetry states of the Mandelbric set and its perturbations by the membership degrees of complex numbers, unlike the existing methods described in the literature.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030157
Authors: Likun Li Liyu Jiang Wenzhang Tu Liquan Jiang Ruhan He
Car-like mobile robots (CLMRs) are extensively utilized in various intricate scenarios owing to their exceptional maneuverability, stability, and adaptability, in which path planning is an important technical basis for their autonomous navigation. However, path planning methods are prone to inefficiently generate unsmooth paths in narrow and large-size scenes, especially considering the chassis model complexity of CLMRs with suspension. To this end, instead of traditional path planning based on an integer order model, this paper proposes fractional-order enhanced path planning using an improved Ant Colony Optimization (ACO) for CLMRs with suspension, which can obtain smooth and efficient paths in narrow and large-size scenes. On one hand, to improve the accuracy of the kinematic model construction of CLMRs with suspension, an accurate fractional-order-based kinematic modelling method is proposed, which considers the dynamic adjustment of the angle constraints. On the other hand, an improved ACO-based path planning method using fractional-order models is introduced by adopting a global multifactorial heuristic function with dynamic angle constraints, adaptive pheromone adjustment, and fractional-order state-transfer models, which avoids easily falling into a local optimum and unsmooth problem in a narrow space while increasing the search speed and success rate in large-scale scenes. Finally, the proposed method’s effectiveness is validated in both large-scale and narrow scenes, confirming its capability to handle various challenging scenarios.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030156
Authors: Hristo Kiskinov Mariyan Milev Milena Petkova Andrey Zahariev
In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030155
Authors: Xiliang He Yu Wang Tianzeng Li Rong Kang Yu Zhao
The synchronization of complex networks, as an important and captivating dynamic phenomenon, has been investigated across diverse domains ranging from social activities to ecosystems and power systems. Furthermore, the synchronization of networks proves instrumental in solving engineering quandaries, such as cryptography and image encryption. And finite-time synchronization (FTS) controls exhibit substantial resistance to interference, accelerating network convergence speed and heightening control efficiency. In this paper, finite-time synchronization (FTS) is investigated for a class of fractional-order nonidentical complex networks under uncertain parameters (FONCNUPs). Firstly, some new FTS criteria for FONCNUPs are proposed based on Lyapunov theory and fractional calculus theory. Then, the new controller is designed based on inequality theory. Compared to the general controller, it controls all nodes and adds additional control to some of them. When compared to other controllers, it has lower control costs and higher efficiency. Finally, a numerical example is presented to validate the effectiveness and rationality of the obtained results.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030154
Authors: Wen-Biao Gao
In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between the DOCLCT and the three-dimensional (3-D) discrete linear canonical transform (DLCT) is obtained. Moreover, based on a new convolution operator, we derive the convolution theorem of the DOCLCT. Finally, the correlation theorem of the DOCLCT is established.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030153
Authors: Sumin Bark Junghyeon Kim Minjae Lee Sungjoon Lim
In this paper, we propose an optically transparent dual-band metamaterial absorber (MMA) that uses Ag nanowire screen-printed fractal structures. The proposed MMA exhibits near-perfect absorption in the C- and K-bands. This dual-band absorption property is achieved through two inductive–capacitive (L-C) resonances located at 6.45 and 21.14 GHz, which are generated by the second-order fractal structures. We analyzed the microwave absorbing mechanisms through the distributions of the surface current and electromagnetic field on the top and bottom layers. The MMA demonstrates an optical transmittance of 63.1% at a wavelength of 550 nm. This high optical transmittance is attained by screen printing transparent Ag nanowire ink onto a transparent PET substrate. Since screen printing is a simple and low-cost fabrication method, the proposed MMA offers the advantages of being low cost while having the properties of optical transparency and effective dual-band absorption. Consequently, it holds great potential for the radar stealth application of C- and K-bands in that it can be attached to the windows of stealth aircraft due to its optical transparency and dual-band near-perfect absorption property.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030152
Authors: Jorge Luis Flores Alarcón Carlos Gabriel Figueroa Víctor Hugo Jacobo Fernando Velázquez Villegas Rafael Schouwenaars
The simulation and characterisation of randomly rough surfaces is an important topic in surface science, tribology, geo- and planetary sciences, image analysis and optics. Extensions to general random processes with two continuous variables are straightforward. Several surface generation algorithms are available, and preference for one or another method often depends on the specific scientific field. The same holds for the methods to estimate the fractal dimension D. This work analyses six algorithms for the determination of D as a function of the size of the domain, variance, and the input value for D, using surfaces generated by Fourier filtering techniques and the random midpoint displacement algorithm. Several of the methods to determine fractal dimension are needlessly complex and severely biased, whereas simple and computationally efficient methods produce better results. A fine-tuned analysis of the power spectral density is very precise and shows how the different surface generation algorithms deviate from ideal fractal behaviour. For large datasets defined on equidistant two-dimensional grids, it is clearly the most sensitive and precise method to determine fractal dimension.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030151
Authors: Zhimo Jian Gang Peng Chaoqian Luo Tianyi Zhou Yajun Yin
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+α2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030150
Authors: Xiaofeng Wang Shaonan Guo
In this paper, a family of fifth-order Chebyshev–Halley-type iterative methods with one parameter is presented. The convergence order of the new iterative method is analyzed. By obtaining rational operators associated with iterative methods, the stability of the iterative method is studied by using fractal theory. In addition, some strange fixed points and critical points are obtained. By using the parameter space related to the critical points, some parameters with good stability are obtained. The dynamic plane corresponding to these parameters is plotted, visualizing the stability characteristics. Finally, the fractal diagrams of several iterative methods on different polynomials are compared. Both numerical results and fractal graphs show that the new iterative method has good convergence and stability when α=12.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030149
Authors: Bhukya Ramadevi Venkata Ramana Kasi Kishore Bingi
Efficient integration of wind energy requires accurate wind power forecasting. This prediction is critical in optimising grid operation, energy trading, and effectively harnessing renewable resources. However, the wind’s complex and variable nature poses considerable challenges to achieving accurate forecasts. In this context, the accuracy of wind parameter forecasts, including wind speed and direction, is essential to enhancing the precision of wind power predictions. The presence of missing data in these parameters further complicates the forecasting process. These missing values could result from sensor malfunctions, communication issues, or other technical constraints. Addressing this issue is essential to ensuring the reliability of wind power predictions and the stability of the power grid. This paper proposes a long short-term memory (LSTM) model to forecast missing wind speed and direction data to tackle these issues. A fractional-order neural network (FONN) with a fractional arctan activation function is also developed to enhance generated wind power prediction. The predictive efficacy of the FONN model is demonstrated through two comprehensive case studies. In the first case, wind direction and forecast wind speed data are used, while in the second case, wind speed and forecast wind direction data are used for predicting power. The proposed hybrid neural network model improves wind power forecasting accuracy and addresses data gaps. The model’s performance is measured using mean errors and R2 values.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030148
Authors: Leila Gholizadeh Zivlaei Angelo B. Mingarelli
In this paper, we provide existence and uniqueness results for the initial value problems associated with mixed Riemann–Liouville/Caputo differential equations in the real domain. We show that, under appropriate conditions in a fractional order, solutions are always square-integrable on the finite interval under consideration. The results are valid for equations that have sign-indefinite leading terms and measurable coefficients. Existence and uniqueness theorem results are also provided for two-point boundary value problems in a closed interval.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030147
Authors: Yi Lu Xiru Wu Yaonan Wang Lihong Huang Qingjin Wei
This paper investigates the H∞ consensus problem of discrete-time Markov jump fractional-order multiagent systems (DTMJFOMASs) under denial-of-service (DoS) attacks. By applying the short-memory principle, we can obtain discrete-time Markov jump multiagent systems with partially unknown probabilities. A novel quantized event-triggering mechanism (QETM), based on a mode-dependent logarithmic quantizer, is proposed to enhance transmission efficiency among multiagents. A distributed controller with quantized output is developed. Sufficient conditions are provided to ensure the system achieves H∞ consensus through Lyapunov stability theory. Finally, two examples are given to verify the effectiveness of the proposed model.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030146
Authors: Wenhao Xu Jing Ba Jianxiong Cao Cong Luo
The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030145
Authors: Xiaoqiang He Yuxin Song Fengmin Yu Huiming Duan
In recent years, global attention to carbon emissions has increased, becoming one of the main drivers of global climate change. Accurate prediction of carbon emission trends in small and medium-sized countries and scientific regulation of carbon emissions can provide theoretical support and policy references for the effective and rational use of energy and the promotion of the coordinated development of energy, environment, and economy. This paper establishes a grey prediction model using the classical Logistic mathematical model in a determined environment to investigate the carbon emission system. At the same time, we use the basic principle of fractional-order accumulation to establish a grey prediction model with fractional-order Logistic and obtain the parameter estimation and time-response equation of the new model by solving the model through the theory related to fractional-order operators. The particle swarm optimization algorithm is used to complete the optimization process of the order of the fractional order grey prediction model and obtain the optimal model order. Then, the new model is applied to predict carbon emissions in five medium-emission countries: Ethiopia, Djibouti, Ghana, Belgium, and Austria. The new model shows better advantages in the validity analysis process, and the simulation results indicate that the new model proposed in this paper has stronger stability and better simulation and prediction accuracy than other comparative models, proving the model’s validity. Finally, the model is used to forecast the carbon emissions of these five countries for the five years of 2021–2025, and the results are analyzed, and relevant policy recommendations are made.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030144
Authors: Zainab Alsheekhhussain Ahmad Gamal Ibrahim Mohammed Mossa Al-Sawalha Yousef Jawarneh
In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ-Hilfer fractional derivative, D0,tσ,v,ψ,w,of order μ∈(1,2), in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D0,tσ,v,ψ,w of order μ ∈(1,2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D0,tσ,v,ψ,w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030143
Authors: Enrique C. Gabrick Paulo R. Protachevicz Diogo L. M. Souza José Trobia Elaheh Sayari Fernando S. Borges Marcelo K. Lenzi Iberê L. Caldas Antonio M. Batista Ervin K. Lenzi
We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u(0,t)−2, our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher’s equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030142
Authors: Baofeng Lan Ruidong Yang Zhonghu Wu Haishen Jiang Xinzheng Li
To better understand the influence of different levels of two-way stress differences on the development of damage in Anchang diametral laminar shale in the northern Qianbei area, a numerical model of laminar shale with a representative fine-scale structure was established by using RFPA3D-CT. A triaxial compression test was conducted; a three-dimensional mesoscale fracture box dimension algorithm based on digital images was generated by using MATLAB R2020b; and the fractal characteristics were quantitatively analyzed. The results showed that under the influence of the horizontal stress ratio and two-way stress, the greater the two-way stress is, the more notable the plastic characteristics of specimen damage are, and the higher the residual strength is. The specimens with lower two-way stress exhibited obvious brittle damage characteristics. The difficulty degree of complex fracture network formation increased with the increase in the horizontal tension ratio, and the degree of increase in the fracture network complexity gradually decreased. At a horizontal stress ratio of 1.25, the fractal dimension was the highest, which indicates that the cracks were the most pronounced. Fracture formation after specimen damage was the most common phenomenon. Under the condition of a lower horizontal stress ratio, a large number of fracture structures could be generated in shale specimens after damage, promoting the expansion of natural fractures.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030141
Authors: Yong Li Xiaodong Zhang Yijuan Sun Zhen Wang Shuo Zhang Binghui Li
CO2 injection in coal seams, which is a significant initiative to mitigate environmental problems caused by greenhouse gases, often leads a sequence of changes in the physical properties of coal reservoirs. To look into how the pore structure changes in the process of CO2 sequestration, we selected fresh coal from Huoerxinhe coal mine in China as the object. Then, acid treatment and SC-CO2 extraction were used to dissolve Organic and inorganic components in coal. Thus, by using SEM, LTGA-N2 apparatus and XRD, the characteristics of pore parameter and fractal dimension variation were discussed. The research results show that, the APS of samples THF, HCL-HF and Y-C increase, while the total PV decreases and the pore connectivity deteriorates. The pore connectivity of Samples THF and HCL-HF is improved (THF-C, HCL-HF-C), but the total pore volume continuously reduces. In addition, solvents treatment and SC-CO2 extraction mainly act on the microporous fraction. After solvents pretreatment, the changes in the pore size distribution curves are mainly manifested in the reduction of number of micropores, especially in the micropores around 3–4 nm. There is a small increase in micropores for samples Y-C and HCL-HF-C, with the pore size mainly concentrated around 4 nm, while the pores of the sample THF-C mainly show an increase within the scope of 3–16 nm. Generally, solvent pretreatment and SC-CO2 extraction help to simplify pore structure. However, the sample HCL-HF-C shows opposite change characteristics. In a short period of time, the larger pore fractal dimension, the less beneficial it is to the flow of CO2, while pore fractal dimension becomes progressively less useful in assessing pore connectivity with increasing time.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030140
Authors: Abdallah Aldosary
Power quality (PQ) is a major issue in today’s electrical system that affects both utilities and customers. The proliferation of power electronics devices, smart grid technology, and renewable energy sources (RES) have all contributed to the emergence of PQ concerns in today’s power system. The Unified Power Quality Conditioner (UPQC) is a versatile tool that can be used to fix distribution grid issues caused by irregular voltage, current, or frequency. Several tuning parameters, however, restrict the effectiveness of the Fractional-Order Proportional Integral Derivative (FOPID) control technique, which is proposed to improve UPQC performance. To move beyond these restrictions and find the optimal solution for the FOPID controller problem, a hybrid optimization strategy called the Hybrid Jellyfish Search Optimizer and Particle Swarm Optimizer (HJSPSO) is employed. To meet the load requirement during PQ issue periods, the suggested model incorporates a renewable energy source into the grid system. Whether the load is linear or non-linear, the design maintains PQ problems to a minimum. Furthermore, the FOPID control technique is compared with other controllers. Results show that grid-connected RES systems using the proposed FOPID control approach for UPQC have fewer PQ problems. The presented UPQC with HJSPSO strategy significantly outperformed, with the shortest computing time of 127.474 s and an objective function value of 1.423.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030139
Authors: Abdallah Aldosary
This paper is dedicated to optimizing the functionality of Microgrid-Integrated Charging Stations (MICCS) through the implementation of a new control strategy, specifically the fractional-order proportional-integral (FPI) controller, aided by a hybrid optimization algorithm. The primary goal is to elevate the efficiency and stability of the MICCS-integrated inverter, ensuring its seamless integration into modern energy ecosystems. The MICCS system considered here comprises a PV array as the primary electrical power source, complemented by a proton exchange membrane fuel cell as a supporting power resource. Additionally, it includes a battery system and an electric vehicle charging station. The optimization model is formulated with the objective of minimizing the integral of square errors in both the DC-link voltage and grid current while also reducing total harmonic distortion. To enhance the precision of control parameter estimation, a hybrid of the one-to-one optimizer and sine cosine algorithm (HOOBSCA) is introduced. This hybrid approach improves the exploitation and exploration characteristics of individual algorithms. Different meta-heuristic algorithms are tested against HOOBSCA in different case studies to see how well it tunes FPI settings. Findings demonstrate that the suggested method improves the integrated inverters’ transient and steady-state performance, confirming its improved performance in generating high-quality solutions. The best fitness value achieved by the proposed optimizer was 3.9109, outperforming the other algorithms investigated in this paper. The HOOBSCA-based FPI successfully improved the response of the DC-link voltage, with a maximum overshooting not exceeding 8.5% compared to the other algorithms employed in this study.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030138
Authors: Chenming Zhang Xiaoying Ping Qindong Fan Chunlin Li
Urban morphology has been empirically demonstrated to be self-organized and can be quantified by fractal dimension. However, the spatial variation rule of fractal features at the sub-zone scale has yet to be uncovered, as well as the relationship between fractal dimension values and road network or land-use patterns. In this study, the urban area is partitioned into 158 grid units, with subsequent calculations conducted to determine the fractal dimensions (using 2D box-counting and 3D voxel-counting methods), road network characteristics, and land-use patterns within each individual unit. The pattern of how architectures fill into the 2D or 3D embedding space at the grid level is revealed. Moreover, the spatial relationship between the road network, land-use, and their impacts on the local architectural layout is elucidated by employing MGWR, a model that incorporates the principles of fitting localized spatial regression. The results are as follows: (1) urban morphology follows fractal laws at a sub-zone scale, both in a 2D plane and 3D volume; (2) the filling degree of architecture is high in the urban center but low in the periphery areas; (3) the selected variables fit well with the regression models; (4) there is spatial heterogeneity regarding the influence of each factor. The research findings provide valuable insights into the theoretical relationship between urban morphology and the composite structure of road networks and land use. This facilitates identifying crucial areas and priority directions for urban renewal construction, as well as optimizing architectural design to improve efficiency and functionality.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030137
Authors: Ju Hyoung Lee Hyun-Cheol Kim
Fractals are widely recognized as one of the best geometric models to depict soil roughness on various scales from tillage to micro-topography smaller than radar wavelength. However, most fractal approaches require an additional geometric description of experimental sites to be analysed by existing radiative transfer models. For example, fractal dimension or spectral parameter is often related to root-mean-square (RMS) height to be characterized as the microwave surface. However, field measurements hardly represent multi-scale roughness. In this study, we rescaled Power Spectral Density with Synthetic Aperture Radar (SAR)-inverted rms height, and estimated non-stationary fractal roughness to accommodate multi-scale roughness into a radiative transfer model structure. As a result, soil moisture was retrieved over the Yanco site in Australia. Local validation shows that the Integral Equation Model (IEM) poorly simulated backscatters using inverted roughness as compared to fractal roughness even in anisotropic conditions. This is considered due to a violation of time-invariance assumption used for inversion. Spatial analysis also shows that multi-scale fractal roughness better illustrated the hydrologically reasonable backscattering partitioning, as compared to inverted roughness. Fractal roughness showed a greater contribution of roughness to backscattering in dry conditions. Differences between IEM backscattering and measurement were lower, even when the isotropic assumption of the fractal model was violated. In wet conditions, the contribution of soil moisture to backscattering was shown more clearly by fractal roughness. These results suggest that the multi-scale fractal roughness can be better adapted to the IEM even in anisotropic conditions than the inversion to assume time-invariance of roughness.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030136
Authors: Yu Wang Tianzeng Li Yu Zhao
The finite difference method is used to solve a new class of unsteady generalized Maxwell fluid models with multi-term time-fractional derivatives. The fractional order range of the Maxwell model index is from 0 to 2, which is hard to approximate with general methods. In this paper, we propose a new finite difference scheme to solve such problems. Based on the discrete H1 norm, the stability and convergence of the considered discrete scheme are discussed. We also prove that the accuracy of the method proposed in this paper is O(τ+h2). Finally, some numerical examples are provided to further demonstrate the superiority of this method through comparative analysis with other algorithms.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030135
Authors: Xinzhi Ren Lili Liu Tianran Zhang Xianning Liu
To provide insights into the spreading speed and propagation dynamics of viruses within a host, in this paper, we investigate the traveling wave solutions and minimal wave speed for a degenerate viral infection dynamical model with a nonlocal dispersal operator and saturated incidence rate. It is found that the minimal wave speed c∗ is the threshold that determines the existence of traveling wave solutions. The existence of traveling fronts connecting a virus-free steady state and a positive steady state with wave speed c≥c∗ is established by using Schauder’s fixed-point theorem, limiting arguments, and the Lyapunov functional. The nonexistence of traveling fronts for c<c∗ is proven by the Laplace transform. In particular, the lower-bound estimation of the traveling wave solutions is provided by adopting a rescaling method and the comparison principle, which is a crucial prerequisite for demonstrating that the traveling semifronts connect to the positive steady state at positive infinity by using the Lyapunov method and is a challenge for some nonlocal models. Moreover, simulations show that the asymptotic spreading speed may be larger than the minimal wave speed and the spread of the virus may be postponed if the diffusion ability or diffusion radius decreases. The spreading speed may be underestimated or overestimated if local dispersal is adopted.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030134
Authors: Tao Li Xiaoting Wu Zhuhui Luo Yanan Chen Caichun He Rongjun Ding Changfan Zhang Jun Yang
A bearing fault is one of the major causes of rotating machinery faults. However, in real industrial scenarios, the harsh and complex environment makes it very difficult to collect sufficient fault data. Due to this limitation, most of the current methods cannot accurately identify the fault type in cases with limited data, so timely maintenance cannot be conducted. In order to solve this problem, a bearing fault diagnosis method based on the fractional order Siamese deep residual shrinkage network (FO-SDRSN) is proposed in this paper. After data collection, all kinds of vibration data are first converted into two-dimensional time series feature maps, and these feature maps are divided into the same or different types of fault sample pairs. Then, a Siamese network based on the deep residual shrinkage network (DRSN) is used to extract the features of the fault sample pairs, and the fault type is determined according to the features. After that, the contrastive loss function and diagnostic loss function of the sample pairs are combined, and the network parameters are continuously optimized using the fractional order momentum gradient descent method to reduce the loss function. This improves the accuracy of fault diagnosis with a small sample training dataset. Finally, four small sample datasets are used to verify the effectiveness of the proposed method. The results show that the FO-SDRSN method is superior to other advanced methods in terms of training accuracy and stability under small sample conditions.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030133
Authors: Xiaofeng Wang Wenshuo Li
In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω-continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030132
Authors: Abdullah M. Noman Mokhtar Aly Mohammed H. Alqahtani Sulaiman Z. Almutairi Ali S. Aljumah Mohamed Ebeed Emad A. Mohamed
An important issue in interconnected microgrids (MGs) is the realization of balance between the generation side and the demand side. Imbalanced generation and load demands lead to security, power quality, and reliability issues. The load frequency control (LFC) is accountable for regulating MG frequency against generation/load disturbances. This paper proposed an optimized fractional order (FO) LFC scheme with cascaded outer and inner control loops. The proposed controller is based on a cascaded one plus tilt derivative (1+TD) in the outer loop and an FO tilt integrator-derivative with a filter (FOTIDF) in the inner loop, forming the cascaded (1+TD/FOTIDF) controller. The proposed 1+TD/FOTIDF achieves better disturbance rejection compared with traditional LFC methods. The proposed 1+TD/FOTIDF scheme is optimally designed using a modified version of the liver cancer optimization algorithm (MLCA). In this paper, a new modified liver cancer optimization algorithm (MLCA) is proposed to overcome the shortcomings of the standard Liver cancer optimization algorithm (LCA), which contains the early convergence to local optima and the debility of its exploration process. The proposed MLCA is based on three improvement mechanisms, including chaotic mutation (CM), quasi-oppositional based learning (QOBL), and the fitness distance balance (FDB). The proposed MLCA method simultaneously adjusts and selects the best 1+TD/FOTIDF parameters to achieve the best control performance of MGs. Obtained results are compared to other designed FOTID, TI/FOTID, and TD/FOTID controllers. Moreover, the contribution of electric vehicles and the high penetration of renewables are considered with power system parameter uncertainty to test the stability of the proposed 1+TD/FOTIDF LFC technique. The obtained results under different possible load/generation disturbance scenarios confirm a superior response and improved performance of the proposed 1+TD/FOTIDF and the proposed MLCA-based optimized LFC controller.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030131
Authors: Kadhavoor R. Karthikeyan Gangadharan Murugusundaramoorthy
Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030130
Authors: Feng Xiong Wentao Huang
In this paper, we investigate the existence of infinitely many small solutions for problem (fφp) involving φp-Laplacian by exploiting critical point theory. Moreover, the present study first attempts to address discrete Dirichlet problems with φp-Laplacian in relation to some relative existing references. As far as we know, this research of the partial discrete bvp involves φp-Laplacian for the first time. Our results are illustrated with three examples.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030129
Authors: Günter Leugering Vaibhav Mehandiratta Mani Mehra
We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030128
Authors: Lin Wang Shijiao Liu Shuning Liang Xuelian Liu Chunyang Wang
The performance of laser beams in tracking Lissajous scan trajectories is severely limited by beam jitter. To enhance the performance of fast steering mirror (FSM) control in tracking Lissajous scan trajectories, this paper proposed a fractional order active disturbance rejection controller (FOADRC) and verified its effectiveness in improving system scanning tracking accuracy. A dynamic mathematical model of a fast steering mirror was studied, and the design of parameters for the control mode of the closed-loop system was determined. A reduced-order linear active disturbance rejection controller suitable for FSM systems was designed, and the corresponding fractional-order proportional differentiation (FOPD) controller was determined according to the mathematical model. The use of the designed controller enabled high-performance tracking of high-frequency Lissajous scanning curves (X-axis 500 Hz, Y-axis 350 Hz) and met the need for high-frequency repetitive scanning. The controller has the characteristics of simple implementation and low computational complexity and is suitable for closed-loop control applications in engineering.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030127
Authors: Xinyi Xie Fei Gao
We investigated the blow-up of the weak solution to a class of fractional nonlinear stochastic differential equations driven by multiplicative noise in this paper. The a priori estimates and Galerkin method were applied to demonstrate the existence and uniqueness of the weak solution. Underlying the hypotheses of the nonlinear function and the initial data, for finite time, we prove that the solution does not blow up. Additionally, under further assumptions, we verified that the presence of multiplicative noise can delay the blow-up of the solution to infinity.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030126
Authors: Ping Wang Xi Chen Yunning Zhang Lei Zhang Yuehua Huang
Modern power systems are confronted with widespread concern on the frequency stability issue due to the widespread integration of randomly fluctuating renewable resources. To address the above concern, this work introduces a load-frequency-control (LFC) scheme based on a parameter tuning strategy for fractional-order proportional–integral–derivative (FOPID) controller. Firstly, a two-area interconnected power system (IPS) model, including thermal, hydro, solar, wind, and gas power generator and a hydrogen-based energy-storage unit, is established. Then, a FOPID controller is designed for this IPS model, and an improved gradient-based optimizer (IGBO) is developed to adaptively regulate the parameters of the FOPID controllers. Finally, the effectiveness of the offered LFC scheme is tested through load disturbance and renewable energy fluctuations test scenarios and provides a comparison and robustness analysis among different schemes. The test results validated that the offered LFC scheme can effectively suppress the frequency fluctuations of the IPS and has excellent robustness.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030125
Authors: Tareq Saeed Eze R. Nwaeze Muhammad Bilal Khan Khalil Hadi Hakami
In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right ℏ-convexity (LR-ℏ-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated ℏ-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes.
]]>Fractal and Fractional doi: 10.3390/fractalfract8030124
Authors: Jian-Gen Liu Yi-Ying Feng
In this article, we analyzed the time fractional higher-dimensional nonlinear modified model of wave propagation, namely the (3 + 1)-dimensional Benjamin–Bona–Mahony-type equation. The fractional sense was defined by the classical Riemann–Liouville fractional derivative. We derived firstly the existence of symmetry of the time fractional higher-dimensional equation. Next, we constructed the one-dimensional optimal system to the time fractional higher-dimensional nonlinear modified model of wave propagation. Subsequently, it was reduced into the lower-dimensional fractional differential equation. Meanwhile, on the basis of the reduced equation, we obtained its similarity solution. Through a series of analyses of the time fractional high-dimensional model and the results of the above obtained, we can gain a further understanding of its essence.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020123
Authors: Alaa Altassan Muhammad Imran
In this article, we compute the irregularity measures of generalized Sierpiński graphs and obtain some bounds on these irregularities. Moreover, we discuss some bounds on connectivity indices for generalized Sierpiński graphs of any arbitrary graph H along with classification of the extremal graphs used to attain them.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020122
Authors: Xiaomeng Zhang Shuo Zhang Furui Xiong Lu Liu Lichuan Zhang Xuan Han Heng Wang Yanzhu Zhang Ranzhen Ren
The vibration of piping systems is one of the most important causes of accelerated equipment wear and reduced work efficiency and safety. In this study, an active vibration control method based on a fractional-order proportional–integral–derivative (PID) controller was proposed to suppress pipeline vibration and reduce pipeline damage. First, a mathematical model of the distributed piping system was established using the finite element analysis method, and the characteristics of the distributed piping system were studied effectively. Further, the time-frequency domain parameter identification method was used to realise the system identification of the cross-point vibration transfer function between the brake and sensor, and the particle swarm optimisation algorithm was utilised to further optimise the transfer function parameters to improve the system identification accuracy. Therefore, a fractional-order PID controller was designed using the D-decomposition method, and the optimal controller parameters were obtained. The experimental and numerical simulation results show that the improved system identification algorithm can significantly improve modelling accuracy. In addition, the designed fractional-order PID controller can effectively reduce the system’s overshoot, oscillation time, and adjustment time, thereby reducing the vibration response of piping systems.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020121
Authors: Xiaofeng Li Yaokun Li
The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero very slowly with a negative power law. The corresponding non-zero ensemble mean during the integration period can exert a pronounced influence on the ensemble-mean dynamics of the RO model. The state-dependent noise, also called the multiplicative noise, can present its influence by reducing the relaxation coefficient and by introducing periodic external forcing. The decreasing relaxation coefficient can enhance the oscillation amplitude and shorten the oscillation period. The forced frequency is close to the natural frequency. The two mechanisms together can further amplify the amplitude and shorten the period, compared with the state-independent noise or additive noise, which only exhibits its influence by introducing non-periodic external forcing. These two mechanisms explicitly elucidate the influence of the stochastic forcing on the ensemble-mean dynamics of the RO model. It provides comprehensive knowledge to better understand the interaction between the fast-varying stochastic forcing and the slow-varying deterministic system and deserves further investigation.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020120
Authors: Yuan Ren Lei Li Weijie Wang Lifen Wang Weikun Pang
To achieve high-precision deflection control of a Magnetically Suspended Control and Sensitive Gyroscope rotor under high dynamic conditions, a deflection decoupling method using Quantum Radial Basis Function Neural Network and fractional-order terminal sliding mode control is proposed. The convergence speed and time complexity of the neural network controller limit the control accuracy and stability of rotor deflection under high-bandwidth conditions. To solve the problem, a quantum-computing-based structure optimization method for the Radial Basis Function Neural Network is proposed for the first time, where the input and the center of hidden layer basis function of the neural network are quantum-coded, and quantum rotation gates are designed to replace the Gaussian function. The parallel characteristic of quantum computing is utilized to reduce the time complexity and improve the convergence speed of the neural network. On top of that, in order to further address the issue of input jitter, a fractional-order terminal sliding mode controller based on the Quantum Radial Basis Function Neural Network is designed, the fractional-order differential sliding mode surface and the fractional-order convergence law are proposed to reduce the input jitter and achieve finite-time convergence of the controller, and the Quantum Radial Basis Function Neural Network is used to approximate the residual coupling and external disturbances of the system, resulting in improving the rotor deflection control accuracy. The semi-physical simulation experiments demonstrate the effectiveness and superiority of the proposed method.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020119
Authors: Jie He Tao Li Yi Rui
The degradation of soil bonding, which can be described by the evolution of bond degradation variables, is essential in the constitutive modeling of cemented soils. A degradation variable with a value of 0/1.0 indicates that the applied stress is completely sustained by bonded particles/unbounded grains. The discrete element method (DEM) was used for cemented soils to analyze the bond degradation evolution and to evaluate the degradation variables at the contact scale. Numerical cemented soil samples with different bonding strengths were first prepared using an advanced contact model (CM). Constant stress ratio compression, one-dimensional compression, conventional triaxial tests (CTTs), and true triaxial tests (TTTs) were then implemented for the numerical samples. After that, the numerical results were adopted to investigate the evolution of the bond degradation variables BN and B0. In the triaxial tests, B0 evolves to be near to or larger than BN due to shearing, which indicates that shearing increases the bearing rate of bond contacts. Finally, an approximate stress-path-independent bond degradation variable Bσ was developed. The evolution of Bσ with the equivalent plastic strain can be effectively described by an exponential function and a hyperbolic function.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020118
Authors: Weibiao Xie Qiuli Yin Jingbo Zeng Fan Yang Pan Zhang Binpeng Yan
Micro-pore structures are an essential factor for the electrical properties of porous rock. Theoretical electrical conductivity models considering pore structure can highly improve the accuracy of reservoir estimation. In this study, a pore structure characterization method based on a multi-fractal theory using capillary pressure is developed. Next, a theoretical electrical conductivity equation is derived based on the new pore structure characterization method. Furthermore, a distinct interrelationship between fractal dimensions of capillary pressure curves (Dv) and of resistivity index curves (Dt and Dr) is obtained. The experimental data of 7 sandstone samples verify that the fitting result by the new pore structure characterization method is highly identical to the experimental capillary pressure curves, and the accuracy of the improved rock resistivity model is higher than the Archie model. In addition, capillary pressure curves can be directly converted to resistivity index curves according to the relationship model between fractal dimensions of capillary pressure curves (Dv) and resistivity index curves (Dt and Dr). This study provides new ideas to improve the accuracy of pore structure characterization and oil saturation calculation; it has good application prospects and guiding significance in reservoir evaluation and rock physical characteristics research.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020117
Authors: Turker Acikgoz Soner Gokten Abdullah Bugra Soylu
Green bonds represent a compelling financial innovation that presents a financial perspective solution to address climate change and promote sustainable development. On the other hand, the recent process of financialisation of commodities disrupts the dynamics of the commodity market, increasing its correlation with financial markets and raising the risks associated with commodities. In this context, understanding the dynamics of the interconnectivity between green bonds and commodity markets is crucial for risk management and portfolio diversification. This study aims to reveal the multifractal cross-correlations between green bonds and commodities by employing methods from statistical physics. We apply multifractal detrended cross-correlation analysis (MFDCCA) to both return and volatility series, demonstrating that green bonds and commodities exhibit multifractal characteristics. The analysis reveals long-range power-law cross-correlations between these two markets. Specifically, volatility cross-correlations persist across various fluctuations, while return series display persistence in small fluctuations and antipersistence in large fluctuations. These findings carry significant practical implications for hedging and risk diversification purposes.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020116
Authors: Rekha Srivastava Asifa Tassaddiq Ruhaila Md Kasmani
Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020115
Authors: Qinghua Feng Bin Zheng
In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include known Kamenev and Philos-type oscillation criteria and are new oscillation results so far in the literature. Some inequalities, the Riccati transformation, the integral technique, and the theory of time scale are used in the establishment of these oscillation criteria. The proposed results unify continuous and discrete analysis, and the process of deduction is further extended to another class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term and a damping term. As applications for the established oscillation criteria, some examples are given.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020114
Authors: Melike Bildirici Özgür Ömer Ersin Blend Ibrahim
Metaverse (MV) technology introduces new tools for users each day. MV companies have a significant share in the total stock markets today, and their size is increasing. However, MV technologies are questioned as to whether they contribute to environmental pollution with their increasing energy consumption (EC). This study explores complex nonlinear contagion with tail dependence and causality between MV stocks, EC, and environmental pollution proxied with carbon dioxide emissions (CO2) with a decade-long daily dataset covering 18 May 2012–16 March 2023. The Mandelbrot–Wallis and Lo’s rescaled range (R/S) tests confirm long-term dependence and fractionality, and the largest Lyapunov exponents, Shannon and Havrda, Charvât, and Tsallis (HCT) entropy tests followed by the Kolmogorov–Sinai (KS) complexity measure confirm chaos, entropy, and complexity. The Brock, Dechert, and Scheinkman (BDS) test of independence test confirms nonlinearity, and White‘s test of heteroskedasticity of nonlinear forms and Engle’s autoregressive conditional heteroskedasticity test confirm heteroskedasticity, in addition to fractionality and chaos. In modeling, the marginal distributions are modeled with Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula (MS-GARCH–Copula) processes with two regimes for low and high volatility and asymmetric tail dependence between MV, EC, and CO2 in all regimes. The findings indicate relatively higher contagion with larger copula parameters in high-volatility regimes. Nonlinear causality is modeled under regime-switching heteroskedasticity, and the results indicate unidirectional causality from MV to EC, from MV to CO2, and from EC to CO2, in addition to bidirectional causality among MV and EC, which amplifies the effects on air pollution. The findings of this paper offer vital insights into the MV, EC, and CO2 nexus under chaos, fractionality, and nonlinearity. Important policy recommendations are generated.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020113
Authors: Guangze Gu Changyang Mu Zhipeng Yang
We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground state solutions exist for the fractional Kirchhoff equation with p=4.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020112
Authors: Xin Zhang Yu Bo Yuanfeng Jin
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and second-order convergence in time. Additionally, we provide a detailed proof for the existence and uniqueness, as well as the stability of scheme, along with a priori error estimates. Finally, we validate our theoretical results through various numerical computations.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020111
Authors: Kaihong Zhao Juqing Liu Xiaojun Lv
The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value conditions. To unify the two types of equations, we investigate a general nonlinear impulsive coupled implicit system. By cleverly constructing relevant operators involving impulsive terms, we establish the coincidence degree theory and obtain the solvability. We explore the stability of solutions using nonlinear analysis and inequality techniques. As the most direct application, we naturally obtained the solvability and stability of the Langevin and Sturm–Liouville equations mentioned above. Finally, an example is provided to demonstrate the validity and availability of our major findings.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020109
Authors: Norelys Aguila-Camacho Javier A. Gallegos
This paper presents the design and analysis of Switched Fractional Order Model Reference Adaptive Controllers (SFOMRAC) for Multiple Input Multiple Output (MIMO) linear systems with unknown parameters. The proposed controller uses adaptive laws whose derivation order switches between a fractional order and the integer order, according to a certain level of control error. The switching aims to use fractional orders when the control error is larger to improve transient response and system performance during large disturbed states, and to obtain smoother control signals, leading to a better control energy usage. Then, it switches to the integer order when the control error is smaller to improve steady state. Boundedness of all the signals in the scheme is analytically proved, as well as convergence of the control error to zero. Moreover, these properties are extended to the case when system states are affected by a bounded non-parametric disturbance. Simulation studies are carried out using different representative plants to be controlled, showing that fractional orders and switching error levels can be found in most of the cases, such as when SFOMRAC achieves a better balance among control energy and system performance than the non-switched equivalent strategies.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020110
Authors: Wei Xu Hui Liu Lijuan Chen Yongtao Zhou
Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020108
Authors: Chengtao Yang Rendong Huang Dunwen Liu Weichao Qiu Ruiping Zhang Yu Tang
To better analyze the fluctuation characteristics and development law of tunnel deformation data, multifractal theory is applied to tunnel deformation analysis. That is, the multifractal detrended fluctuation analysis (MF-DFA) model is first utilized to carry out the multifractal characterization of tunnel deformation data. Further, Mann–Kendall (M–K) analysis is utilized to construct the dual criterion (∆α indicator criterion and ∆f(α) indicator criterion) for the tunnel deformation early warning study. In addition, the particle swarm optimization long-short-term memory (PSO-LSTM) prediction model is used for predicting tunnel settlement. The results show that, in reference to the tunnel warning level criteria and based on the Z-value results of the indicator criterion, the warning level of all four sections is class II. At the same time, through the analysis of tunnel settlement predictions, the PSO-LSTM model has a better prediction effect and stability for tunnel settlement. The predicted results show a slow increase in tunnel settlement over the next 5 days. Finally, the tunnel warning level and the predicted results of tunnel settlement are analyzed in a comprehensive manner. The deformation will increase slowly in the future. Therefore, monitoring and measurement should be strengthened, and disaster preparedness plans should be prepared.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020107
Authors: Miguel Vivas-Cortez Muhammad Zakria Javed Muhammad Uzair Awan Silvestru Sever Dragomir Ahmed M. Zidan
Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties of both operators as well. Using these concepts, we deliver new variants of Young’s inequality, Hölder’s inequality, Minkowski’s inequality, Hermite–Hadamard’s inequality, Ostrowski’s inequality, and Gruss–Chebysev inequality. We report the Hermite–Hadamard’s inequalities by taking into account the differentiability of convex mappings. These fundamental results are pivotal to studying the various other problems in the field of inequalities. The validation of results is also supported with some visuals.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020106
Authors: Zihao An Chaobao Huang
This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a weak singularity at the initial time. Using the order reduction method, the fourth-order fractional diffusion-wave equation can be rewritten as a coupled system of low order, which is approximated by the nonuniform Alikhanov scheme in time and the finite difference method in space. Furthermore, the H2-norm stability result is obtained. With the help of this result and a priori bounds of the solution, an α-robust error estimate with optimal convergence order is derived. In order to further verify the accuracy of our theoretical analysis, some numerical results are provided.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020105
Authors: Chengyi Wang Shichao Yi
In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α(x,t)∈(1,2) and space-fractional order s∈(0,1), where v=u′(t) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v. The convergence order can reach O(τ2+h2), different from the common L1,2−α schemes with convergence rate O(τ2,3−α(x,t)) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020104
Authors: Bachir Nail Mahedi Abdelghani Atoussi Slami Saadi Imad Eddine Tibermacine Christian Napoli
In this paper, we use two Fractional-Order Chaotic Systems (FOCS)—one for the sender and one for the receiver—to determine the optimal synchronisation for a secure communication technique. With the help of the Step-By-Step Sliding-Mode Observer (SBS-SMO), this synchronisation is accomplished. An innovative optimisation method, known as the artificial Harris hawks optimisation (HHO), was employed to enhance the observer’s performance. This method eliminates the random parameter selection process and instead selects the optimal values for the observer. In a short amount of time, the secret message that could have been in the receiver portion (signal, voice, etc.) was successfully recovered using the proposed scheme. The experimental validation of the numerical results was carried out with the assistance of an Arduino microcontroller and several electronic components. In addition, the findings of the experiments were compared with the theoretical calculations, revealing a satisfactory level of agreement.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020103
Authors: Guozhi Zheng Naitian Zhang Songtao Lv
This paper aims to incorporate the fractional derivative viscoelastic model into a finite element analysis. Firstly, based on the constitutive equation of the fractional derivative three-parameter solid model (FTS), the constitutive equation is discretized by using the Grünwald–Letnikov definition of the fractional derivative, and the stress increment and strain increment relationship and Jacobian matrix are obtained by using the difference method. Subsequently, we degrade the model to establish stress increment and strain increment relationships and Jacobian matrices for the fractional derivative Kelvin model (FK) and fractional derivative Maxwell model (FM). Finally, we further degrade the fractional derivative viscoelastic model to derive stress increment and strain increment relationships and Jacobian matrices for a three-component solid model and Kelvin and Maxwell models. Based on these developments, a UMAT subroutine is implemented in ABAQUS 6.14 finite element software. Three different loading modes, including static load, dynamic load, and mobile load, are analyzed and calculated. The calculations primarily involve a convergence analysis, verification of numerical solutions, and comparative analysis of responses among different viscoelastic models.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020102
Authors: Junyu He Ming Li
Gaining insights into the space–time variations in the long-range dependence of sea surface chlorophyll is crucial for the early detection of environmental issues in oceans. To this end, 12 locations were selected along the Yangtze River and Pearl River estuaries, varying in distances from the Chinese coastline. Daily satellite-observed sea surface chlorophyll concentration data at these 12 locations were collected from the Copernicus Marine Service website, spanning from December 1997 to November 2023. The main objective of the current study is to introduce a multi-fractional generalized Cauchy model for calculating the values of Hurst exponents and quantitatively assessing the long-range dependence strength of sea surface chlorophyll at different spatial locations and time instants during the study period. Furthermore, ANOVA was utilized to detect the differences of calculated Hurst exponent values among the locations during various months and seasons. From a spatial perspective, the findings reveal a significantly stronger long-range dependence of sea surface chlorophyll in offshore regions compared to nearshore areas, with Hurst exponent values > 0.5 versus <0.5. It is noteworthy that the values of Hurst exponents at each location exhibit significant differences during various seasons, from a temporal perspective. Specifically, the long-range dependence of sea surface chlorophyll in summer in the nearshore region is weaker than in other seasons, whereas that in the offshore region is stronger than in other seasons. The study concludes that long-range dependence is inversely related to the distance from the coastline, and anthropogenic activity plays a dominant role in shaping the long-range dependence of sea surface chlorophyll in the coastal regions of China.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020101
Authors: Hongmei Zhang Mengchen Zhang Fawang Liu Ming Shen
The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020100
Authors: Albert Sabban
Future communication, 5G, medical, and IoT systems need compact, green, efficient wideband sensors, and antennas. Novel linear and dual-polarized antennas for 5G, 6G, medical devices, Internet of Things (IoT) systems, and healthcare monitoring sensors are presented in this paper. One of the major goals in the evaluation of medical, 5G, and smart wireless communication devices is the development of efficient, compact, low-cost antennas and sensors. Moreover, passive and active sensors may be self-powered by connecting an energy-harvesting unit to the antenna to collect electromagnetic radiation and charge the wearable sensor battery. Wearable sensors and antennas can be employed in smart grid applications that provide communication between neighbors, localized management, bidirectional power transfer, and effective demand response. A low-cost wearable antenna may be developed by etching the printed feed and matching the network on the same substrate in the printed antenna. Active modules may be placed on the same dielectric board. The antenna design parameters and a comparison between the computation and measured electrical performance of the antennas are presented in this paper. The electrical characteristics of the new compact antennas in the vicinity of the patient’s body were simulated by using electromagnetic simulation techniques. Fractal and metamaterial efficient antennas and sensors were evaluated to maximize the electrical characteristics of smart communication and medical devices. The dual- and circularly polarized antennas developed in this paper are crucial to the evaluation of wideband and multiband compact 5G, 6G, and IoT advanced systems. The new efficient sensors and antennas maximize the system’s dynamic range and electrical characteristics. The new efficient wearable antennas and sensors are compact, wideband, and low-cost. The operating resonant frequency of the metamaterial antennas with circular split-ring resonators (CSRRs) may be 5% to 9% lower than the resonant frequency of the sensor without CSRRs. The directivity and gain of the metamaterial fractal antennas with CSRRs may be up to 3 dB higher than the antennas without CSRRs. The directivity and gain of the metamaterial fractal passive sensors with CSRRs may be up to 8.5 dBi. This study presents new wideband active meta-fractal antennas and sensors. The bandwidth of the new sensors is around 9% to 20%. At 2.83 GHz, the receiving active sensor gain is 13.5 dB and drops to 8 dB at 3.2 GHz. The receiving module noise figure with TAV541 LNA is around 1dB.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020099
Authors: Guangxin Jiao Zhengcai An Shuyi Shao Dong Sun
In this paper, a fractional-order control method based on the twin-delayed deep deterministic policy gradient (TD3) algorithm in reinforcement learning is proposed. A fractional-order disturbance observer is designed to estimate the disturbances, and the radial basis function network is selected to approximate system uncertainties in the system. Then, a fractional-order sliding-mode controller is constructed to control the system, and the parameters of the controller are tuned using the TD3 algorithm, which can optimize the control effect. The results show that the fractional-order control method based on the TD3 algorithm can not only improve the closed-loop system performance under different operating conditions but also enhance the signal tracking capability.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020098
Authors: Jida Liu Changqi Dong
Regional integration and pairing assistance are two forms of cross-regional emergency collaboration practice carried out by the Chinese government. Based on the Chinese government’s emergency management practice, evolutionary game models of cross-regional emergency collaboration were constructed. Further, the traditional evolutionary game model was improved by introducing the stochastic process, and Gaussian white noise was introduced as a random disturbance. The stochastic evolutionary game model was constructed, and the existence and stability of the equilibrium solutions of the two kinds of stochastic evolutionary game systems for cross-regional emergency collaboration were verified based on the stability discrimination theorem of stochastic differential equations. We used numerical simulations to simulate the evolution trajectories of the regional integration and the pairing assistance stochastic evolutionary game system. In the regional integration game system, when the efficiency of emergency collaboration, the emergency capital stock, and the externality coefficients are higher, positive emergency strategies are more likely to become the stable state of the game subjects’ strategy selection. In the pairing assistance game system, the efficiency of emergency collaboration, the rewards and benefits from the central government, and the matching degree between governments all had positive effects on the formation of the positive emergency strategies of the game subjects. In addition, the pairing assistance mechanism for sustainable development requires external support from the central government.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020097
Authors: Yeguo Sun Yihong Liu Lei Liu
Due to the widespread application of neural networks (NNs), and considering the respective advantages of fractional calculus (FC), inertial neural networks (INNs), cellular neural networks (CNNs), and fuzzy neural networks (FNNs), this paper investigates the fixed-time synchronization (FDTS) issues for a particular category of fractional-order cellular-inertial fuzzy neural networks (FCIFNNs) that involve mixed time-varying delays (MTDs), including both discrete and distributed delays. Firstly, we establish an appropriate transformation variable to reformulate FCIFNNs with MTD into a differential first-order system. Then, utilizing the finite-time stability (FETS) theory and Lyapunov functionals (LFs), we establish some new effective criteria for achieving FDTS of the response system (RS) and drive system (DS). Eventually, we offer two numerical examples to display the effectiveness of our proposed synchronization strategies. Moreover, we also demonstrate the benefits of our approach through an application in image encryption.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020096
Authors: Haji Ahmed Faheem Aslam Paulo Ferreira
Financial stress can have significant implications for individuals, businesses, asset prices and the economy as a whole. This study examines the nonlinear structure and dynamic changes in the multifractal behavior of cross-correlation between the financial stress index (FSI) and four well-known commodity indices, namely Commodity Research Bureau Index (CRBI), Baltic Dry Index (BDI), London Metal Index (LME) and Brent Oil prices (BROIL), using multifractal detrended cross correlation analysis (MFDCCA). For analysis, we utilized daily values of FSI and commodity index prices from 16 June 2016 to 9 July 2023. The following are the most important empirical findings: (I) All of the chosen commodity market indices show cross correlations with the FSI and have notable multifractal characteristics. (II) The presence of power law cross-correlation implies that a noteworthy shift in FSI is likely to coincide with a considerable shift in the commodity indices. (III) The multifractal cross-correlation is highest between FSI and Brent Oil (BROIL) and lowest with LME. (IV) The rolling windows analysis reveals a varying degree of persistency between FSI and commodity markets. The findings of this study have a number of important implications for commodity market investors and policymakers.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020095
Authors: Fawaz K. Alalhareth Mohammed H. Alharbi Noura Laksaci Ahmed Boudaoui Meroua Medjoudja
The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders (γ) and fractal dimensions (θ), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020094
Authors: Hicham Saber Abdelkader Braik Noureddine Bahri Abderrahmane Beniani Tariq Alraqad Yousef Jawarneh Khaled Zennir
We consider a damped transmission problem in a bounded domain where the damping is effective in a neighborhood of a suitable subset of the boundary. Using the semigroup approach together with Hille–Yosida theorem, we prove the existence and uniqueness of global solution. Under suitable assumption on the geometrical conditions on the damping, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020093
Authors: Usama Ghafoor Muhammad Abbas Tayyaba Akram Emad K. El-Shewy Mahmoud A. E. Abdelrahman Noura F. Abdo
The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo’s formula, while the CBS technique with the help of a θ-weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020092
Authors: Yuan-Min Li Mingjie Jiang Deyun Wei Yang Deng
In this paper, we propose a secure image encryption method using compressive sensing (CS) and a two-dimensional linear canonical transform (2D LCT). First, the SHA256 of the source image is used to generate encryption security keys. As a result, the suggested technique is able to resist selected plaintext attacks and is highly sensitive to plain images. CS simultaneously encrypts and compresses a plain image. Using a starting value correlated with the sum of the image pixels, the Mersenne Twister (MT) is used to control a measurement matrix in compressive sensing. Then, the scrambled image is permuted by Lorenz’s hyper-chaotic systems and encoded by chaotic and random phase masks in the 2D LCT domain. In this case, chaotic systems increase the output complexity, and the independent parameters of the 2D LCT expand the key space of the suggested technique. Ultimately, diffusion based on addition and modulus operations yields a cipher-text image. Simulations showed that this cryptosystem was able to withstand common attacks and had adequate cryptographic features.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020091
Authors: Yahong Wang Wenmin Wang Cheng Yu Hongbo Sun Ruimin Zhang
The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier–Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020090
Authors: Fengyun Hu Keneng Zhang Kaofei Zhu Bintian Li Zhao Zhang Yong He
Soil-rock mixtures (SRM) are extensively utilized as filling materials in engineering slopes and roadbeds. A comprehensive understanding of the crushing characteristics of SRM during compaction is essential for precisely controlling its mechanical properties, particularly when dealing with SRM comprising soft rock blocks. This study conducted heavy compaction and screening tests to investigate the crushing and compaction behaviors of soil-soft rock mixture (SSRM) with varying coarse particle content (P5 content), the primary focus was primarily on analyzing the double fractal characteristics of coarse and fine particles. The research findings are as follows: with the increase of P5 content, the maximum dry density of SSRM initially rises and then declines, reaching its peak when P5 content is 70%. Soft rock blocks in SSRM exhibit extreme fragility during compaction, the crushing index of coarse particles exhibits a linear increase with the rise in P5 content, whereas the crushing index of fine particles displays a “double peak” characteristic. After compaction, a linear positive correlation is observed between the fractal dimension and the crushing index of coarse and fine particles. With the increase in P5 content, the slope of the relationship curve between the fractal dimension and the crushing index of coarse particles remains relatively constant, while the intercept gradually decreases. Moreover, the fractal dimension of fine particles effectively reflects the compaction characteristics of SSRM, and the relationship between the fractal dimension of fine particles and dry density aligns with the compaction curve of SSRM.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020089
Authors: Jing Gao Huaiguang Chen
We develop and analyze an explicit finite difference scheme that satisfies a bound-preserving principle for space–time fractional advection equations with the orders of 0<α and β≤1. The stability (and convergence) of the method is discussed. Due to the nonlocal property of the fractional operators, the numerical method generates dense coefficient matrices with complex structures. In order to increase the effectiveness of the method, we use Toeplitz-like structures in the full coefficient matrix in a sparse form to reduce the costs of computation, and we also apply a fast evaluation method for the time–fractional derivative. Therefore, an efficient solver is constructed. Numerical experiments are provided for the utility of the method.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020088
Authors: Dimiter Prodanov
The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020087
Authors: Qi Fan Yu Ma Pengzhi Wang Fenghua Bai
To solve the shortcomings of the Otsu image segmentation algorithm based on traditional Moth–Flame Optimization (MFO), such as its poor segmentation accuracy, slow convergence, and tendency to fall into local optimum, this paper proposes fractional order moth–flame optimization with the Otsu image segmentation algorithm. Utilizing the advantages of memorability and heritability in fractional order differentiation, the position updating of moths is controlled by fractional order. Using the adaptive fractional order, the positions of moths are used to adjust the fractional order adaptively to improve the convergence speed. Combining the improved MFO algorithm with the two-dimensional Otsu algorithm, the optimization objective function is achieved by using its dispersion matrix. The experimental results indicate that, compared with traditional MFO, the convergence rate of the proposed algorithm is improved by about 74.62%. Furthermore, it has better segmentation accuracy and a higher fitness value than traditional MFO.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020086
Authors: Ahmad Al-Omari Hanan Al-Saadi Fawaz Alharbi
This study aims to prove the existence and uniqueness of the (ω,c)-periodic solution as a specific solution to Hadamard impulsive boundary value integro-differential equations with fixed lower limits. The results are proven using the Banach contraction, Schaefer’s fixed point theorem, and the Arzelà–Ascoli theorem. Furthermore, we establish the necessary conditions for a set of solutions to the explored boundary values with impulsive fractional differentials. Finally, we present two examples as applications for our results.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020085
Authors: Haotian Chen Ming He Wei Han Sicong Liu Chenyue Wei
In this paper, a discrete-time fractional flocking control algorithm of multi-agent systems is put forward to address the slow convergence issue of multi-agent systems. Firstly, by introducing Grünwald-Letnikov (G-L) fractional derivatives, the algorithm allows agents to utilize historical information when updating their states. Secondly, based on the Lyapunov stability theory, the convergence of the algorithm is proven. Finally, simulations are conducted to verify the effectiveness of the proposed algorithm. Comparisons are made between the proposed algorithm and other methods. The results show that the proposed algorithm can effectively improve the convergence speed of multi-agent systems.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020084
Authors: Shimaa H. Abel-Gaid Ahlam Hasan Qamlo Bahaa Gaber Mohamed
In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020083
Authors: Chengqiang Wang Xiangqing Zhao Qiuyue Mai Zhiwei Lv
We are concerned in this paper with the stability and bifurcation problems for three-neuron-based bi-directional associative memory neural networks that are involved with time delays in transmission terms and possess Caputo fractional derivatives of non-commensurate orders. For the fractional bi-directional associative memory neural networks that are dealt with in this paper, we view the time delays as the bifurcation parameters. Via a standard contraction mapping argument, we establish the existence and uniqueness of the state trajectories of the investigated fractional bi-directional associative memory neural networks. By utilizing the idea and technique of linearization, we analyze the influence of time delays on the dynamical behavior of the investigated neural networks, as well as establish and prove several stability/bifurcation criteria for the neural networks dealt with in this paper. According to each of our established criteria, the equilibrium states of the investigated fractional bi-directional associative memory neural networks are asymptotically stable when some of the time delays are less than strictly specific positive constants, i.e., when the thresholds or the bifurcation points undergo Hopf bifurcation in the concerned networks at the aforementioned threshold constants. In the meantime, we provide several illustrative examples to numerically and visually validate our stability and bifurcation results. Our stability and bifurcation theoretical results in this paper yield some insights into the cause mechanism of the bifurcation phenomena for some other complex phenomena, and this is extremely helpful for the design of feedback control to attenuate or even to remove such complex phenomena in the dynamics of fractional bi-directional associative memory neural networks with time delays.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020082
Authors: Jingshuo Zhang Xiaoming Ni Xiaolei Liu Erlei Su
The effect of different acids on the pore structure and fractal characteristics of micropores and mesopores was determined with the help of low-temperature liquid nitrogen adsorption, X-ray diffraction, and the Frenkel–Halsey–Hill (FHH) model by using Yuwu coal as a sample and placing it in acidic environments, such as HF, HCl, HNO3, and CH3COOH. The results show that the acidization effects of HF and CH3COOH are separately dominated by the micropore and mesopore formation effects, while HCl and HNO3 mainly play their roles in expanding mesopores. After acidization, the surface fractal dimensions D1 and D1′ of micropores and mesopores in coal are always negatively correlated with the total specific surface area SBET, specific surface area Smic of micropores, and specific surface area Smes of mesopores. After being acidized by HF, D2 is negatively correlated with the total volume Vtot and the corresponding micropore volume Vmic, while acidization with HCl and HNO3 leads to the opposite result. After being acidized by CH3COOH, D2 has a negative correlation with Vtot and a positive correlation with Vmic. The structural fractal dimensions D2′ of mesopores in samples acidized by HF and CH3COOH are positively correlated with both the volume Vtot and mesopore volume Vmes, while it is the opposite for samples acidized by HNO3. D2′ of coal samples acidized by HCl is negatively correlated with Vtot while positively correlated with Vmes.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020081
Authors: Donghui Yu Xiaozhong Liao Yong Wang
This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model of a fractional-order Boost converter by taking advantage of the state space averaging method, providing accurate analytical solutions for the quiescent operating point and the ripple parameters of the circuit with three state variables. Secondly, this paper constructs a small signal model of the C-F definition-based fractional-order Boost converter by small signal linearization, providing the transfer function of the fractional-order system with three state variables. Finally, this paper conducts circuit-oriented simulation experiments where the steady-state parameters and the transfer function of the circuit are obtained, and then the effect of the order of capacitor, induced inductor, and load inductor on the quiescent operating point and ripple parameters is analyzed. The experimental results show that the simulation results are consistent with those obtained by the proposed mathematical model and that the three fractional orders in the fractional model with three state variables have a significant impact on the DC component and steady-state characteristics of the fractional-order Boost converter. In conclusion, the proposed mathematical model can more comprehensively analyze the system characteristics of the C-F definition-based fractional-order Boost converter with fractional-order inductive loads, benefiting the circuit design of Boost converters.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020080
Authors: SAIRA Wen-Xiu Ma Guidong Liu
The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020079
Authors: Yuliya Mishura Kostiantyn Ralchenko
Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering. We construct least-square estimators for the unknown drift parameters within Vasicek models that are driven by these processes. To demonstrate their strong consistency, we establish asymptotic bounds with probability 1 for the rate of growth of trajectories of tempered fractional processes.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020078
Authors: Feng You Hong-An Tang Yanhong Wang Zi-Yi Xia Jin-Wei Li
This article discusses the adaptive output synchronization problem of coupled fractional-order memristive reaction-diffusion neural networks (CFOMRDNNs) with multiple output couplings or multiple output derivative couplings. Firstly, by using Lyapunov functional and inequality techniques, an adaptive output synchronization criterion for CFOMRDNNs with multiple output couplings is proposed. Then, an adaptive controller is designed for ensuring the output synchronization of CFOMRDNNs with multiple output derivative couplings. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.
]]>Fractal and Fractional doi: 10.3390/fractalfract8020077
Authors: Yuanda Lv Jin-Xi Zhang Xuefeng Zhang
In this paper, new stability criteria for linear time-invariant fractional-order systems (LTIFOSs) based on linear matrix inequalities (LMIs) are derived. The solved variable of the existing LMI formulations is generalized to a complex one. In addition, based on the congruent transformation, a new LMI formulation is obtained, which is different from those in the existing literature. To deal with the above LMIs more conveniently with simulation software, the complex LMIs are converted to equivalent real LMIs. Finally, numerical examples are presented to validate the effectiveness of our theoretical results.
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