Journal Description
Fractal and Fractional
Fractal and Fractional
is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering published monthly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, and other databases.
- Journal Rank: JCR - Q1 (Mathematics, Interdisciplinary Applications) / CiteScore - Q1 (Analysis)
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 18.9 days after submission; acceptance to publication is undertaken in 3.5 days (median values for papers published in this journal in the second half of 2023).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Impact Factor:
5.4 (2022);
5-Year Impact Factor:
4.7 (2022)
Latest Articles
Structural Characterization of Toxoplasma gondii Brain Cysts in a Model of Reactivated Toxoplasmosis Using Computational Image Analysis
Fractal Fract. 2024, 8(3), 175; https://doi.org/10.3390/fractalfract8030175 - 18 Mar 2024
Abstract
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
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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.
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(This article belongs to the Section Life Science, Biophysics)
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Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil
by
Shanyuan Qin, Jidong Yang, Ning Qin, Jianping Huang and Kun Tian
Fractal Fract. 2024, 8(3), 174; https://doi.org/10.3390/fractalfract8030174 - 18 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Fractional Equations and Calculation Methods in Exploration Seismology)
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Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations
by
Mengru Liu and Lihong Zhang
Fractal Fract. 2024, 8(3), 173; https://doi.org/10.3390/fractalfract8030173 - 16 Mar 2024
Abstract
This article mainly studies the double index logarithmic nonlinear fractional Laplacian parabolic equations with the Marchaud fractional time derivatives . Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double
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This article mainly studies the double index logarithmic nonlinear fractional Laplacian parabolic equations with the Marchaud fractional time derivatives . 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 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 Laplacian parabolic equations is studied.
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(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)
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Fractional Fuzzy Neural System: Fractional Differential-Based Compensation Prediction for Reputation Infringement Cases
by
Ni Zhang, Wu-Yang Zhu, Peng Jin, Guo Huang and Yi-Fei Pu
Fractal Fract. 2024, 8(3), 172; https://doi.org/10.3390/fractalfract8030172 - 16 Mar 2024
Abstract
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.
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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.
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(This article belongs to the Special Issue Fractional Calculus in Signal, Imaging Processing and Machine Learning)
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Identification of the Dynamic Trade Relationship between China and the United States Using the Quantile Grey Lotka–Volterra Model
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Zheng-Xin Wang, Yue-Ting Li and Ling-Fei Gao
Fractal Fract. 2024, 8(3), 171; https://doi.org/10.3390/fractalfract8030171 - 15 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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Accelerated Gradient Descent Driven by Lévy Perturbations
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Yuquan Chen, Zhenlong Wu, Yixiang Lu, Yangquan Chen and Yong Wang
Fractal Fract. 2024, 8(3), 170; https://doi.org/10.3390/fractalfract8030170 - 14 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Fractional Calculus in Signal, Imaging Processing and Machine Learning)
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Utilizing a Fractional-Order Grey Model to Predict the Development Trends of China’s Electronic Commerce Service Industry
by
Jianhong Guo, Che-Jung Chang and Yingyi Huang
Fractal Fract. 2024, 8(3), 169; https://doi.org/10.3390/fractalfract8030169 - 14 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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Convergence Analysis of Iterative Learning Control for Initialized Fractional Order Systems
by
Xiaofeng Xu, Jiangang Lu and Jinshui Chen
Fractal Fract. 2024, 8(3), 168; https://doi.org/10.3390/fractalfract8030168 - 14 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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The Classification and Evaluation of an Interlayer Shale Oil Reservoir Based on the Fractal Characteristics of Pore Systems: A Case Study in the HSN Area, China
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Changsheng Lu, Xixin Wang, Shuwei Ma, Shaohua Li, Ting Xue and Qiangqiang Li
Fractal Fract. 2024, 8(3), 167; https://doi.org/10.3390/fractalfract8030167 - 14 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Flow and Transport in Fractal Models of Rock Mechanics)
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On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces
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James Abah Ugboh, Joseph Oboyi, Mfon Okon Udo, Hossam A. Nabwey, Austine Efut Ofem and Ojen Kumar Narain
Fractal Fract. 2024, 8(3), 166; https://doi.org/10.3390/fractalfract8030166 - 14 Mar 2024
Abstract
In this paper, we consider a faster iterative method for approximating the fixed points of generalized -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
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In this paper, we consider a faster iterative method for approximating the fixed points of generalized -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.
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(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)
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New Results on
by
Adel Salim Tayyah and Waggas Galib Atshan
Fractal Fract. 2024, 8(3), 165; https://doi.org/10.3390/fractalfract8030165 - 13 Mar 2024
Abstract
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
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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.
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(This article belongs to the Section General Mathematics, Analysis)
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An α-Robust Galerkin Spectral Method for the Nonlinear Distributed-Order Time-Fractional Diffusion Equations with Initial Singularity
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Haiyu Liu and Shujuan Lü
Fractal Fract. 2024, 8(3), 164; https://doi.org/10.3390/fractalfract8030164 - 13 Mar 2024
Abstract
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
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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 -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 -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.
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(This article belongs to the Section Numerical and Computational Methods)
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Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain
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Fengzhou Tian, Yulan Wang and Zhiyuan Li
Fractal Fract. 2024, 8(3), 163; https://doi.org/10.3390/fractalfract8030163 - 12 Mar 2024
Abstract
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
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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.
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(This article belongs to the Topic AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity)
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A New Approach to Multiroot Vectorial Problems: Highly Efficient Parallel Computing Schemes
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Mudassir Shams, Naila Rafiq, Bruno Carpentieri and Nazir Ahmad Mir
Fractal Fract. 2024, 8(3), 162; https://doi.org/10.3390/fractalfract8030162 - 12 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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Analysis of Pore Characterization and Energy Evolution of Granite by Microwave Radiation
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Keping Zhou, Yifan Zhang, Chun Yang, Niange Yang and Zheng Pan
Fractal Fract. 2024, 8(3), 161; https://doi.org/10.3390/fractalfract8030161 - 12 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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Event-Triggered Adaptive Fuzzy Control for Strict-Feedback Nonlinear FOSs Subjected to Finite-Time Full-State Constraints
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Changhui Wang, Wencheng Li and Mei Liang
Fractal Fract. 2024, 8(3), 160; https://doi.org/10.3390/fractalfract8030160 - 12 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Fractional Order Controllers for Non-linear Systems)
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A Fractal-Based Quantitative Method for the Study of Fracture Evolution of Coal under Different Confining Pressures
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Ancheng Wang and Lei Wang
Fractal Fract. 2024, 8(3), 159; https://doi.org/10.3390/fractalfract8030159 - 11 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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Fuzzy Mandelbric Set and Its Perturbations by Dynamical Noises
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Nikola Popović, Soley Ersoy, İbrahim İnce, Ana Savić and Vladimir Baltić
Fractal Fract. 2024, 8(3), 158; https://doi.org/10.3390/fractalfract8030158 - 11 Mar 2024
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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
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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.
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Smooth and Efficient Path Planning for Car-like Mobile Robot Using Improved Ant Colony Optimization in Narrow and Large-Size Scenes
by
Likun Li, Liyu Jiang, Wenzhang Tu, Liquan Jiang and Ruhan He
Fractal Fract. 2024, 8(3), 157; https://doi.org/10.3390/fractalfract8030157 - 10 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Applications of Fractional-Order Calculus in Robotics)
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Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions
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Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev
Fractal Fract. 2024, 8(3), 156; https://doi.org/10.3390/fractalfract8030156 - 10 Mar 2024
Abstract
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
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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.
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(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)
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