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Fractal Fract., Volume 8, Issue 4 (April 2024) – 67 articles

Cover Story (view full-size image): This study uses a hyper-heuristic approach to design heuristic-based tuners for fractional-order PID (FOPID) controllers within Automatic Voltage Regulator (AVR) systems. This approach allows for the automated generation of metaphor-less metaheuristics by combining population-based search operators, reducing human bias in the tuning process. The automatically tailored metaheuristic effectively fine-tunes FOPID controller parameters to enhance control performance, achieving faster response times and virtually eliminating overshoot. This research also explores the impact of fractional orders on conventional PID controllers. It assesses and highlights the tuned FOPID controller’s resilience against disturbances, providing substantial advancements in electrical engineering and control systems. View this paper
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23 pages, 17881 KiB  
Article
Heterogeneity of Micro- and Nanopore Structure of Lacustrine Shales with Complex Lamina Structure
by Shuning Liu, Juncheng Qiao, Jianhui Zeng, Chuanming Li, Yazhou Liu, Zheng Kong and Xinlong Liu
Fractal Fract. 2024, 8(4), 245; https://doi.org/10.3390/fractalfract8040245 - 22 Apr 2024
Viewed by 627
Abstract
Thin sections, AIM-SEM, MICP, and nitrogen adsorption were performed on laminated and layered shales to characterize their complex pore and fracture structure. Combining the MICP model with the FHH model, this work proposes a new fractal method for lacustrine shales with complex lamina [...] Read more.
Thin sections, AIM-SEM, MICP, and nitrogen adsorption were performed on laminated and layered shales to characterize their complex pore and fracture structure. Combining the MICP model with the FHH model, this work proposes a new fractal method for lacustrine shales with complex lamina structure. The fractal characteristics presented four zones, representing the heterogeneity of fractures, macropores, mesopores, and micropores. The pores and fractures of shale have strong heterogeneity. Laminated shale has strong heterogeneity in mesopores and moderate heterogeneity in micropores. Layered shale has strong heterogeneity in fractures and moderate heterogeneity in micropores. The lamina structure and content of organic and mineral composition has a great influence on heterogeneity. The mineral laminae in laminated shale change frequently; lamellation fractures are mainly developed, and the structures are similar. Layered shales develop fractures between layers and structural fractures; the structural differences are significant. Macropores are mostly interparticle pores between quarts with similar structures. The wider lamina thickness of layered shale provides sufficient crystallization space for minerals, so the mesopores of layered shale are more homogeneous. Micropores are less developed, mainly consisting of intraparticle pores between clay minerals, which are complex but similar in structure in the two types of shale. The heterogeneity of mesopores and micropores is not conducive to hydrocarbon migration. Fractures and macropores need to be connected with meso–micropores to form a transport system. So, mesopores and micropores play decisive roles in hydrocarbon migration. Based on the above understanding, this paper points out that hydrocarbon in laminated shale with more carbonate minerals and a high thermal evolution degree has better availability. Full article
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17 pages, 8493 KiB  
Article
Robust Fractional-Order PI/PD Controllers for a Cascade Control Structure of Servo Systems
by Vo Lam Chuong, Ngo Hong Nam, Le Hieu Giang and Truong Nguyen Luan Vu
Fractal Fract. 2024, 8(4), 244; https://doi.org/10.3390/fractalfract8040244 - 22 Apr 2024
Viewed by 539
Abstract
In this paper, a cascade control structure is suggested to control servo systems that normally include a servo motor in coupling with two kinds of mechanism elements, a translational or rotational movement. These kinds of systems have high demands for performance in terms [...] Read more.
In this paper, a cascade control structure is suggested to control servo systems that normally include a servo motor in coupling with two kinds of mechanism elements, a translational or rotational movement. These kinds of systems have high demands for performance in terms of fastest response and no overshoot/oscillation to a ramp function input. The fractional-order proportional integral (FOPI) and proportional derivative (FOPD) controllers are addressed to deal with those control problems due to their flexibility in tuning rules and robustness. The tuning rules are designed in the frequency domain based on the concept of the direct synthesis method and also ensure the robust stability of controlled systems by using the maximum sensitivity function. The M-Δ structure, using multiplicative output uncertainties for both control loops simultaneously, is addressed to justify the robustness of the controlled systems. Simulation studies are considered for two kinds of plants that prove the effectiveness of the proposed method, with good tracking of the ramp function input under the effects of the disturbances. In addition, the robustness of the controlled system is illustrated by a structured singular value (µ) plot in which its value is less than 1 over the frequency range. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)
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15 pages, 301 KiB  
Article
Fixed Point Theorems: Exploring Applications in Fractional Differential Equations for Economic Growth
by Afrah Ahmad Noman Abdou
Fractal Fract. 2024, 8(4), 243; https://doi.org/10.3390/fractalfract8040243 - 22 Apr 2024
Viewed by 592
Abstract
The aim of this research is to introduce two new notions, Θ-(Ξ,h)-contraction and rational (α,η)-ψ-interpolative contraction, in the setting of F-metric space and to establish corresponding fixed point theorems. To [...] Read more.
The aim of this research is to introduce two new notions, Θ-(Ξ,h)-contraction and rational (α,η)-ψ-interpolative contraction, in the setting of F-metric space and to establish corresponding fixed point theorems. To reinforce understanding and highlight the novelty of our findings, we provide a non-trivial example that not only supports the obtained results but also illuminates the established theory. Finally, we apply our main result to discuss the existence and uniqueness of solutions for a fractional differential equation describing an economic growth model. Full article
18 pages, 2500 KiB  
Article
Using Fractal Theory to Study the Influence of Movable Oil on the Pore Structure of Different Types of Shale: A Case Study of the Fengcheng Formation Shale in Well X of Mahu Sag, Junggar Basin, China
by Hong Zhang, Zhengchen Zhang, Zhenlin Wang, Yamin Wang, Rui Yang, Tao Zhu, Feifei Luo and Kouqi Liu
Fractal Fract. 2024, 8(4), 242; https://doi.org/10.3390/fractalfract8040242 - 20 Apr 2024
Viewed by 587
Abstract
This study investigated the influence of movable oil on the pore structure of various shale types, analyzing 19 shale samples from Well X in the Mahu Sag of the Junggar Basin. Initially, X-ray diffraction (XRD) analysis classified the shale samples. Subsequently, the geochemical [...] Read more.
This study investigated the influence of movable oil on the pore structure of various shale types, analyzing 19 shale samples from Well X in the Mahu Sag of the Junggar Basin. Initially, X-ray diffraction (XRD) analysis classified the shale samples. Subsequently, the geochemical properties and pore structures of the samples, both pre and post oil Soxhlet extraction, were comparatively analyzed through Total Organic Carbon (TOC) content measurement, Rock-Eval pyrolysis, and nitrogen adsorption experiments. Additionally, fractal theory quantitatively described the impact of movable oil on the pore structure of different shale types. Results indicated higher movable oil content in siliceous shale compared to calcareous shale. Oil extraction led to a significant increase in specific surface area and pore volume in all samples, particularly in siliceous shale. Calcareous shale predominantly displays H2–H3 type hysteresis loops, indicating a uniform pore structure with ink-bottle-shaped pores. Conversely, siliceous shale exhibited diverse hysteresis loops, reflecting its complex pore structure. The fractal dimension in calcareous shale correlated primarily with pore structure, exhibiting no significant correlation with TOC content before or after oil extraction. Conversely, the fractal dimension changes in siliceous shale samples do not have a clear correlation with either TOC content or pore structure, suggesting variations may result from both TOC and pore structure. Full article
(This article belongs to the Special Issue Flow and Transport in Fractal Models of Rock Mechanics)
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19 pages, 5158 KiB  
Article
Unravelling the Fractal Complexity of Temperature Datasets across Indian Mainland
by Adarsh Sankaran, Thomas Plocoste, Arathy Nair Geetha Raveendran Nair and Meera Geetha Mohan
Fractal Fract. 2024, 8(4), 241; https://doi.org/10.3390/fractalfract8040241 - 20 Apr 2024
Viewed by 520
Abstract
Studying atmospheric temperature characteristics is crucial under climate change, as it helps us to understand the changing patterns in temperature that have significant implications for the environment, ecosystems, and human well-being. This study presents the comprehensive analysis of the spatiotemporal variability of scaling [...] Read more.
Studying atmospheric temperature characteristics is crucial under climate change, as it helps us to understand the changing patterns in temperature that have significant implications for the environment, ecosystems, and human well-being. This study presents the comprehensive analysis of the spatiotemporal variability of scaling behavior of daily temperature series across the whole Indian mainland, using a Multifractal Detrended Fluctuation Analysis (MFDFA). The analysis considered 1° × 1° datasets of maximum temperature (Tmax), minimum temperature (Tmin), mean temperature (Tmean), and diurnal temperature range (DTR) (TDTR = TmaxTmin) from 1951 to 2016 to compare their scaling behavior for the first time. Our results indicate that the Tmin series exhibits the highest persistence (with the Hurst exponent ranging from 0.849 to unity, and a mean of 0.971), and all four-temperature series display long-term persistence and multifractal characteristics. The variability of the multifractal characteristics is less significant in North–Central India, while it is highest along the western coast of India. Moreover, the assessment of multifractal characteristics of different temperature series during the pre- and post-1976–1977 period of the Pacific climate shift reveals a notable decrease in multifractal strength and persistence in the post-1976–1977 series across all regions. Moreover, for the detection of climate change and its dominant driver, we propose a new rolling window multifractal (RWM) framework by evaluating the temporal evolution of the spectral exponents and the Hurst exponent. This study successfully captured the regime shifts during the periods of 1976–1977 and 1997–1998. Interestingly, the earlier climatic shift primarily mitigated the persistence of the Tmax series, whereas the latter shift significantly influenced the persistence of the Tmean series in the majority of temperature-homogeneous regions in India. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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18 pages, 7062 KiB  
Article
Compressive Failure Characteristics of a Coal–Rock Combination at Different Angles: Experimental Study and Fractal Analysis
by Long Tang, Shihao Tu, Hongsheng Tu, Kaijun Miao, Wenlong Li, Hongbin Zhao, Jieyang Ma and Lei Zhang
Fractal Fract. 2024, 8(4), 240; https://doi.org/10.3390/fractalfract8040240 - 20 Apr 2024
Viewed by 585
Abstract
In order to explore the influence of dip angles on the deformation and failure of a coal–rock combination, uniaxial compression experiments were carried out on a coal–rock combination with different dip angles, and the acoustic emissions (hereinafter referred to as AE) responses during [...] Read more.
In order to explore the influence of dip angles on the deformation and failure of a coal–rock combination, uniaxial compression experiments were carried out on a coal–rock combination with different dip angles, and the acoustic emissions (hereinafter referred to as AE) responses during loading were collected. Based on the damage mechanics theory and fractal theory, the fractal dimensions of different damage degrees were calculated. The results show that, with the increase in the inclination angle, the compressive strength and elastic modulus of the coal–rock combination gradually decreased, while the AE ringing count gradually increased first and then decreased. At the initial loading stage of the specimen, the greater the damage degree of the coal–rock combination under the same strain condition, the larger the value of its overall fractal dimension. The AE fractal dimension of the coal–rock combination increases gradually between 10% and 20% of the damage degree. It suddenly decreased between 50% and 60%, then increased slightly before gradually decreasing to the minimum between 80% and 100%. The sudden decrease in fractal dimension, a slight increase, and then a continuous decrease can be used as the precursor information for the instability and failure of the coal–rock combination. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)
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25 pages, 26345 KiB  
Article
Auto- and Cross-Correlation Multifractal Analysis of Sea Surface Temperature Variability
by Gyuchang Lim and Jong-Jin Park
Fractal Fract. 2024, 8(4), 239; https://doi.org/10.3390/fractalfract8040239 - 19 Apr 2024
Viewed by 622
Abstract
In this study, we investigate multiscale auto- and cross-correlation structural characteristics of sea surface temperature (SST) variability using our new methodology, called the multifractal asymmetric cross-correlation analysis (MF-ACCA), incorporating signs of a segment’s detrended covariance and linear trend. SST is greatly affected by [...] Read more.
In this study, we investigate multiscale auto- and cross-correlation structural characteristics of sea surface temperature (SST) variability using our new methodology, called the multifractal asymmetric cross-correlation analysis (MF-ACCA), incorporating signs of a segment’s detrended covariance and linear trend. SST is greatly affected by air–sea interactions and the advection of water masses with a wide range of spatiotemporal scales. Since these force factors are imprinted on SST variability, their features can be revealed in terms of long-range auto- and cross-correlation structures of SST variability via a multifractal analysis. By applying the MF-ACCA methodology to SST variability in the East/Japan Sea, we have found the following features: (1) the auto- and cross-correlation multifractal features are dependent on several parameters, such as the location, linear trends (rising or falling), level of fluctuations, and temporal scales; (2) there are crossover behaviors that are discrete for small scales (less than 1000 days) but continuous for large scales (more than 1000 days); (3) long-range persistence of auto- and cross-correlations is random for large scales during the falling phase; (4) long-range persistence is stronger during the rising phase than during the falling phase; (5) the degree of asymmetry is greater for large scales than for small scales. Full article
(This article belongs to the Section Complexity)
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21 pages, 620 KiB  
Article
Spectral and Oscillation Theory for an Unconventional Fractional Sturm–Liouville Problem
by Mohammad Dehghan and Angelo B. Mingarelli
Fractal Fract. 2024, 8(4), 238; https://doi.org/10.3390/fractalfract8040238 - 19 Apr 2024
Viewed by 549
Abstract
Here, we investigate the spectral and oscillation theory for a class of fractional differential equations subject to specific boundary conditions. By transforming the problem into a modified version with a classical structure, we establish the orthogonality properties of eigenfunctions and some major comparison [...] Read more.
Here, we investigate the spectral and oscillation theory for a class of fractional differential equations subject to specific boundary conditions. By transforming the problem into a modified version with a classical structure, we establish the orthogonality properties of eigenfunctions and some major comparison theorems for solutions. We also derive a new type of integration by using parts of formulas for modified fractional integrals and derivatives. Furthermore, we analyze the variational characterization of the first eigenvalue, revealing its non-zero first eigenfunction within the interior. Our findings demonstrate the potential for novel definitions of fractional derivatives to mirror the classical Sturm–Liouville theory through simple isospectral transformations. Full article
(This article belongs to the Section Numerical and Computational Methods)
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17 pages, 5814 KiB  
Article
COVID-19 Diagnosis by Extracting New Features from Lung CT Images Using Fractional Fourier Transform
by Ali Nokhostin and Saeid Rashidi
Fractal Fract. 2024, 8(4), 237; https://doi.org/10.3390/fractalfract8040237 - 18 Apr 2024
Viewed by 833
Abstract
COVID-19 is a lung disease caused by a coronavirus family virus. Due to its extraordinary prevalence and associated death rates, it has spread quickly to every country in the world. Thus, achieving peaks and outlines and curing different types of relapses is extremely [...] Read more.
COVID-19 is a lung disease caused by a coronavirus family virus. Due to its extraordinary prevalence and associated death rates, it has spread quickly to every country in the world. Thus, achieving peaks and outlines and curing different types of relapses is extremely important. Given the worldwide prevalence of coronavirus and the participation of physicians in all countries, information has been gathered regarding the properties of the virus, its diverse types, and the means of analyzing it. Numerous approaches have been used to identify this evolving virus. It is generally considered the most accurate and acceptable method of examining the patient’s lungs and chest through a CT scan. As part of the feature extraction process, a method known as fractional Fourier transform (FrFT) has been applied as one of the time-frequency domain transformations. The proposed method was applied to a database consisting of 2481 CT images. Following the transformation of all images into equal sizes and the removal of non-lung areas, multiple combination windows are used to reduce the number of features extracted from the images. In this paper, the results obtained for KNN and SVM classification have been obtained with accuracy values of 99.84% and 99.90%, respectively. Full article
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29 pages, 464 KiB  
Article
On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam
by Sina Etemad, Sotiris K. Ntouyas, Ivanka Stamova and Jessada Tariboon
Fractal Fract. 2024, 8(4), 236; https://doi.org/10.3390/fractalfract8040236 - 17 Apr 2024
Viewed by 695
Abstract
Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, [...] Read more.
Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned (p;q)-difference Navier problems in two single-valued and set-valued versions. We use the main properties of the (p;q)-operators in this regard. Application of the fixed points of the ρ-θ-contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential (p;q)-difference Navier problems. Full article
19 pages, 346 KiB  
Article
Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay
by Aeshah Abdullah Muhammad Al-Dosari
Fractal Fract. 2024, 8(4), 235; https://doi.org/10.3390/fractalfract8040235 - 17 Apr 2024
Viewed by 608
Abstract
This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area [...] Read more.
This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (Eα,β), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
21 pages, 1581 KiB  
Article
A Low Power Analog Integrated Fractional Order Type-2 Fuzzy PID Controller
by Vassilis Alimisis, Nikolaos P. Eleftheriou, Evangelos Georgakilas, Christos Dimas, Nikolaos Uzunoglu and Paul P. Sotiriadis
Fractal Fract. 2024, 8(4), 234; https://doi.org/10.3390/fractalfract8040234 - 16 Apr 2024
Viewed by 620
Abstract
This paper introduces an analog integrated fractional order type-2 fuzzy PID control system. Current approaches frequently depend on energy-intensive embedded digital systems, consuming substantial energy levels ranging from a few μW to mW. To address this limitation we propose a fully analog design [...] Read more.
This paper introduces an analog integrated fractional order type-2 fuzzy PID control system. Current approaches frequently depend on energy-intensive embedded digital systems, consuming substantial energy levels ranging from a few μW to mW. To address this limitation we propose a fully analog design offering insights into the potential of analog circuits for powerefficient robust control in complex and uncertain environments. It consists of Gaussian function, min/max, Operational transcoductance amplifier circuits and Resistor-Capacitor networks for the implementation of the fractional-order components. Crafted for operation under a reduced voltage supply (0.6 V), the controller attains minimal power usage (861.8 nW), facilitating uninterrupted, extended-term functioning. Post-layout simulation results confirm the proper operation of the proposed design. The proposed system is designed and simulated using the Cadence IC Suite in a TSMC 90 nm CMOS process. Full article
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5 pages, 172 KiB  
Editorial
Fractional Calculus and Hypergeometric Functions in Complex Analysis
by Gheorghe Oros and Georgia Irina Oros
Fractal Fract. 2024, 8(4), 233; https://doi.org/10.3390/fractalfract8040233 - 16 Apr 2024
Viewed by 594
Abstract
Fractional calculus has had a powerful impact on recent research, with many applications in different branches of science and engineering [...] Full article
(This article belongs to the Special Issue Fractional Calculus and Hypergeometric Functions in Complex Analysis)
22 pages, 4577 KiB  
Article
Stability and Bifurcation Control for a Generalized Delayed Fractional Food Chain Model
by Qing Li, Hongxia Liu, Wencai Zhao and Xinzhu Meng
Fractal Fract. 2024, 8(4), 232; https://doi.org/10.3390/fractalfract8040232 - 16 Apr 2024
Viewed by 580
Abstract
In this paper, a generalized fractional three-species food chain model with delay is investigated. First, the existence of a positive equilibrium is discussed, and the sufficient conditions for global asymptotic stability are given. Second, through selecting the delay as the bifurcation parameter, we [...] Read more.
In this paper, a generalized fractional three-species food chain model with delay is investigated. First, the existence of a positive equilibrium is discussed, and the sufficient conditions for global asymptotic stability are given. Second, through selecting the delay as the bifurcation parameter, we obtain the sufficient condition for this non-control system to generate Hopf bifurcation. Then, a nonlinear delayed feedback controller is skillfully applied to govern the system’s Hopf bifurcation. The results indicate that adjusting the control intensity or the control target’s age can effectively govern the bifurcation dynamics behavior of this system. Last, through application examples and numerical simulations, we confirm the validity and feasibility of the theoretical results, and find that the control strategy is also applicable to eco-epidemiological systems. Full article
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11 pages, 1599 KiB  
Article
Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo
by Mustafa Turkyilmazoglu and A. S. Alofi
Fractal Fract. 2024, 8(4), 231; https://doi.org/10.3390/fractalfract8040231 - 16 Apr 2024
Viewed by 723
Abstract
This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an [...] Read more.
This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an undisturbed initial state. Our aim is to compare the time evolutions of the physical system in fractional and integer order models by examining the torque transmission from the rotating body to the surrounding liquid. This is achieved through closed-form, time-developing solutions expressed in terms of Mittag–Leffler and Bessel functions. Analysis reveals that the rotational velocity and, consequently, the vortex structure of the liquid are influenced by three distinct time zones that differ between integer and noninteger models. Anomalous diffusion, favoring noninteger fractions, dominates at early times but gradually gives way to the integer derivative model behavior as time progresses through a transitional regime. Our derived vortex formula clearly demonstrates how the liquid vortex is regulated in time for each considered fractional model. Full article
(This article belongs to the Section Engineering)
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15 pages, 318 KiB  
Article
Further Fractional Hadamard Integral Inequalities Utilizing Extended Convex Functions
by Areej A. Almoneef, Mohamed A. Barakat and Abd-Allah Hyder
Fractal Fract. 2024, 8(4), 230; https://doi.org/10.3390/fractalfract8040230 - 16 Apr 2024
Viewed by 826
Abstract
This work investigates novel fractional Hadamard integral inequalities by utilizing extended convex functions and generalized Riemann-Liouville operators. By carefully using extended integral formulations, we not only find novel inequalities but also improve the accuracy of error bounds related to fractional Hadamard integrals. Our [...] Read more.
This work investigates novel fractional Hadamard integral inequalities by utilizing extended convex functions and generalized Riemann-Liouville operators. By carefully using extended integral formulations, we not only find novel inequalities but also improve the accuracy of error bounds related to fractional Hadamard integrals. Our study broadens the applicability of these inequalities and shows that they are useful for a variety of convexity cases. Our results contribute to the advancement of mathematical analysis and provide useful information for theoretical comprehension as well as practical applications across several scientific directions. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional, 2nd Edition)
38 pages, 1035 KiB  
Article
Novel Hopf Bifurcation Exploration and Control Strategies in the Fractional-Order FitzHugh–Nagumo Neural Model Incorporating Delay
by Yunzhang Zhang and Changjin Xu
Fractal Fract. 2024, 8(4), 229; https://doi.org/10.3390/fractalfract8040229 - 15 Apr 2024
Viewed by 688
Abstract
In this article, we propose a new fractional-order delay-coupled FitzHugh–Nagumo neural model. Taking advantage of delay as a bifurcation parameter, we explore the stability and bifurcation of the formulated fractional-order delay-coupled FitzHugh–Nagumo neural model. A delay-independent stability and bifurcation conditions for the fractional-order [...] Read more.
In this article, we propose a new fractional-order delay-coupled FitzHugh–Nagumo neural model. Taking advantage of delay as a bifurcation parameter, we explore the stability and bifurcation of the formulated fractional-order delay-coupled FitzHugh–Nagumo neural model. A delay-independent stability and bifurcation conditions for the fractional-order delay-coupled FitzHugh–Nagumo neural model is acquired. By designing a proper PDp controller, we can efficaciously control the stability domain and the time of emergence of the bifurcation phenomenon of the considered fractional delay-coupled FitzHugh–Nagumo neural model. By exploiting a reasonable hybrid controller, we can successfully adjust the stability domain and the bifurcation onset time of the involved fractional delay-coupled FitzHugh–Nagumo neural model. This study shows that when the delay crosses a critical value, a Hopf bifurcation will arise. When we adjust the control parameter, we can find other critical values to enlarge or narrow the stability domain of the fractional-order delay-coupled FitzHugh–Nagumo neural model. In order to check the correctness of the acquired outcomes of this article, we present some simulation outcomes via Matlab 7.0 software. The obtained theoretical fruits in this article have momentous theoretical significance in running and constructing networks. Full article
(This article belongs to the Special Issue Analysis and Control of Fractional-Order Delay Coupling Networks)
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16 pages, 6073 KiB  
Article
Fractal Characteristics and Microstructure of Coal with Impact of Starch-Polymerized Aluminum Sulfate Fracturing Fluids
by Feng Cai, Qian Zhang and Lingling Yang
Fractal Fract. 2024, 8(4), 228; https://doi.org/10.3390/fractalfract8040228 - 15 Apr 2024
Viewed by 725
Abstract
The degree of irregularity and complexity of the pore structure are comprehensively reflected in the fractal dimension. The porosity of coal was determined by its fractal dimension, where a larger dimension indicates a lower porosity. Fractal theory and the Frenkel–Halsey–Hill (FHH) model were [...] Read more.
The degree of irregularity and complexity of the pore structure are comprehensively reflected in the fractal dimension. The porosity of coal was determined by its fractal dimension, where a larger dimension indicates a lower porosity. Fractal theory and the Frenkel–Halsey–Hill (FHH) model were applied to explore the variation rules of concentration on functional groups and pore structure in this study. Combined with infrared spectroscopy (FTIR) and low-temperature nitrogen adsorption, a starch-polymerized aluminum sulfate composite fracturing fluid was prepared, which plays an important role in methane adsorption and permeability of coal samples. The test results showed that, compared with the original coal, the pore volume and specific surface area of each group of coal samples were reduced, the average pore diameter was initially enlarged and then declined, and fractal dimension D1 dropped by 5.4% to 15.4%, while fractal dimension D2 gained 1.2% to 7.9%. Moreover, the nitrogen adsorption of each group of coal samples was obviously lower than the original coal, and the concentration of starch-polymerized aluminum sulfate solution existed at a critical optimal concentration for the modification of the coal samples, and the nitrogen adsorption reached a minimum value of 0.6814 cm3/g at a concentration of 10%. The novel composite solution prepared by the combination of starch and flocculant in this paper enhanced the permeability of the coal seam, which is of great significance in improving the efficiency of coalbed methane mining. Full article
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19 pages, 1775 KiB  
Article
The Four-Dimensional Natural Transform Adomian Decomposition Method and (3+1)-Dimensional Fractional Coupled Burgers’ Equation
by Huda Alsaud and Hassan Eltayeb
Fractal Fract. 2024, 8(4), 227; https://doi.org/10.3390/fractalfract8040227 - 15 Apr 2024
Viewed by 645
Abstract
This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers’ equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In [...] Read more.
This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers’ equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In order to observe the influence of time-Caputo fractional derivatives on the outcomes of the aforementioned models, two examples are illustrated along with their three-dimensional figures. The effectiveness and reliability of this approach are validated through the analysis of these examples related to the (3+1)-dimensional time-singular fractional coupled Burgers’ equations. This study underscores the method’s applicability and effectiveness in addressing the complex mathematical models encountered in various scientific and engineering domains. Full article
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28 pages, 6558 KiB  
Article
Field-Programmable Analog Array Implementation of Neuromorphic Silicon Neurons with Fractional Dynamics
by Andrés J. Serrano-Balbontín, Inés Tejado and Blas M. Vinagre
Fractal Fract. 2024, 8(4), 226; https://doi.org/10.3390/fractalfract8040226 - 15 Apr 2024
Viewed by 783
Abstract
Silicon neurons are bioinspired circuits with the capability to reproduce the modulation through pulse-frequency observed in real neurons. They are of particular interest in closed-loop schemes to encode the control signal into pulses. This paper proposes the analog realization of neuromorphic silicon neurons [...] Read more.
Silicon neurons are bioinspired circuits with the capability to reproduce the modulation through pulse-frequency observed in real neurons. They are of particular interest in closed-loop schemes to encode the control signal into pulses. This paper proposes the analog realization of neuromorphic silicon neurons with fractional dynamics. In particular, the fractional-order (FO) operator is introduced into classical neurons with the intention of reproducing the adaptation that has been observed experimentally in real neurons, which is the variation in the firing frequency even when considering a constant or periodic incoming stimulus. For validation purposes, simulations using a field-programmable analog array (FPAA) are performed to verify the behavior of the circuits. Full article
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20 pages, 3992 KiB  
Article
A Fractional-Order Archimedean Spiral Moth–Flame Optimization Strategy to Solve Optimal Power Flows
by Abdul Wadood, Ejaz Ahmed, Sang Bong Rhee and Babar Sattar Khan
Fractal Fract. 2024, 8(4), 225; https://doi.org/10.3390/fractalfract8040225 - 13 Apr 2024
Viewed by 882
Abstract
This research utilizes the innovative fractional-order Archimedean spiral moth–flame optimization (FO-AMFO) technique to address the issues of the optimal reactive power dispatch (ORPD) problem. The formulated fitness function aims to minimize power losses and determine the ideal flow of reactive power for the [...] Read more.
This research utilizes the innovative fractional-order Archimedean spiral moth–flame optimization (FO-AMFO) technique to address the issues of the optimal reactive power dispatch (ORPD) problem. The formulated fitness function aims to minimize power losses and determine the ideal flow of reactive power for the IEEE 30- and 57-bus test systems. The extensive functions of the fractional evolutionary computing strategy are utilized to address the minimization problem of ORPD. This involves determining the control variables, such as VAR compensators, bus voltages, and the tap setting of the transformers. The effective incorporation of reactive compensation devices into traditional power grids has greatly reduced power losses; however, it has resulted in an increase in the complexity of optimization problems. A comparison of the findings indicates that swarming fractional intelligence using FO-AMFO surpassed the state-of-the-art competitors in terms of minimizing power losses. Full article
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34 pages, 28362 KiB  
Article
Fractal-Based Multi-Criteria Feature Selection to Enhance Predictive Capability of AI-Driven Mineral Prospectivity Mapping
by Tao Sun, Mei Feng, Wenbin Pu, Yue Liu, Fei Chen, Hongwei Zhang, Junqi Huang, Luting Mao and Zhiqiang Wang
Fractal Fract. 2024, 8(4), 224; https://doi.org/10.3390/fractalfract8040224 - 12 Apr 2024
Viewed by 725
Abstract
AI-driven mineral prospectivity mapping (MPM) is a valid and increasingly accepted tool for delineating the targets of mineral exploration, but it suffers from noisy and unrepresentative input features. In this study, a set of fractal and multifractal methods, including box-counting calculation, concentration–area fractal [...] Read more.
AI-driven mineral prospectivity mapping (MPM) is a valid and increasingly accepted tool for delineating the targets of mineral exploration, but it suffers from noisy and unrepresentative input features. In this study, a set of fractal and multifractal methods, including box-counting calculation, concentration–area fractal modeling, and multifractal analyses, were employed to excavate the underlying nonlinear mineralization-related information from geological features. Based on these methods, multiple feature selection criteria, namely prediction–area plot, K-means clustering, information gain, chi-square, and the Pearson correlation coefficient, were jointly applied to rank the relative importance of ore-related features and their fractal representations, so as to choose the optimal input feature dataset readily used for training predictive AI models. The results indicate that fault density, the multifractal spectrum width (∆α) of the Yanshanian intrusions, information dimension (D1) of magnetic anomalies, correlation dimension (D2) of iron-oxide alteration, and the D2 of argillic alteration serve as the most effective predictor features representative of the corresponding ore-controlling elements. The comparative results of the model assessment suggest that all the AI models trained by the fractal datasets outperform their counterparts trained by raw datasets, demonstrating a significant improvement in the predictive capability of fractal-trained AI models in terms of both classification accuracy and predictive efficiency. A Shapley additive explanation was employed to trace the contributions of these features and to explain the modeling results, which imply that fractal representations provide more discriminative and definitive feature values that enhance the cognitive capability of AI models trained by these data, thereby improving their predictive performance, especially for those indirect predictor features that show subtle correlations with mineralization in the raw dataset. In addition, fractal-trained models can benefit practical mineral exploration by outputting low-risk exploration targets that achieve higher capturing efficiency and by providing new mineralization clues extracted from remote sensing data. This study demonstrates that the fractal representations of geological features filtered by multi-criteria feature selection can provide a feasible and promising means of improving the predictive capability of AI-driven MPM. Full article
(This article belongs to the Special Issue Fractals in Geology and Geochemistry)
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21 pages, 4157 KiB  
Article
Designing Heuristic-Based Tuners for Fractional-Order PID Controllers in Automatic Voltage Regulator Systems Using a Hyper-Heuristic Approach
by Daniel Fernando Zambrano-Gutierrez, Gerardo Humberto Valencia-Rivera, Juan Gabriel Avina-Cervantes, Ivan Amaya and Jorge Mario Cruz-Duarte
Fractal Fract. 2024, 8(4), 223; https://doi.org/10.3390/fractalfract8040223 - 12 Apr 2024
Viewed by 845
Abstract
This work introduces an alternative approach for developing a customized Metaheuristic (MH) tailored for tuning a Fractional-Order Proportional-Integral-Derivative (FOPID) controller within an Automatic Voltage Regulator (AVR) system. Leveraging an Automated Algorithm Design (AAD) methodology, our strategy generates MHs by utilizing a population-based Search [...] Read more.
This work introduces an alternative approach for developing a customized Metaheuristic (MH) tailored for tuning a Fractional-Order Proportional-Integral-Derivative (FOPID) controller within an Automatic Voltage Regulator (AVR) system. Leveraging an Automated Algorithm Design (AAD) methodology, our strategy generates MHs by utilizing a population-based Search Operator (SO) domain, thus minimizing human-induced bias. This approach eliminates the need for manual coding or the daunting task of selecting an optimal algorithm from a vast collection of the current literature. The devised MH consists of two distinct SOs: a dynamic swarm perturbator succeeded by a Metropolis-type selector and a genetic crossover perturbator, followed by another Metropolis-type selector. This MH fine-tunes the FOPID controller’s parameters, aiming to enhance control performance by reducing overshoot, rise time, and settling time. Our research includes a comparative analysis with similar studies, revealing that our tailored MH significantly improves the FOPID controller’s speed by 1.69 times while virtually eliminating overshoot. Plus, we assess the tuned FOPID controller’s resilience against internal disturbances within AVR subsystems. The study also explores two facets of control performance: the impact of fractional orders on conventional PID controller efficiency and the delineating of a confidence region for stable and satisfactory AVR operation. This work’s main contributions are introducing an innovative method for deriving efficient MHs in electrical engineering and control systems and demonstrating the substantial benefits of precise controller tuning, as evidenced by the superior performance of our customized MH compared to existing solutions. Full article
(This article belongs to the Special Issue Inverse Problems for Fractional Differential Equations)
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14 pages, 661 KiB  
Article
Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation
by Mohammad Alshammari, Khaled Moaddy, Muhammad Naeem, Zainab Alsheekhhussain, Saleh Alshammari and M. Mossa Al-Sawalha
Fractal Fract. 2024, 8(4), 222; https://doi.org/10.3390/fractalfract8040222 - 11 Apr 2024
Cited by 1 | Viewed by 662
Abstract
The Zoomeron equation plays a significant role in many fields of physics, especially in soliton theory, such as helping to reveal new distinctive properties in different physical phenomena such as fluid dynamics, laser physics, and nonlinear optics. By using the Riccati–Bernoulli sub-ODE approach [...] Read more.
The Zoomeron equation plays a significant role in many fields of physics, especially in soliton theory, such as helping to reveal new distinctive properties in different physical phenomena such as fluid dynamics, laser physics, and nonlinear optics. By using the Riccati–Bernoulli sub-ODE approach and the Backlund transformation, we search for soliton solutions of the fractional Zoomeron nonlinear equation. A number of solutions have been put forth, such as kink, anti-kink, cuspon kink, lump-type kink solitons, single solitons, and others defined in terms of pseudo almost periodic functions. The (2 + 1)-dimensional fractional Zoomeron equation given in a form undergoes precise dynamics. We use the computational software, Matlab 19, to express these solutions graphically by changing the value of various parameters involved. A detailed analysis of their dynamics allows us to obtain completely better insights necessarily with the elementary physical phenomena controlled by the fractional Zoomeron equation. Full article
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35 pages, 18605 KiB  
Article
Predator–Prey Interaction with Fear Effects: Stability, Bifurcation and Two-Parameter Analysis Incorporating Complex and Fractal Behavior
by Qamar Din, Raja Atif Naseem and Muhammad Sajjad Shabbir
Fractal Fract. 2024, 8(4), 221; https://doi.org/10.3390/fractalfract8040221 - 11 Apr 2024
Viewed by 1419
Abstract
This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: [...] Read more.
This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models. Full article
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14 pages, 321 KiB  
Article
Ozaki-Type Bi-Close-to-Convex and Bi-Concave Functions Involving a Modified Caputo’s Fractional Operator Linked with a Three-Leaf Function
by Kaliappan Vijaya, Gangadharan Murugusundaramoorthy, Daniel Breaz, Georgia Irina Oros and Sheza M. El-Deeb
Fractal Fract. 2024, 8(4), 220; https://doi.org/10.3390/fractalfract8040220 - 10 Apr 2024
Viewed by 774
Abstract
The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with [...] Read more.
The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, |a2| and |a3|, are discussed. Several new implications of the findings are also highlighted as corollaries. Full article
13 pages, 344 KiB  
Article
On Some Impulsive Fractional Integro-Differential Equation with Anti-Periodic Conditions
by Ymnah Alruwaily, Kuppusamy Venkatachalam and El-sayed El-hady
Fractal Fract. 2024, 8(4), 219; https://doi.org/10.3390/fractalfract8040219 - 10 Apr 2024
Viewed by 674
Abstract
We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions [...] Read more.
We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions (BCs) established in this study are of a more general type and can be reduced to numerous specific examples by defining the parameters involved in the conditions. In this way, we extend some recent nice results. At the end, we use an example to verify our results. Full article
1 pages, 149 KiB  
Correction
Correction: Sirpal et al. Association between Opioid Dependence and Scale Free Fractal Brain Activity: An EEG Study. Fractal Fract. 2023, 7, 659
by Parikshat Sirpal, William A. Sikora, Desiree R. Azizoddin, Hazem H. Refai and Yuan Yang
Fractal Fract. 2024, 8(4), 218; https://doi.org/10.3390/fractalfract8040218 - 8 Apr 2024
Viewed by 537
Abstract
Desiree R. Azizoddin PsyD from the TSET Health Promotion Research Center, Stephenson Cancer Center at the University of Oklahoma Health Sciences Center, USA, was not included as an author in the original publication [...] Full article
(This article belongs to the Section Complexity)
17 pages, 1678 KiB  
Article
Dynamic Fractional-Order Grey Prediction Model with GWO and MLP for Forecasting Overseas Talent Mobility in China
by Geng Wu, Haiwei Fu, Peng Jiang, Rui Chi and Rongjiang Cai
Fractal Fract. 2024, 8(4), 217; https://doi.org/10.3390/fractalfract8040217 - 8 Apr 2024
Viewed by 709
Abstract
International students play a crucial role in China’s talent development strategy. Thus, predicting overseas talent mobility is essential for formulating scientifically reasonable talent introduction policies, optimizing talent cultivation systems, and fostering international talent cooperation. In this study, we proposed a novel fractional-order grey [...] Read more.
International students play a crucial role in China’s talent development strategy. Thus, predicting overseas talent mobility is essential for formulating scientifically reasonable talent introduction policies, optimizing talent cultivation systems, and fostering international talent cooperation. In this study, we proposed a novel fractional-order grey model based on the Multi-Layer Perceptron (MLP) and Grey Wolf Optimizer (GWO) algorithm to forecast the movement of overseas talent, namely MGDFGM(1,1). Compared to the traditional grey model FGM(1,1), which utilizes the same fractional order at all time points, the proposed MGDFGM(1,1) model dynamically adjusts the fractional-order values based on the time point. This dynamic adjustment enables our model to better capture the changing trends in the data, thereby enhancing the model’s fitting capability. To validate the effectiveness of the MGDFGM(1,1) model, we primarily utilize Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) as the evaluation criteria for the prediction accuracy, as well as standard deviation (STD) as an indicator of the model stability. Furthermore, we perform experimental analysis to evaluate the predictive performance of the MGDFGM(1,1) model in comparison to NAÏVE, ARIMA, GM(1,1), FGM(1,1), LSSVR, MLP, and LSTM. The research findings demonstrate that the MGDFGM(1,1) model achieves a remarkably high level of prediction accuracy and stability for forecasting overseas talent mobility in China. The implications of this study offer valuable insights and assistance to government departments involved in overseas talent management. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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9 pages, 256 KiB  
Article
A Note on Averaging Principles for Fractional Stochastic Differential Equations
by Jiankang Liu, Haodian Zhang, Jinbin Wang, Chen Jin, Jing Li and Wei Xu
Fractal Fract. 2024, 8(4), 216; https://doi.org/10.3390/fractalfract8040216 - 8 Apr 2024
Viewed by 694
Abstract
Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form [...] Read more.
Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is first proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements. Full article
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