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Fractal Fract., Volume 7, Issue 11 (November 2023) – 59 articles

Cover Story (view full-size image): Pore structure features govern the capacity of gas storage and migration in shales and are highly dependent on the types of pores, i.e., interparticle (InterP) pores, intraparticle (IntraP) pores, and organic matter (OM)-hosted pores. On the basis of high-resolution imaging, this work focuses on the distinctive structural and fractal features inherent to pore types and their respective contributions to bulk permeability. Overall, clustered and well-connected pores hosted by OM correspond to the smallest pore diameter and the highest fractal dimension (D), while rounded and scarce IntraP pores translate into the lowest D. A considerable enhancement in overall permeability can be expected when OM-hosted pores interconnect with crack-like pores (e.g., InterP pores) or other effective pore networks. View this paper
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28 pages, 1633 KiB  
Article
Asymptotic and Mittag–Leffler Synchronization of Fractional-Order Octonion-Valued Neural Networks with Neutral-Type and Mixed Delays
Fractal Fract. 2023, 7(11), 830; https://doi.org/10.3390/fractalfract7110830 - 20 Nov 2023
Viewed by 865
Abstract
Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They [...] Read more.
Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example. Full article
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16 pages, 4550 KiB  
Article
The Formative Factors of a Rock Burst Based on Energy Calculations and the Experimental Verification of Butterfly-Shaped Plastic Zones
Fractal Fract. 2023, 7(11), 829; https://doi.org/10.3390/fractalfract7110829 - 20 Nov 2023
Viewed by 765
Abstract
The research on the formation factors of rock burst is one of the main research directions of rock mechanics in recent years, which is helpful to solve the problem of rock burst accidents. So, in this study, the calculation method of energy released [...] Read more.
The research on the formation factors of rock burst is one of the main research directions of rock mechanics in recent years, which is helpful to solve the problem of rock burst accidents. So, in this study, the calculation method of energy released during rock burst is first obtained by using different medium models, and then, the formation factors of rock bursts are obtained by comparing the calculation energy with the actual accident energy. The method of energy calculation utilizes the difference between elastoplastic and pure elastic models to innovatively quantify the specific values of energy released before and after the occurrence of the rock burst. It is considered that the stress and plastic zone state before the occurrence of rock burst have an important influence on the occurrence of the accident and are one of the formation factors, while the deviatoric stress field and butterfly-shaped plastic zone create conditions for greater energy release. In addition, the trigger stress constitutes another formation factor. The plastic zone state before rock failure is verified by the experimental test; the location distribution shape of acoustic emission (AE) events during the later stage of compression failure is approximately the same as theoretical result. The results also preliminarily indicated the fractal characteristics of acoustic emission events distribution before sample failure. The study obtained the formative factors of rock burst accident, which provides a new ideas and references for the research on the formation of rock bursts. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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13 pages, 427 KiB  
Article
General Methods to Synchronize Fractional Discrete Reaction–Diffusion Systems Applied to the Glycolysis Model
Fractal Fract. 2023, 7(11), 828; https://doi.org/10.3390/fractalfract7110828 - 20 Nov 2023
Viewed by 862
Abstract
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this [...] Read more.
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this study. Furthermore, we investigate the complex synchronization dynamics of a suggested discrete master–slave reaction–diffusion system using the accuracy of linear control techniques combined with a fractional discrete Lyapunov approach. This study’s deviation from the behavior of equivalents with integer orders makes it very fascinating. Like the non-local nature inherent in Caputo fractional derivatives, it creates a memory Lyapunov function that is closely linked to the historical background of the system. The investigation provides a strong basis to the theoretical results. Full article
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19 pages, 4278 KiB  
Article
Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon
Fractal Fract. 2023, 7(11), 827; https://doi.org/10.3390/fractalfract7110827 - 17 Nov 2023
Viewed by 926
Abstract
The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with [...] Read more.
The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with the RUL, which is based on adaptive fractional Lévy stable motion (AfLSM) and integrated with the Mellin–Stieltjes transform and Monte Carlo simulation. The proposed degradation model exhibits flexibility for capturing long-range dependence, has a non-Gaussian distribution, and accurately describes heavy-tailed properties. Additionally, the nonlinear drift coefficients of the model can be adaptively updated on the basis of the degradation trajectory. The performance of the proposed RUL prediction model was verified by using the University of Maryland CALEC dataset. Our forecasting results demonstrate the high accuracy of the method and its superiority over other state-of-the-art methods. Full article
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18 pages, 2141 KiB  
Article
Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz–Rutishauser Algorithm
Fractal Fract. 2023, 7(11), 826; https://doi.org/10.3390/fractalfract7110826 - 16 Nov 2023
Cited by 1 | Viewed by 861
Abstract
Orthogonal generalized Laguerre moments of fractional orders (FrGLMs) are signal and image descriptors. The utilization of the FrGLMs in the analysis of big-size signals encounters three challenges. First, calculating the high-order moments is a time-consuming process. Second, accumulating numerical errors leads to numerical [...] Read more.
Orthogonal generalized Laguerre moments of fractional orders (FrGLMs) are signal and image descriptors. The utilization of the FrGLMs in the analysis of big-size signals encounters three challenges. First, calculating the high-order moments is a time-consuming process. Second, accumulating numerical errors leads to numerical instability and degrades the reconstructed signals’ quality. Third, the QR decomposition technique is needed to preserve the orthogonality of the higher-order moments. In this paper, the authors derived a new recurrence formula for calculating the FrGLMs, significantly reducing the computational CPU times. We used the Schwarz–Rutishauser algorithm as an alternative to the QR decomposition technique. The proposed method for computing FrGLMs for big-size signals is accurate, simple, and fast. The proposed algorithm has been tested using the MIT-BIH arrhythmia benchmark dataset. The results show the proposed method’s superiority over existing methods in terms of processing time and reconstruction capability. Concerning the reconstructed capability, it has achieved superiority with average values of 25.3233 and 15.6507 with the two metrics PSNR and MSE, respectively. Concerning the elapsed reconstruction time, it also achieved high superiority with an efficiency gain of 0.8. The proposed method is suitable for utilization in the Internet of Healthcare Things. Full article
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15 pages, 408 KiB  
Article
New Approach to Quasi-Synchronization of Fractional-Order Delayed Neural Networks
Fractal Fract. 2023, 7(11), 825; https://doi.org/10.3390/fractalfract7110825 - 16 Nov 2023
Viewed by 860
Abstract
This article investigates quasi-synchronization for a class of fractional-order delayed neural networks. By utilizing the properties of the Laplace transform, the Caputo derivative, and the Mittag–Leffler function, a new fractional-order differential inequality is introduced. Furthermore, an adaptive controller is designed, resulting in the [...] Read more.
This article investigates quasi-synchronization for a class of fractional-order delayed neural networks. By utilizing the properties of the Laplace transform, the Caputo derivative, and the Mittag–Leffler function, a new fractional-order differential inequality is introduced. Furthermore, an adaptive controller is designed, resulting in the derivation of an effective criterion to ensure the aforementioned synchronization. Finally, a numerical illustration is provided to demonstrate the validity of the presented theoretical findings. Full article
(This article belongs to the Special Issue Recent Advances in Fractional-Order Time Delay Systems)
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11 pages, 1036 KiB  
Article
Dilatancy Equation Based on the Property-Dependent Plastic Potential Theory for Geomaterials
Fractal Fract. 2023, 7(11), 824; https://doi.org/10.3390/fractalfract7110824 - 15 Nov 2023
Cited by 10 | Viewed by 959
Abstract
The dilatancy equation ignores the noncoaxiality of granular soil for the coaxial assumption of the direction of the stress and strain rate in conventional plastic potential theory, which is inconsistent with extensive laboratory tests. To reasonably describe the noncoaxial effects on dilatancy, the [...] Read more.
The dilatancy equation ignores the noncoaxiality of granular soil for the coaxial assumption of the direction of the stress and strain rate in conventional plastic potential theory, which is inconsistent with extensive laboratory tests. To reasonably describe the noncoaxial effects on dilatancy, the energy dissipation of plastic flow is derived based on the property-dependent plastic potential theory for geomaterials and integrates the noncoaxiality, the potential theory links the plastic strain of granular materials with its fabric, and the noncoaxiality is naturally related to the mesoscopic properties of materials. When the fabric is isotropic, the dilatancy equation degenerates into the form of the critical state theory, and when the fabric is anisotropic, it naturally describes the effects of noncoaxiality. In the plane stress state, a comparison between a simple shear test and prediction of the dilatancy equation shows that the equation can reasonably describe the effect of noncoaxiality on dilatancy with the introduction of microscopic fabric parameters, and its physical significance is clear. This paper can provide a reference for the theoretical description of the macro and micro mechanical properties of geomaterials. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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15 pages, 979 KiB  
Article
Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains
Fractal Fract. 2023, 7(11), 823; https://doi.org/10.3390/fractalfract7110823 - 15 Nov 2023
Cited by 1 | Viewed by 856
Abstract
Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long [...] Read more.
Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long distance in either the forward or backward direction along preferential channels during the transport. The classical advection–diffusion (ADE) model has been widely used to describe tracer transport; however, they often fall short in capturing the intricacies of nonlocal diffusion processes. The fractional operator has gained recognition among hydrologists due to its potential to capture distinct mechanisms of transport and storage for tracer particles exhibiting nonlocal dynamics. However, the hypersingularity of the fractional Laplacian operator presents considerable difficulties in its numerical approximation in bounded domains. This study focuses on the development and application of the fractional Laplacian-based model to characterize nonlocal tracer transport behavior, specifically considering both forward and backward diffusion processes on bounded domains. The Riesz fractional Laplacian provides a mathematical framework for describing tracer diffusion processes on a bounded domain, and a novel finite difference method (FDM) is introduced as an effective numerical solver for addressing the fractional Laplacian-based model. Applications reveal that the fractional Laplacian-based model can effectively capture the observed nonlocal tracer transport behavior in a heterogeneous system, and nonlocal tracer transport exhibits dynamic characteristics, influenced by the evolving heterogeneity of the media at various temporal scales. Full article
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21 pages, 10940 KiB  
Article
Parameter Estimation of LFM Signals Based on FOTD-CFRFT under Impulsive Noise
Fractal Fract. 2023, 7(11), 822; https://doi.org/10.3390/fractalfract7110822 - 15 Nov 2023
Viewed by 668
Abstract
Due to the short duration and high amplitude characteristics of impulsive noise, these parameter estimation methods based on Gaussian assumptions are ineffective in the presence of impulsive noise. To address this issue, a LFM signal parameter estimation method is proposed based on FOTD [...] Read more.
Due to the short duration and high amplitude characteristics of impulsive noise, these parameter estimation methods based on Gaussian assumptions are ineffective in the presence of impulsive noise. To address this issue, a LFM signal parameter estimation method is proposed based on FOTD and CFRFT. Firstly, the mathematical expression of FOTD is presented and its tracking performance is verified. Secondly, the tracked signal is subjected to discrete time CFRFT, and a mathematical optimization model for LFM signal parameter estimation is established on the fractional spectrum characteristic. Finally, a correction method for non-standard SαS distributed noise is proposed, and the performance of parameter estimation under both standard and non-standard SαS distributions are analyzed. The simulation results show that this method not only effectively suppresses the impact of impulsive noise on the fractional spectrum of LFM signal, but also has better parameter estimation accuracy and stability in the low GSNR. The proposed method is particularly effective under the measured noise environment, as it successfully suppresses the impact of impulsive noise and achieves high-precision parameter estimation. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)
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10 pages, 2685 KiB  
Article
Experimental Study on the Cross-Scale Relationship of Cemented Backfill under the Action of an Air-Entraining Agent
Fractal Fract. 2023, 7(11), 821; https://doi.org/10.3390/fractalfract7110821 - 15 Nov 2023
Viewed by 739
Abstract
Air-entraining agents have the function of optimizing pores and improving the performance of backfill. In this study, we used tailings and cement as the main raw materials and added different amounts of air-entraining agents to make backfill samples. By testing the uniaxial compressive [...] Read more.
Air-entraining agents have the function of optimizing pores and improving the performance of backfill. In this study, we used tailings and cement as the main raw materials and added different amounts of air-entraining agents to make backfill samples. By testing the uniaxial compressive strength (UCS) and microstructure, macro- and micro characteristics were studied. Nuclear magnetic resonance technology was used to explore pore characteristics, and fractal theory was used to quantitatively discuss the complexity of pore structure. Finally, a cross-scale relationship model between UCS and pores was established. The main conclusions are as follows: (1) Adding the appropriate amount of air-entraining agents can optimize pore structure and increase the UCS of backfill materials, which is beneficial to backfill materials. (2) The pores of backfill materials have fractal characteristics, the fractal effects of pores with different pore size ranges are different, and the air-entraining agent has a certain influence on the fractal characteristics of the pores. (3) There are inverse relationships between UCS and different pore size ranges. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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16 pages, 5547 KiB  
Article
An Algorithm for Crack Detection, Segmentation, and Fractal Dimension Estimation in Low-Light Environments by Fusing FFT and Convolutional Neural Network
Fractal Fract. 2023, 7(11), 820; https://doi.org/10.3390/fractalfract7110820 - 14 Nov 2023
Viewed by 845
Abstract
The segmentation of crack detection and severity assessment in low-light environments presents a formidable challenge. To address this, we propose a novel dual encoder structure, denoted as DSD-Net, which integrates fast Fourier transform with a convolutional neural network. In this framework, we incorporate [...] Read more.
The segmentation of crack detection and severity assessment in low-light environments presents a formidable challenge. To address this, we propose a novel dual encoder structure, denoted as DSD-Net, which integrates fast Fourier transform with a convolutional neural network. In this framework, we incorporate an information extraction module and an attention feature fusion module to effectively capture contextual global information and extract pertinent local features. Furthermore, we introduce a fractal dimension estimation method into the network, seamlessly integrated as an end-to-end task, augmenting the proficiency of professionals in detecting crack pathology within low-light settings. Subsequently, we curate a specialized dataset comprising instances of crack pathology in low-light conditions to facilitate the training and evaluation of the DSD-Net algorithm. Comparative experimentation attests to the commendable performance of DSD-Net in low-light environments, exhibiting superlative precision (88.5%), recall (85.3%), and F1 score (86.9%) in the detection task. Notably, DSD-Net exhibits a diminutive Model Size (35.3 MB) and elevated Frame Per Second (80.4 f/s), thereby endowing it with the potential to be seamlessly integrated into edge detection devices, thus amplifying its practical utility. Full article
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21 pages, 918 KiB  
Article
On Mixed Fractional Lifting Oscillation Spaces
Fractal Fract. 2023, 7(11), 819; https://doi.org/10.3390/fractalfract7110819 - 13 Nov 2023
Viewed by 585
Abstract
We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on Rd, with d2. Contrary to Besov spaces and fractional Sobolev spaces with dominating mixed smoothness, the [...] Read more.
We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on Rd, with d2. Contrary to Besov spaces and fractional Sobolev spaces with dominating mixed smoothness, the new spaces take into account the geometric disposition of the hyperbolic wavelet coefficients at each scale (j1,,jd), and are therefore suitable for a multifractal analysis of rectangular regularity. We prove that hyperbolic oscillation spaces are closely related to hyperbolic variation spaces, and consequently do not almost depend on the chosen hyperbolic wavelet basis. Therefore, the so-called rectangular multifractal analysis, related to hyperbolic oscillation spaces, is somehow ‘robust’, i.e., does not change if the analyzing wavelets were changed. We also study optimal relationships between hyperbolic and mixed fractional lifting oscillation spaces and Besov spaces with dominating mixed smoothness. In particular, we show that, for some indices, hyperbolic and mixed fractional lifting oscillation spaces are not always sharply imbedded between Besov spaces or fractional Sobolev spaces with dominating mixed smoothness, and thus are new spaces of a really different nature. Full article
(This article belongs to the Section General Mathematics, Analysis)
14 pages, 2507 KiB  
Article
Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network
Fractal Fract. 2023, 7(11), 818; https://doi.org/10.3390/fractalfract7110818 - 13 Nov 2023
Viewed by 1165
Abstract
Studying the firing dynamics and phase synchronization behavior of heterogeneous coupled networks helps us understand the mechanism of human brain activity. In this study, we propose a novel small heterogeneous coupled network in which the 2D Hopfield neural network (HNN) and the 2D [...] Read more.
Studying the firing dynamics and phase synchronization behavior of heterogeneous coupled networks helps us understand the mechanism of human brain activity. In this study, we propose a novel small heterogeneous coupled network in which the 2D Hopfield neural network (HNN) and the 2D Hindmarsh–Rose (HR) neuron are coupled through a locally active memristor. The simulation results show that the network exhibits complex dynamic behavior and is different from the usual phase synchronization. More specifically, the membrane potential of the 2D HR neuron exhibits five stable firing modes as the coupling parameter k1 changes. In addition, it is found that in the local region of k1, the number of spikes in bursting firing increases with the increase in k1. More interestingly, the network gradually changes from synchronous to asynchronous during the increase in the coupling parameter k1 but suddenly becomes synchronous around the coupling parameter k1 = 1.96. As far as we know, this abnormal synchronization behavior is different from the existing findings. This research is inspired by the fact that the episodic synchronous abnormal firing of excitatory neurons in the hippocampus of the brain can lead to diseases such as epilepsy. This helps us further understand the mechanism of brain activity and build bionic systems. Finally, we design the simulation circuit of the network and implement it on an STM32 microcontroller. Full article
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14 pages, 311 KiB  
Article
Solving a Nonlinear Fractional Differential Equation Using Fixed Point Results in Orthogonal Metric Spaces
Fractal Fract. 2023, 7(11), 817; https://doi.org/10.3390/fractalfract7110817 - 12 Nov 2023
Cited by 2 | Viewed by 749
Abstract
This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal [...] Read more.
This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
23 pages, 393 KiB  
Article
Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions
Fractal Fract. 2023, 7(11), 816; https://doi.org/10.3390/fractalfract7110816 - 11 Nov 2023
Viewed by 782
Abstract
We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply [...] Read more.
We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative. Full article
20 pages, 389 KiB  
Article
Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient
Fractal Fract. 2023, 7(11), 815; https://doi.org/10.3390/fractalfract7110815 - 11 Nov 2023
Viewed by 738
Abstract
Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces [...] Read more.
Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension dimp(B). On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients. Full article
19 pages, 1729 KiB  
Article
Anisotropic Fractional Cosmology: K-Essence Theory
Fractal Fract. 2023, 7(11), 814; https://doi.org/10.3390/fractalfract7110814 - 09 Nov 2023
Viewed by 760
Abstract
In the particular configuration of the scalar field k-essence in the Wheeler–DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is [...] Read more.
In the particular configuration of the scalar field k-essence in the Wheeler–DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is β=2α2α1. This fractional equation belongs to different intervals depending on the value of the barotropic parameter; when ωX[0,1], the order belongs to the interval 1β2, and when ωX[1,0), the order belongs to the interval 0<β1. In the quantum scheme, we introduce the factor ordering problem in the variables (Ω,ϕ) and its corresponding momenta (ΠΩ,Πϕ), obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time T(τ); however, in the dust era we found a closed solution in the gauge time τ. Full article
(This article belongs to the Special Issue Fractional Gravity/Cosmology in Classical and Quantum Regimes)
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21 pages, 2498 KiB  
Article
Fixed-Time Distributed Time-Varying Optimization for Nonlinear Fractional-Order Multiagent Systems with Unbalanced Digraphs
Fractal Fract. 2023, 7(11), 813; https://doi.org/10.3390/fractalfract7110813 - 09 Nov 2023
Viewed by 880
Abstract
This paper investigates the problem of fixed-time distributed time-varying optimization of a nonlinear fractional-order multiagent system (FOMAS) over a weight-unbalanced directed graph (digraph), where the heterogeneous unknown nonlinear functions and disturbances are involved. The aim is to cooperatively minimize a convex time-varying global [...] Read more.
This paper investigates the problem of fixed-time distributed time-varying optimization of a nonlinear fractional-order multiagent system (FOMAS) over a weight-unbalanced directed graph (digraph), where the heterogeneous unknown nonlinear functions and disturbances are involved. The aim is to cooperatively minimize a convex time-varying global cost function produced by a sum of time-varying local cost functions within a fixed time, where each time-varying local cost function does not have to be convex. Using a three-step design procedure, a fully distributed fixed-time optimization algorithm is constructed to achieve the objective. The first step is to design a fully distributed fixed-time estimator to estimate some centralized optimization terms within a fixed time T0. The second step is to develop a novel discontinuous fixed-time sliding mode algorithm with nominal controller to derive all the agents to the sliding-mode surface within a fixed time T1, and meanwhile the dynamics of each agent is described by a single-integrator MAS with nominal controller. In the third step, a novel estimator-based fully distributed fixed-time nominal controller for the single-integrator MAS is presented to guarantee all agents reach consensus within a fixed time T2, and afterwards minimize the convex time-varying global cost function within a fixed time T3. The upper bound of each fixed time Tm(m=0,1,2,3) is given explicitly, which is independent of the initial states. Finally, a numerical example is provided to validate the results. Full article
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20 pages, 1176 KiB  
Article
Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform
Fractal Fract. 2023, 7(11), 812; https://doi.org/10.3390/fractalfract7110812 - 09 Nov 2023
Cited by 1 | Viewed by 839
Abstract
The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy [...] Read more.
The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues. Full article
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13 pages, 1217 KiB  
Article
Research on Information Identification of Chaotic Map with Multi-Stability
Fractal Fract. 2023, 7(11), 811; https://doi.org/10.3390/fractalfract7110811 - 09 Nov 2023
Viewed by 816
Abstract
Influenced by the rapid development of artificial intelligence, the identification of chaotic systems with intelligent optimization algorithms has received widespread attention in recent years. This paper focuses on the intelligent information identification of chaotic maps with multi-stability properties, and an improved sparrow search [...] Read more.
Influenced by the rapid development of artificial intelligence, the identification of chaotic systems with intelligent optimization algorithms has received widespread attention in recent years. This paper focuses on the intelligent information identification of chaotic maps with multi-stability properties, and an improved sparrow search algorithm is proposed as the identification algorithm. Numerical simulations show that different initial values can lead to the same dynamic behavior, making it impossible to stably and accurately identify the initial values of multi-stability chaotic maps. An identification scheme without considering the initial values is proposed for solving this problem, and simulations demonstrate that the proposed method has the highest identification precision among seven existing intelligent algorithms and a certain degree of noise resistance. In addition, the above research reveals that chaotic systems with multi-stability may have more potential applications in fields such as secure communication. Full article
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20 pages, 1733 KiB  
Article
Disturbance Observer-Based Event-Triggered Adaptive Command Filtered Backstepping Control for Fractional-Order Nonlinear Systems and Its Application
Fractal Fract. 2023, 7(11), 810; https://doi.org/10.3390/fractalfract7110810 - 09 Nov 2023
Cited by 2 | Viewed by 737
Abstract
This paper considers the disturbance observer-based event-triggered adaptive fuzzy tracking control issue for a class of fractional-order nonlinear systems (FONSs) with quantized signals and unknown disturbances. To improve the disturbance rejection ability, a fractional-order nonlinear disturbance observer (FONDO) is designed to estimate the [...] Read more.
This paper considers the disturbance observer-based event-triggered adaptive fuzzy tracking control issue for a class of fractional-order nonlinear systems (FONSs) with quantized signals and unknown disturbances. To improve the disturbance rejection ability, a fractional-order nonlinear disturbance observer (FONDO) is designed to estimate the unknown composite disturbances. Furthermore, by combining an improved fractional-order command-filtered backstepping control technique and an event-triggered control mechanism, an event-triggered adaptive fuzzy quantized control scheme is established, which guarantees the desired tracking performance can be achieved even in the presence of network constraint. Finally, the validity and superiority of the theoretic results are verified by a fractional-order horizontal platform system. Full article
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28 pages, 9236 KiB  
Article
The Fractional Soliton Wave Propagation of Non-Linear Volatility and Option Pricing Systems with a Sensitive Demonstration
Fractal Fract. 2023, 7(11), 809; https://doi.org/10.3390/fractalfract7110809 - 09 Nov 2023
Cited by 1 | Viewed by 792
Abstract
In this study, we explore a fractional non-linear coupled option pricing and volatility system. The model under consideration can be viewed as a fractional non-linear coupled wave alternative to the Black–Scholes option pricing governing system, introducing a leveraging effect where stock volatility corresponds [...] Read more.
In this study, we explore a fractional non-linear coupled option pricing and volatility system. The model under consideration can be viewed as a fractional non-linear coupled wave alternative to the Black–Scholes option pricing governing system, introducing a leveraging effect where stock volatility corresponds to stock returns. Employing the inverse scattering transformation, we find that the Cauchy problem for this model is insolvable. Consequently, we utilize the Φ6-expansion algorithm to generate generalized novel solitonic analytical wave structures within the system. We present graphical representations in contour, 3D, and 2D formats to illustrate how the system’s behavior responds to the propagation of pulses, enabling us to predict suitable parameter values that align with the data. Finally, a conclusion is given. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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19 pages, 4068 KiB  
Article
Multi-Machine Power System Transient Stability Enhancement Utilizing a Fractional Order-Based Nonlinear Stabilizer
Fractal Fract. 2023, 7(11), 808; https://doi.org/10.3390/fractalfract7110808 - 07 Nov 2023
Viewed by 1044
Abstract
Given the intricate nature of contemporary energy systems, addressing the control and stability analysis of these systems necessitates the consideration of highly large-scale models. Transient stability analysis stands as a crucial challenge in enhancing energy system efficiency. Power System Stabilizers (PSSs), integrated within [...] Read more.
Given the intricate nature of contemporary energy systems, addressing the control and stability analysis of these systems necessitates the consideration of highly large-scale models. Transient stability analysis stands as a crucial challenge in enhancing energy system efficiency. Power System Stabilizers (PSSs), integrated within excitation control for synchronous generators, offer a cost-effective means to bolster power systems’ stability and reliability. In this study, we propose an enhanced nonlinear control strategy based on synergetic control theory for PSSs. This strategy aims to mitigate electromechanical oscillations and rectify the limitations associated with linear approximations within large-scale energy systems that incorporate thyristor-controlled series capacitors (TCSCs). To dynamically adjust the coefficients of the nonlinear controller, we employ the Fractional Order Fish Migration Optimization (FOFMO) algorithm, rooted in fractional calculus (FC) theory. The FOFMO algorithm adapts by updating position and velocity within fractional-order structures. To assess the effectiveness of the improved controller, comprehensive numerical simulations are conducted. Initially, we examine its performance in a single machine connected to the infinite bus (SMIB) power system under various fault conditions. Subsequently, we extend the application of the proposed nonlinear stabilizer to a two-area, four-machine power system. Our numerical results reveal highly promising advancements in both control accuracy and the dynamic characteristics of controlled power systems. Full article
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44 pages, 17085 KiB  
Article
First-of-Its-Kind Frequency Enhancement Methodology Based on an Optimized Combination of FLC and TFOIDFF Controllers Evaluated on EVs, SMES, and UPFC-Integrated Smart Grid
Fractal Fract. 2023, 7(11), 807; https://doi.org/10.3390/fractalfract7110807 - 06 Nov 2023
Viewed by 906
Abstract
The most recent advancements in renewable energy resources, as well as their broad acceptance in power sectors, have created substantial operational, security, and management concerns. As a result of the continual decrease in power system inertia, it is critical to maintain the normal [...] Read more.
The most recent advancements in renewable energy resources, as well as their broad acceptance in power sectors, have created substantial operational, security, and management concerns. As a result of the continual decrease in power system inertia, it is critical to maintain the normal operating frequency and reduce tie-line power changes. The preceding issues sparked this research, which proposes the Fuzzy Tilted Fractional Order Integral Derivative with Fractional Filter (FTFOIDFF), a unique load frequency controller. The FTFOIDFF controller described here combines the benefits of tilt, fuzzy logic, FOPID, and fractional filter controllers. Furthermore, the prairie dog optimizer (PDO), a newly developed metaheuristic optimization approach, is shown to efficiently tune the suggested controller settings as well as the forms of the fuzzy logic membership functions in the two-area hybrid power grid investigated in this paper. When the PDO results are compared to those of the Seagull Optimization Algorithm, the Runge Kutta optimizer, and the Chaos Game Optimizer for the same hybrid power system, PDO prevails. The system model incorporates physical constraints such as communication time delays and generation rate constraints. In addition, a unified power flow controller (UPFC) is put in the tie-line, and SMES units have been planned in both regions. Furthermore, the contribution of electric vehicles (EVs) is considered in both sections. The proposed PDO-based FTFOIDFF controller outperformed many PDO-based traditional (such as proportional integral derivative (PID), proportional integral derivative acceleration (PIDA), and TFOIDFF) and intelligent (such as Fuzzy PID and Fuzzy PIDA) controllers from the literature. The suggested PDO-based FTFOIDFF controller has excellent performance due to the usage of various load patterns such as step load perturbation, multi-step load perturbation, random load perturbation, random sinusoidal load perturbation, and pulse load perturbation. Furthermore, a variety of scenarios have been implemented to demonstrate the advantageous effects that SMES, UPFC, and EV units have on the overall performance of the system. The sensitivity of a system is ascertained by modifying its parameters from their standard configurations. According to the simulation results, the suggested PDO-based FTFOIDFF controller can improve system stability despite the multiple difficult conditions indicated previously. According to the MATLAB/Simulink data, the proposed method decreased the total fitness function to 0.0875, representing a 97.35% improvement over PID, 95.84% improvement over PIDA, 92.45% improvement over TFOIDFF, 83.43% improvement over Fuzzy PID, and 37.9% improvement over Fuzzy PIDA. Full article
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20 pages, 13348 KiB  
Article
Design and Performance Evaluation of a Second-Order Iterated Circular Minkowski Fractal Antenna for Ultra-Wideband Applications
Fractal Fract. 2023, 7(11), 806; https://doi.org/10.3390/fractalfract7110806 - 06 Nov 2023
Viewed by 948
Abstract
In this article, a second-order iterated Circular Minkowski fractal antenna (CMFA) tailored for ultra-wideband (UWB) applications is designed and developed. Leveraging the power of Minkowski fractal geometry, this antenna design achieves a high gain across the UWB frequency spectrum. The design utilizes a [...] Read more.
In this article, a second-order iterated Circular Minkowski fractal antenna (CMFA) tailored for ultra-wideband (UWB) applications is designed and developed. Leveraging the power of Minkowski fractal geometry, this antenna design achieves a high gain across the UWB frequency spectrum. The design utilizes a circular groove on the ground plane and an arc slot on the radiating element for improving the antenna performance. The proposed antenna is fabricated using cost-effective material, an FR-4 substrate. The antenna is simulated and optimized. The fabricated optimized antenna undergoes real-world testing. Measured results reveal an impressive 120.6% impedance bandwidth spanning from 3.37 GHz to 13.6 GHz, with resonant frequencies at 4.43 GHz, 6.07 GHz, and 9.3 GHz. Meanwhile, the simulated results indicate an impedance bandwidth of 118% ranging from 3.17 GHz to 12.44 GHz. Real-world measurements validate the anticipated UWB traits, closely aligning with the simulation data, and confirming efficient impedance matching with a VSWR of less than 2 across the 3.37 GHz to 13.6 GHz frequency range. The radiation pattern analysis demonstrates a robust bidirectional E-plane pattern and a nearly omnidirectional H-plane pattern. This research introduces a highly promising circular Minkowski fractal antenna for UWB applications, offering exceptional bandwidth and resonance characteristics. This antenna design holds excellent potential for multi-functional wireless systems and opens avenues for enhanced UWB communication and sensing capabilities in diverse applications. Full article
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14 pages, 624 KiB  
Article
Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative
Fractal Fract. 2023, 7(11), 805; https://doi.org/10.3390/fractalfract7110805 - 06 Nov 2023
Cited by 2 | Viewed by 1159
Abstract
Breast cancer ranks among the most prevalent malignancies affecting the female population and is a prominent contributor to cancer-related mortality. Mathematical modeling is a significant tool that can be employed to comprehend the dynamics of breast cancer progression and dissemination and to formulate [...] Read more.
Breast cancer ranks among the most prevalent malignancies affecting the female population and is a prominent contributor to cancer-related mortality. Mathematical modeling is a significant tool that can be employed to comprehend the dynamics of breast cancer progression and dissemination and to formulate novel therapeutic approaches. This paper introduces a mathematical model of breast cancer that utilizes the Caputo–Fabrizio fractal-fractional derivative. The aim is to elucidate and comprehend the intricate dynamics governing breast cancer cells and cytotoxic T lymphocytes in the context of the fractional derivative. The derivative presented herein offers a broader perspective than the conventional derivative, as it incorporates the intricate fractal characteristics inherent in the process of tumor proliferation. The significance of this study lies in its contribution to a novel mathematical model for breast cancer, which incorporates the fractal characteristics of tumor development. The present model possesses the capability to investigate the impacts of diverse treatment strategies on the proliferation of breast cancer, as well as to formulate novel treatment strategies that exhibit enhanced efficacy. Full article
(This article belongs to the Section Life Science, Biophysics)
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18 pages, 578 KiB  
Article
Qualitative Analysis of RLC Circuit Described by Hilfer Derivative with Numerical Treatment Using the Lagrange Polynomial Method
Fractal Fract. 2023, 7(11), 804; https://doi.org/10.3390/fractalfract7110804 - 04 Nov 2023
Viewed by 784
Abstract
This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and [...] Read more.
This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and Banach’s contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples. Full article
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0 pages, 5303 KiB  
Article
Pore-Type-Dependent Fractal Features of Shales and Implications on Permeability
Fractal Fract. 2023, 7(11), 803; https://doi.org/10.3390/fractalfract7110803 - 04 Nov 2023
Viewed by 836
Abstract
Pore structure features govern the capacity of gas storage and migration in shales and are highly dependent on the types of pores, i.e., interparticle (InterP) pores, intraparticle (IntraP) pores and organic matter (OM)-hosted pores. However, fractal features in terms of pore types and [...] Read more.
Pore structure features govern the capacity of gas storage and migration in shales and are highly dependent on the types of pores, i.e., interparticle (InterP) pores, intraparticle (IntraP) pores and organic matter (OM)-hosted pores. However, fractal features in terms of pore types and their respective contributions to permeability have been rarely addressed. On the basis of high-resolution imaging, fractal dimensions (Ds) have been determined from both pore size distributions and digital rock to quantify the heterogeneity in pore morphology and spatial textures. Overall, OM-hosted pores are smaller in size and more abundant in quantity, corresponding to a relatively high D, while IntraP pores are mainly isolated and scarce, translating into lower D values. Additionally, crack-like InterP pores with a moderate level of porosity and the D can play a pivotal role in shale seepage potential. A comparison of the estimated permeability among different pore types highlights that the contribution of interconnected OM pores to the overall permeability remains constrained unless they can link neighboring pore clusters, as commonly observed in organo-clay composites. Furthermore, the pore morphology and fractal features of shale rocks can exhibit noteworthy variations subjected to sedimentology, mineralogy, diagenesis and OM maturation. Full article
(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering)
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19 pages, 646 KiB  
Article
An Investigation on Fractal Characteristics of the Superposition of Fractal Surfaces
Fractal Fract. 2023, 7(11), 802; https://doi.org/10.3390/fractalfract7110802 - 04 Nov 2023
Viewed by 791
Abstract
In this paper, we conduct research on the fractal characteristics of the superposition of fractal surfaces from the view of fractal dimension. We give the upper bound of the lower and upper box dimensions of the graph of the sum of two bivariate [...] Read more.
In this paper, we conduct research on the fractal characteristics of the superposition of fractal surfaces from the view of fractal dimension. We give the upper bound of the lower and upper box dimensions of the graph of the sum of two bivariate continuous functions and calculate the exact values of them under some particular conditions. Further, it has been proven that the superposition of two continuous surfaces cannot keep the fractal dimensions invariable unless both of them are two-dimensional. A concrete example of a numerical experiment has been provided to verify our theoretical results. This study can be applied to the fractal analysis of metal fracture surfaces or computer image surfaces. Full article
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21 pages, 10065 KiB  
Article
Parallel Algorithm for Solving the Inverse Two-Dimensional Fractional Diffusion Problem of Identifying the Source Term
Fractal Fract. 2023, 7(11), 801; https://doi.org/10.3390/fractalfract7110801 - 02 Nov 2023
Viewed by 800
Abstract
This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each [...] Read more.
This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each iteration of the method, we need to solve the auxilliary direct initial-boundary value problem. By using the finite difference scheme, this problem is reduced to solving a large system of a linear algebraic equation with a block-tridiagonal matrix at each time step. Solving the system takes almost the entire computation time. To solve this system, we construct and implement the direct parallel matrix sweep algorithm. We establish stability and correctness for this algorithm. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to study the performance of parallel implementations. Full article
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