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Fractal Fract., Volume 8, Issue 1 (January 2024) – 73 articles

Cover Story (view full-size image): This study has demonstrated fractal modeling to be a useful data-driven tool for identifying weak and mixed anomalies from geochemical data of the deep-sea sediments in the Pacific Ocean. Multifractal spectrum analysis shows that the spatial patterns of rare-earth elements (REEs) in the Northwestern Pacific Ocean Basin (NPOB) and Central Pacific Ocean Basin (CPOB) were similar, whereas they were different to those in the Eastern Pacific Ocean Basin (EPOB). Fractal filtering indicates that the vertical distribution of REEs in the NPOB and CPOB shows a downward trend from top to bottom, while that in the EPOB shows an upward trend. These results suggest that phosphorus was the main factor controlling REE enrichment in the EPOB, while phosphorus and calcium were the main factors in the NPOB and CPOB. View this paper
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22 pages, 360 KiB  
Article
Boundary Value Problem for a Coupled System of Nonlinear Fractional q-Difference Equations with Caputo Fractional Derivatives
by Saleh S. Redhwan, Maoan Han, Mohammed A. Almalahi, Mona Alsulami and Maryam Ahmed Alyami
Fractal Fract. 2024, 8(1), 73; https://doi.org/10.3390/fractalfract8010073 - 22 Jan 2024
Cited by 2 | Viewed by 1236
Abstract
This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic [...] Read more.
This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling. Full article
17 pages, 331 KiB  
Article
Matkowski-Type Functional Contractions under Locally Transitive Binary Relations and Applications to Singular Fractional Differential Equations
by Faizan Ahmad Khan, Nidal H. E. Eljaneid, Ahmed Alamer, Esmail Alshaban, Fahad Maqbul Alamrani and Adel Alatawi
Fractal Fract. 2024, 8(1), 72; https://doi.org/10.3390/fractalfract8010072 - 22 Jan 2024
Viewed by 1051
Abstract
This article presents a few fixed-point results under Matkowski-type functional contractive mapping using locally J-transitive binary relations. Our results strengthen, enhance, and consolidate numerous existent fixed-point results. To argue for the efficacy of our results, several illustrated examples are supplied. With the [...] Read more.
This article presents a few fixed-point results under Matkowski-type functional contractive mapping using locally J-transitive binary relations. Our results strengthen, enhance, and consolidate numerous existent fixed-point results. To argue for the efficacy of our results, several illustrated examples are supplied. With the help of our findings, we deal with the existence and uniqueness theorems pertaining to the solution of a variety of singular fractional differential equations. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
20 pages, 382 KiB  
Article
New Applications of Fractional q-Calculus Operator for a New Subclass of q-Starlike Functions Related with the Cardioid Domain
by Mohammad Faisal Khan and Mohammed AbaOud
Fractal Fract. 2024, 8(1), 71; https://doi.org/10.3390/fractalfract8010071 - 22 Jan 2024
Cited by 3 | Viewed by 1122
Abstract
Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical [...] Read more.
Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator Dq,λmρ,σ and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc U. Furthermore, we analyze novel findings with respect to the inverse function (μ1) within the class of q-starlike functions in U. The findings in this paper are easy to understand and show a connection between present and past studies. Full article
28 pages, 412 KiB  
Article
Positive Solutions for a System of Fractional q-Difference Equations with Multi-Point Boundary Conditions
by Rodica Luca
Fractal Fract. 2024, 8(1), 70; https://doi.org/10.3390/fractalfract8010070 - 21 Jan 2024
Cited by 1 | Viewed by 920
Abstract
We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. [...] Read more.
We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. The proofs of our principal findings employ a range of fixed-point theorems, including the Guo–Krasnosel’skii fixed-point theorem, the Leggett–Williams fixed-point theorem, the Schauder fixed-point theorem, and the Banach contraction mapping principle. Full article
13 pages, 7880 KiB  
Article
Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order
by Marius-F. Danca
Fractal Fract. 2024, 8(1), 69; https://doi.org/10.3390/fractalfract8010069 - 19 Jan 2024
Viewed by 1582
Abstract
This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the [...] Read more.
This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined. Full article
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26 pages, 22673 KiB  
Article
New Generalized Jacobi Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Multi-Term Variable-Order Time-Fractional Diffusion-Wave Equations
by Hany Mostafa Ahmed
Fractal Fract. 2024, 8(1), 68; https://doi.org/10.3390/fractalfract8010068 - 18 Jan 2024
Cited by 1 | Viewed by 1094
Abstract
The current study discusses a novel approach for numerically solving MTVO-TFDWEs under various conditions, such as IBCs and DBCs. It uses a class of GSJPs that satisfy the given conditions (IBCs or DBCs). One of the important parts of our method is establishing [...] Read more.
The current study discusses a novel approach for numerically solving MTVO-TFDWEs under various conditions, such as IBCs and DBCs. It uses a class of GSJPs that satisfy the given conditions (IBCs or DBCs). One of the important parts of our method is establishing OMs for Ods and VOFDs of GSJPs. The second part is using the SCM by utilizing these OMs. This algorithm enables the extraction of precision and efficacy in numerical solutions. We provide theoretical assurances of the treatment’s efficacy by validating its convergent and error investigations. Four examples are offered to clarify the approach’s practicability and precision; in each one, the IBCs and DBCs are considered. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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18 pages, 331 KiB  
Article
Exploring the Characteristics of Δh Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators
by Musawa Yahya Almusawa
Fractal Fract. 2024, 8(1), 67; https://doi.org/10.3390/fractalfract8010067 - 18 Jan 2024
Viewed by 1023
Abstract
The objective of this article is to introduce the h bivariate Appell polynomials hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the [...] Read more.
The objective of this article is to introduce the h bivariate Appell polynomials hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of h bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the h bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials h are also proved by demonstrating that the h bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of h bivariate Appell polynomials is provided, and symmetric identities for the h bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for h bivariate Appell polynomials. Additionally, generating relations for the h bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials. Full article
16 pages, 2096 KiB  
Article
The Caputo Nonlocal Structural Derivative Ultraslow Diffusion Model of Language Change and the Microscopic Mechanism
by Wei Xu, Hui Liu, Yingjie Liang and Shijun Zhao
Fractal Fract. 2024, 8(1), 66; https://doi.org/10.3390/fractalfract8010066 - 17 Jan 2024
Viewed by 1066
Abstract
Numerous studies have observed and analyzed the dynamics of language change from a diffusion perspective. As a complex and changeable system, the process of language change is characterized by a long memory that conforms to ultraslow diffusion. However, it is not perfectly suited [...] Read more.
Numerous studies have observed and analyzed the dynamics of language change from a diffusion perspective. As a complex and changeable system, the process of language change is characterized by a long memory that conforms to ultraslow diffusion. However, it is not perfectly suited for modeling with the traditional diffusion model. The Caputo nonlocal structural derivative is a further development of the classic Caputo fractional derivative. Its kernel function, characterized as an arbitrary function, proves highly effective in dealing with ultraslow diffusion. In this study, we utilized an extended logarithmic function to formulate a Caputo nonlocal structural derivative diffusion model for qualitatively analyzing the evolution process of language. The mean square displacement that grows logarithmically was derived through the Tauberian theorem and the Fourier–Laplace transform. Its effectiveness and credibility were verified by the appearance of already popular words on Japanese blogs. Compared to the random diffusion model, the Caputo nonlocal structural derivative diffusion model proves to be more precise in simulating the process of language change. The microscopic mechanism of ultraslow diffusion was explored using the continuous time random walk model, which involves a logarithmic function with a long tail. Both models incorporate memory effects, which can provide useful guidance for modeling diffusion behavior in other social phenomena. Full article
(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)
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25 pages, 4345 KiB  
Article
A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications
by Seyeon Lee, Hyunju Kim and Bongsoo Jang
Fractal Fract. 2024, 8(1), 65; https://doi.org/10.3390/fractalfract8010065 - 17 Jan 2024
Viewed by 1426
Abstract
In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of [...] Read more.
In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of time steps N grows. A fast algorithm to reduce the computational complexity of the memory term is investigated utilizing a sum-of-exponentials (SOEs) approximation. The conventional PCM is equipped with a fast algorithm, and it only requires linear time complexity O(N). Truncation and global error analyses are provided, achieving a uniform accuracy order O(h2) regardless of the fractional order for both the conventional and fast PCMs. We demonstrate numerical examples for nonlinear initial value problems and linear and nonlinear reaction-diffusion fractional-order partial differential equations (FPDEs) to numerically verify the efficiency and error estimates. Finally, the fast PCM is applied to the fractional-order Rössler dynamical system, and the numerical results prove that the computational cost consumed to obtain the bifurcation diagram is significantly reduced using the proposed fast algorithm. Full article
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22 pages, 3961 KiB  
Article
Local Singularity Spectrum: An Innovative Graphical Approach for Analyzing Detrital Zircon Geochronology Data in Provenance Analysis
by Wenlei Wang, Yingru Pei, Qiuming Cheng and Wenjun Wang
Fractal Fract. 2024, 8(1), 64; https://doi.org/10.3390/fractalfract8010064 - 17 Jan 2024
Cited by 1 | Viewed by 1195
Abstract
Detrital zircon geochronology plays a crucial role in provenance analysis, serving as one of the fundamental strategies. The age spectrum of detrital zircons collected from the sedimentary unit of interest is often compared or correlated with that of potential source terranes. However, biases [...] Read more.
Detrital zircon geochronology plays a crucial role in provenance analysis, serving as one of the fundamental strategies. The age spectrum of detrital zircons collected from the sedimentary unit of interest is often compared or correlated with that of potential source terranes. However, biases in the age data can arise due to factors related to detrital sampling, analysis techniques, and nonlinear geological mechanisms. The current study reviewed two sets of detrital zircon datasets established in 2011 and 2021 to discuss the origins of the Tibetan Plateau. These datasets collected from different media effectively demonstrate a progressive understanding of provenance affinity among the main terranes on the Tibetan Plateau. This highlights issues regarding weak and unclear temporal connections identified through analyzing the age spectrum for provenance analysis. Within this context, a local singularity analysis approach is currently employed to address issues associated with unclear and weak provenance information by characterizing local variations in nonlinear behaviors and enhancing detection sensitivity towards subtle anomalies. This new graphical approach effectively quantifies temporal variations in detrital zircon age populations and enhances identification of weak provenance information that may not be readily apparent on conventional age spectra. Full article
(This article belongs to the Special Issue Fractals in Geology and Geochemistry)
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15 pages, 320 KiB  
Article
Solutions for Hilfer-Type Linear Fractional Integro-Differential Equations with a Variable Coefficient
by Sigang Zhu, Huiwen Wang and Fang Li
Fractal Fract. 2024, 8(1), 63; https://doi.org/10.3390/fractalfract8010063 - 17 Jan 2024
Viewed by 953
Abstract
In this paper, we derive an explicit formula of solutions to Hilfer linear fractional integro-differential equations with a variable coefficient in a weighted space, and obtain the existence and uniqueness of solutions for fractional kinetic equations and fractional integro-differential equations with a generalized [...] Read more.
In this paper, we derive an explicit formula of solutions to Hilfer linear fractional integro-differential equations with a variable coefficient in a weighted space, and obtain the existence and uniqueness of solutions for fractional kinetic equations and fractional integro-differential equations with a generalized Mittag–Leffler function. An example is given to illustrate the result obtained. Full article
17 pages, 335 KiB  
Article
Advancements in Best Proximity Points: A Study in F-Metric Spaces with Applications
by Badriah Alamri
Fractal Fract. 2024, 8(1), 62; https://doi.org/10.3390/fractalfract8010062 - 16 Jan 2024
Viewed by 1057
Abstract
The purpose of this study is to explore the existence and uniqueness of the best proximity points for α,Θ-proximal contractions, a novel concept introduced in the context of F-metric spaces. Moreover, we provide an example to show the usability [...] Read more.
The purpose of this study is to explore the existence and uniqueness of the best proximity points for α,Θ-proximal contractions, a novel concept introduced in the context of F-metric spaces. Moreover, we provide an example to show the usability of the obtained results. To broaden the scope of this research area, we leverage our best proximity point results to demonstrate the existence and uniqueness of solutions for differential equations (equation of motion) and also for fractional differential equations. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
25 pages, 9327 KiB  
Article
Application of Tilt Integral Derivative for Efficient Speed Control and Operation of BLDC Motor Drive for Electric Vehicles
by Khairy Sayed, Hebatallah H. El-Zohri, Adel Ahmed and Mohamed Khamies
Fractal Fract. 2024, 8(1), 61; https://doi.org/10.3390/fractalfract8010061 - 16 Jan 2024
Cited by 1 | Viewed by 1397
Abstract
This study presents the tilt integral derivative (TID) controller technique for controlling the speed of BLDC motors in order to improve the real-time control of brushless direct current motors in electric vehicles. The TID controller is applied to the considered model to enhance [...] Read more.
This study presents the tilt integral derivative (TID) controller technique for controlling the speed of BLDC motors in order to improve the real-time control of brushless direct current motors in electric vehicles. The TID controller is applied to the considered model to enhance its performance, e.g., torque and speed. This control system manages the torque output, speed, and position of the motor to ensure precise and efficient operation in EV applications. Brushless direct current motors are becoming more and more popular due to their excellent torque, power factor, efficiency, and controllability. The differences between PID, TID, and PI controllers are compared. The outcomes demonstrated that the TID control enhanced the torque and current stability in addition to the BLDC system’s capacity to regulate speed. TID controllers provide better input power for BLDC (brushless DC) drives than PI and PID controllers do. Better transient responsiveness and robustness to disturbances are features of TID controller design, which can lead to more effective use of input power. TID controllers are an advantageous choice for BLDC drive applications because of their increased performance, which can result in increased system responsiveness and overall efficiency. In an experimental lab, a BLDC motor drive prototype is implemented in this study. To fully enhance the power electronic subsystem and the brushless DC motor’s real-time performance, a test bench was also built. Full article
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16 pages, 6131 KiB  
Article
Fractal Analysis of the Centrifuge Vibrograms
by Iaroslav Lavrenko, Anton Popov, Ivan Seleznov and Ken Kiyono
Fractal Fract. 2024, 8(1), 60; https://doi.org/10.3390/fractalfract8010060 - 16 Jan 2024
Viewed by 982
Abstract
This paper presents a new approach to analyzing the components of centrifuge rotor vibrograms using a 2D trajectory fractal analysis based on the Detrended Moving Average method. The method identifies the different noise oscillatory behavior of the rotor depending on the rotation frequencies, [...] Read more.
This paper presents a new approach to analyzing the components of centrifuge rotor vibrograms using a 2D trajectory fractal analysis based on the Detrended Moving Average method. The method identifies the different noise oscillatory behavior of the rotor depending on the rotation frequencies, ranging from non-stationary unbounded and 1/f pink noise to correlated and uncorrelated noise. Fractal characteristics of the vibrograms were computed for the first time and demonstrated differences for rotation frequencies close to the eigenfrequencies and far from them. This paper also discusses the influence of gyroscopic effects on the natural frequencies of centrifuge oscillations and the excitation of second harmonics when the centrifuge rotates at higher frequencies. The main cause of rotor vibration is identified as the mass imbalance of the rotors, and this paper proposes a vibration classification according to source nodes to diagnose serviceable and faulty technical systems. Finally, the possibility of anisotropy of the vibrogram is discussed, and the oriented fractal scaling components analysis method is applied to pave the way for further investigation. Full article
(This article belongs to the Section Engineering)
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4 pages, 160 KiB  
Editorial
Numerical and Analytical Methods for Differential Equations and Systems
by Burcu Gürbüz and Arran Fernandez
Fractal Fract. 2024, 8(1), 59; https://doi.org/10.3390/fractalfract8010059 - 16 Jan 2024
Viewed by 1158
Abstract
The theory and applications of differential equations have played an essential role both in the development of mathematics and in exploring new horizons in the applied sciences [...] Full article
18 pages, 920 KiB  
Article
Solutions to Fractional q-Kinetic Equations Involving Quantum Extensions of Generalized Hyper Mittag-Leffler Functions
by Mohammed Z. Alqarni, Mohamed Akel and Mohamed Abdalla
Fractal Fract. 2024, 8(1), 58; https://doi.org/10.3390/fractalfract8010058 - 16 Jan 2024
Cited by 1 | Viewed by 1030
Abstract
This manuscript focuses on new generalizations of q-Mittag-Leffler functions, called generalized hyper q-Mittag-Leffler functions, and discusses their regions of convergence and various fractional q operators. Moreover, the solutions to the q-fractional kinetic equations in terms of the investigated generalized hyper [...] Read more.
This manuscript focuses on new generalizations of q-Mittag-Leffler functions, called generalized hyper q-Mittag-Leffler functions, and discusses their regions of convergence and various fractional q operators. Moreover, the solutions to the q-fractional kinetic equations in terms of the investigated generalized hyper q-Mittag-Leffler functions are obtained by applying the q-Sumudu integral transform. Furthermore, we present solutions obtained as numerical graphs using the MATLAB 2018 program. Full article
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17 pages, 686 KiB  
Article
Some Fractional Stochastic Models for Neuronal Activity with Different Time-Scales and Correlated Inputs
by Enrica Pirozzi
Fractal Fract. 2024, 8(1), 57; https://doi.org/10.3390/fractalfract8010057 - 15 Jan 2024
Viewed by 1024
Abstract
In order to describe neuronal dynamics on different time-scales, we propose a stochastic model based on two coupled fractional stochastic differential equations, with different fractional orders. For the specified choice of involved functions and parameters, we provide three specific models, with/without leakage, with [...] Read more.
In order to describe neuronal dynamics on different time-scales, we propose a stochastic model based on two coupled fractional stochastic differential equations, with different fractional orders. For the specified choice of involved functions and parameters, we provide three specific models, with/without leakage, with fractional/non-fractional correlated inputs. We give explicit expressions of the process representing the voltage variation in the neuronal membrane. Expectation values and covariances are given and compared. Numerical evaluations of the average behaviors of involved processes are presented and discussed. Full article
(This article belongs to the Special Issue Fractional Models and Statistical Applications)
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14 pages, 4522 KiB  
Article
Multivariate Multiscale Higuchi Fractal Dimension and Its Application to Mechanical Signals
by Yuxing Li, Shuai Zhang, Lili Liang and Qiyu Ding
Fractal Fract. 2024, 8(1), 56; https://doi.org/10.3390/fractalfract8010056 - 15 Jan 2024
Cited by 4 | Viewed by 1216
Abstract
Fractal dimension, as a common nonlinear dynamics metric, is extensively applied in biomedicine, fault diagnosis, underwater acoustics, etc. However, traditional fractal dimension can only analyze the complexity of the time series given a single channel at a particular scale. To characterize the complexity [...] Read more.
Fractal dimension, as a common nonlinear dynamics metric, is extensively applied in biomedicine, fault diagnosis, underwater acoustics, etc. However, traditional fractal dimension can only analyze the complexity of the time series given a single channel at a particular scale. To characterize the complexity of multichannel time series, multichannel information processing was introduced, and multivariate Higuchi fractal dimension (MvHFD) was proposed. To further analyze the complexity at multiple scales, multivariate multiscale Higuchi fractal dimension (MvmHFD) was proposed by introducing multiscale processing algorithms as a technology that not only improved the use of fractal dimension in the analysis of multichannel information, but also characterized the complexity of the time series at multiple scales in the studied time series data. The effectiveness and feasibility of MvHFD and MvmHFD were verified by simulated signal experiments and real signal experiments, in which the simulation experiments tested the stability, computational efficiency, and signal separation performance of MvHFD and MvmHFD, and the real signal experiments tested the effect of MvmHFD on the recognition of multi-channel mechanical signals. The experimental results show that compared to other indicators, A achieves a recognition rate of 100% for signals in three features, which is at least 17.2% higher than for other metrics. Full article
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14 pages, 4262 KiB  
Article
Predicting the Remaining Useful Life of Turbofan Engines Using Fractional Lévy Stable Motion with Long-Range Dependence
by Deyu Qi, Zijiang Zhu, Fengmin Yao, Wanqing Song, Aleksey Kudreyko, Piercarlo Cattani and Francesco Villecco
Fractal Fract. 2024, 8(1), 55; https://doi.org/10.3390/fractalfract8010055 - 15 Jan 2024
Viewed by 1157
Abstract
Remaining useful life prediction guarantees a reliable and safe operation of turbofan engines. Long-range dependence (LRD) and heavy-tailed characteristics of degradation modeling make this method advantageous for the prediction of RUL. In this study, we propose fractional Lévy stable motion for degradation modeling. [...] Read more.
Remaining useful life prediction guarantees a reliable and safe operation of turbofan engines. Long-range dependence (LRD) and heavy-tailed characteristics of degradation modeling make this method advantageous for the prediction of RUL. In this study, we propose fractional Lévy stable motion for degradation modeling. First, we define fractional Lévy stable motion simulation algorithms. Then, we demonstrate the LRD and heavy-tailed property of fLsm to provide support for the model. The proposed method is validated with the C-MAPSS dataset obtained from the turbofan engine. Principle components analysis (PCA) is conducted to extract sources of variance. Experimental data show that the predictive model based on fLsm with exponential drift exhibits superior accuracy relative to the existing methods. Full article
(This article belongs to the Special Issue Spectral Methods for Fractional Functional Models)
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20 pages, 2743 KiB  
Article
A Fractional-Order ADRC Architecture for a PMSM Position Servo System with Improved Disturbance Rejection
by Shaohua Wang, He Gan, Ying Luo, Xin Luo and Yangquan Chen
Fractal Fract. 2024, 8(1), 54; https://doi.org/10.3390/fractalfract8010054 - 14 Jan 2024
Cited by 1 | Viewed by 1192
Abstract
This paper proposes an active disturbance rejection control (ADRC) architecture for a permanent magnet synchronous motor (PMSM) position servo system. The presented method achieved enhanced tracking and disturbance rejection performance with a limited observer bandwidth. The model-aided extended state observer (MESO)-based ADRC was [...] Read more.
This paper proposes an active disturbance rejection control (ADRC) architecture for a permanent magnet synchronous motor (PMSM) position servo system. The presented method achieved enhanced tracking and disturbance rejection performance with a limited observer bandwidth. The model-aided extended state observer (MESO)-based ADRC was designed for the current, speed, and position loops of the PMSM position servo system. By integrating known plant information, the MESO improved disturbance estimation with a limited observer bandwidth without amplifying the noise. Additionally, a fractional-order proportional-derivative (FOPD) controller was designed as the feedback controller for the speed loop to further enhance the disturbance rejection. A simulation and experimental tests were conducted on a PMSM servo platform. The results demonstrate not only that the proposed method achieved superior tracking performance but also that the position error of the proposed strategy decreases to 2.25% when the constant disturbance was input, significantly improving the disturbance rejection performance. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)
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17 pages, 1337 KiB  
Article
The Finite Volume Element Method for Time Fractional Generalized Burgers’ Equation
by Jingru Zhang and Qing Yang
Fractal Fract. 2024, 8(1), 53; https://doi.org/10.3390/fractalfract8010053 - 14 Jan 2024
Viewed by 958
Abstract
In this paper, we use the finite volume element method (FVEM) to approximate a one-dimensional, time fractional generalized Burgers’ equation. We construct the fully discrete finite volume element scheme for this equation by approximating the time fractional derivative term by the L1 [...] Read more.
In this paper, we use the finite volume element method (FVEM) to approximate a one-dimensional, time fractional generalized Burgers’ equation. We construct the fully discrete finite volume element scheme for this equation by approximating the time fractional derivative term by the L1 formula and approximating the spatial terms using FVEM. The convergence of the scheme is proven. Finally, numerical examples are provided to confirm the scheme’s validity. Full article
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21 pages, 481 KiB  
Article
Processing the Controllability of Control Systems with Distinct Fractional Derivatives via Kalman Filter and Gramian Matrix
by Muath Awadalla, Abir Chaouk, Maher Jneid, Kinda Abuasbeh and Jihan Alahmadi
Fractal Fract. 2024, 8(1), 52; https://doi.org/10.3390/fractalfract8010052 - 13 Jan 2024
Cited by 1 | Viewed by 947
Abstract
In this paper, we investigate the controllability conditions of linear control systems involving distinct local fractional derivatives. Sufficient conditions for controllability using Kalman rank conditions and the Gramian matrix are presented. We show that the controllability of the local fractional system can be [...] Read more.
In this paper, we investigate the controllability conditions of linear control systems involving distinct local fractional derivatives. Sufficient conditions for controllability using Kalman rank conditions and the Gramian matrix are presented. We show that the controllability of the local fractional system can be determined by the invertibility of the Gramian matrix and the full rank of the Kalman matrix. We also show that the local fractional system involving distinct orders is controllable if and only if the Gramian matrix is invertible. Illustrative examples and an application are provided to demonstrate the validity of the theoretical findings. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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20 pages, 372 KiB  
Article
A Mixed Finite Element Method for the Multi-Term Time-Fractional Reaction–Diffusion Equations
by Jie Zhao, Shubin Dong and Zhichao Fang
Fractal Fract. 2024, 8(1), 51; https://doi.org/10.3390/fractalfract8010051 - 12 Jan 2024
Viewed by 965
Abstract
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution [...] Read more.
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete L(L2(Ω)) norm and for the auxiliary variable λ in the discrete L((L2(Ω))2) and L(H(div,Ω)) norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results. Full article
39 pages, 5244 KiB  
Article
Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion
by Yen-Ching Chang
Fractal Fract. 2024, 8(1), 50; https://doi.org/10.3390/fractalfract8010050 - 12 Jan 2024
Viewed by 984
Abstract
The fractal dimension (D) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional Brownian [...] Read more.
The fractal dimension (D) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional Brownian motion (2D FBM) is commonly used because it has specific physical meaning and only contains the finite-valued parameter (a real value from 0 to 1) of the Hurst exponent (H). More usefully, H and D possess the relation of D = 3 − H. The accuracy of the maximum likelihood estimator (MLE) is the best among estimators, but its efficiency is appreciably low. Lately, an efficient MLE for the Hurst exponent was produced to greatly improve its efficiency, but it still incurs much higher computational costs. Therefore, in the paper, we put forward a deep-learning estimator through classification models. The trained deep-learning models for images of 2D FBM not only incur smaller computational costs but also provide smaller mean-squared errors than the efficient MLE, except for size 32 × 32 × 1. In particular, the computational times of the efficient MLE are up to 129, 3090, and 156248 times those of our proposed simple model for sizes 32 × 32 × 1, 64 × 64 × 1, and 128 × 128 × 1. Full article
(This article belongs to the Section Numerical and Computational Methods)
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25 pages, 23198 KiB  
Article
Fractal-Based Pattern Quantification of Mineral Grains: A Case Study of Yichun Rare-Metal Granite
by Yue Liu, Tao Sun, Kaixing Wu, Hongwei Zhang, Jingwei Zhang, Xinwen Jiang, Quanwei Lin and Mei Feng
Fractal Fract. 2024, 8(1), 49; https://doi.org/10.3390/fractalfract8010049 - 12 Jan 2024
Cited by 3 | Viewed by 1070
Abstract
The quantification of the irregular morphology and distribution pattern of mineral grains is an essential but challenging task in ore-related mineralogical research, allowing for tracing the footprints of pattern-forming geological processes that are crucial to understanding mineralization and/or diagenetic systems. In this study, [...] Read more.
The quantification of the irregular morphology and distribution pattern of mineral grains is an essential but challenging task in ore-related mineralogical research, allowing for tracing the footprints of pattern-forming geological processes that are crucial to understanding mineralization and/or diagenetic systems. In this study, a large model, namely, the Segmenting Anything Model (SAM), was employed to automatically segment and annotate quartz, lepidolite and albite grains derived from Yichun rare-metal granite (YCRMG), based on which a series of fractal and multifractal methods, including box-counting calculation, perimeter–area analysis and multifractal spectra, were implemented. The results indicate that the mineral grains from YCRMG show great scaling invariance within the range of 1.04~52,300 μm. The automatic annotation of mineral grains from photomicrographs yields accurate fractal dimensions with an error of only 0.6% and thus can be utilized for efficient fractal-based grain quantification. The resultant fractal dimensions display a distinct distribution pattern in the diagram of box-counting fractal dimension (Db) versus perimeter–area fractal dimension (DPA), in which lepidolites are sandwiched between greater-valued quartz and lower-valued albites. Snowball-textured albites, i.e., concentrically arranged albite laths in quartz and K-feldspar, exhibit characteristic Db values ranging from 1.6 to 1.7, which coincide with the fractal indices derived from the fractal growth model. The zonal albites exhibit a strictly increasing trend regarding the values of fractal and multifractal exponents from core to rim, forming a featured “fractal-index banding” in the radar diagram. This pattern suggests that the snowball texture gradually evolved from rim to core, thus leading to greater fractal indices of outer zones, which represent higher complexity and maturity of the evolving system, which supports a metasomatic origin of the snowball texture. Our study demonstrates that fractal analyses with the aid of a large model are effective and efficient in characterizing and understanding complex patterns of mineral grains. Full article
(This article belongs to the Special Issue Fractals in Geology and Geochemistry)
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27 pages, 13900 KiB  
Article
Full-Scale Pore Structure Characterization and Its Impact on Methane Adsorption Capacity and Seepage Capability: Differences between Shallow and Deep Coal from the Tiefa Basin in Northeastern China
by Na Zhang, Shuaidong Wang, Jiaqi Wu, Zheng Li and Xinyue Wang
Fractal Fract. 2024, 8(1), 48; https://doi.org/10.3390/fractalfract8010048 - 12 Jan 2024
Viewed by 998
Abstract
Deep low-rank coalbed methane (CBM) resources are numerous and widely distributed in China, although their exploration remains in its infancy. In this work, gas adsorption (N2/CO2), mercury intrusion porosimetry, and 3D CT reconstruction were performed on five coal samples [...] Read more.
Deep low-rank coalbed methane (CBM) resources are numerous and widely distributed in China, although their exploration remains in its infancy. In this work, gas adsorption (N2/CO2), mercury intrusion porosimetry, and 3D CT reconstruction were performed on five coal samples of deep and shallow low-rank coal from northeast China to analyze their pore structure. The impact of the features in the pore structure at full scale on the capacity for methane adsorption and seepage is discussed. The results indicate that there are significant differences between deep low-rank coal and shallow low-rank coal in terms of porosity, permeability, composition, and adsorption capacity. The full-scale pore distribution was dispersed over a broad range and exhibited a multi-peak distribution, with the majority of the peak concentrations occurring between 0.45–0.7 nm and 3–4 nm. Mesopores are prevalent in shallow coal rock, whereas micropores are the most numerous in deep coal rock. The primary contributors to the specific surface area of both deep and superficial coal rock are micropores. Three-dimensional CT reconstruction can characterize pores with pore size greater than 1 μm, while the dominating equivalent pore diameters (Deq) range from 1 to 10 μm. More mini-scale pores and fissures are observed in deep coal rock, while shallow coal rock processes greater total and connection porosity. Multifractal features are prevalent in the fractal qualities of all the numbered samples. An enhancement in pore structure heterogeneity occurs with increasing pore size. The pore structure of deep coal rock is more heterogeneous. Furthermore, methane adsorption capacity is favorably connected with D1 (0.4 nm < pore diameter ≤ 2 nm), D2 (2 nm < pore diameter ≤ 5 nm), micropore volume, and specific surface area and negatively correlated with D3 (5 nm < pore diameter ≤ 50 nm), showing that methane adsorption capability is primarily controlled by micropores and mesopores. Methane seepage capacity is favorably connected with the pore volume and connected porosity of macropores and negatively correlated with D4 (pore diameter > 50 nm), indicating that the macropores are the primary factor influencing methane seepage capacity. Full article
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10 pages, 330 KiB  
Article
Stability in the Sense of Hyers–Ulam–Rassias for the Impulsive Volterra Equation
by El-sayed El-hady, Süleyman Öğrekçi, Tania A. Lazăr and Vasile L. Lazăr
Fractal Fract. 2024, 8(1), 47; https://doi.org/10.3390/fractalfract8010047 - 12 Jan 2024
Viewed by 964
Abstract
This article aims to use various fixed-point techniques to study the stability issue of the impulsive Volterra integral equation in the sense of Ulam–Hyers (sometimes known as Hyers–Ulam) and Hyers–Ulam–Rassias. By eliminating key assumptions, we are able to expand upon and enhance some [...] Read more.
This article aims to use various fixed-point techniques to study the stability issue of the impulsive Volterra integral equation in the sense of Ulam–Hyers (sometimes known as Hyers–Ulam) and Hyers–Ulam–Rassias. By eliminating key assumptions, we are able to expand upon and enhance some recent findings. Full article
13 pages, 506 KiB  
Article
Privacy Preservation of Nabla Discrete Fractional-Order Dynamic Systems
by Jiayue Ma, Jiangping Hu and Zhinan Peng
Fractal Fract. 2024, 8(1), 46; https://doi.org/10.3390/fractalfract8010046 - 11 Jan 2024
Viewed by 969
Abstract
This article investigates the differential privacy of the initial state for nabla discrete fractional-order dynamic systems. A novel differentially private Gaussian mechanism is developed which enhances the system’s security by injecting random noise into the output state. Since the existence of random noise [...] Read more.
This article investigates the differential privacy of the initial state for nabla discrete fractional-order dynamic systems. A novel differentially private Gaussian mechanism is developed which enhances the system’s security by injecting random noise into the output state. Since the existence of random noise gives rise to the difficulty of analyzing the nabla discrete fractional-order systems, to cope with this challenge, the observability of nabla discrete fractional-order systems is introduced, establishing a connection between observability and differential privacy of initial values. Based on it, the noise magnitude required for ensuring differential privacy is determined by utilizing the observability Gramian matrix of systems. Furthermore, an optimal Gaussian noise distribution that maximizes algorithmic performance while simultaneously ensuring differential privacy is formulated. Finally, a numerical simulation is provided to validate the effectiveness of the theoretical analysis. Full article
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19 pages, 4841 KiB  
Article
Geochemical Characteristics of Deep-Sea Sediments in Different Pacific Ocean Regions: Insights from Fractal Modeling
by Yan Zhang, Gaowen He, Fan Xiao, Yong Yang, Fenlian Wang and Yonggang Liu
Fractal Fract. 2024, 8(1), 45; https://doi.org/10.3390/fractalfract8010045 - 11 Jan 2024
Cited by 1 | Viewed by 1130
Abstract
Exploration of mineral resources in the deep sea has become an international trend. However, deep-sea mineral exploration faces challenges such as complex offshore drilling and the weak and mixed signals of ore deposits. Therefore, studying methods for identifying weak and mixed anomalies and [...] Read more.
Exploration of mineral resources in the deep sea has become an international trend. However, deep-sea mineral exploration faces challenges such as complex offshore drilling and the weak and mixed signals of ore deposits. Therefore, studying methods for identifying weak and mixed anomalies and extracting composite information in the deep sea is crucial for innovative prediction and evaluation of deep-sea mineral resources. In this study, the Central Pacific Ocean, Northwestern Pacific Ocean, and Eastern Pacific Ocean were selected as research areas. Drawing upon the fractal self-similarity exhibited by rare earth minerals in the deep-sea sediments within the Pacific Ocean, we conducted an analysis and comparison of the fractal geochemical characteristics in various regions of the Pacific Ocean’s deep-sea sediments. Thereafter, we studied the spatial distribution of rare earth elements (REEs) in deep-sea sediments in these regions to explore the mechanisms responsible for rare earth enrichment in the Pacific Ocean. The results revealed that the geochemical fractal characteristics of deep-sea sediments in the Northwestern Pacific Ocean Basin and the Central Pacific Ocean Basin were similar, whereas there were slight differences in the fractal characteristics observed in the Eastern Pacific Ocean Basin. By calculating the singularity index of CaO/P2O5, it was found that the singularity index in the Central and Northwestern Pacific Ocean basins was lower than that in the Eastern Pacific Ocean Basin, suggesting that the phosphorus content in the Eastern Pacific Ocean Basin was lower than that in the Central and Northwestern Pacific Ocean basins. In the Eastern Pacific Ocean, we found that phosphorus content in deep-sea sediments was the primary controlling factor for REE enrichment. Conversely, in the Central and Northwestern Pacific Ocean, both the phosphorus and calcium content in deep-sea sediments played significant roles in REE enrichment. Full article
(This article belongs to the Special Issue Fractals in Geology and Geochemistry)
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19 pages, 1196 KiB  
Article
Fractional Optimal Control Model and Bifurcation Analysis of Human Syncytial Respiratory Virus Transmission Dynamics
by Muath Awadalla, Jihan Alahmadi, Kumama Regassa Cheneke and Sania Qureshi
Fractal Fract. 2024, 8(1), 44; https://doi.org/10.3390/fractalfract8010044 - 11 Jan 2024
Cited by 4 | Viewed by 1251
Abstract
In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction [...] Read more.
In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction number is calculated by using the next-generation matrix’s spectral radius. The fractional optimal control model includes the control functions of vaccination and treatment to illustrate the impact of these interventions on the dynamics of virus transmission. In addition, the order of the derivative in the fractional optimal control problem indicates that encouraging vaccination and treatment early on can slow the spread of RSV. The overall analysis and the simulated behavior of the fractional optimum control model are in good agreement, and this is due in large part to the use of the MATLAB platform. Full article
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