Journal Description
Fractal and Fractional
Fractal and Fractional
is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering published monthly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, and other databases.
- Journal Rank: JCR - Q1 (Mathematics, Interdisciplinary Applications) / CiteScore - Q1 (Analysis)
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 18.9 days after submission; acceptance to publication is undertaken in 3.5 days (median values for papers published in this journal in the second half of 2023).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Impact Factor:
5.4 (2022);
5-Year Impact Factor:
4.7 (2022)
Latest Articles
Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach
Fractal Fract. 2024, 8(5), 298; https://doi.org/10.3390/fractalfract8050298 (registering DOI) - 18 May 2024
Abstract
This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is
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This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is examined using the transition between phase plane orbits. Their bifurcation and dependence on initial conditions are investigated. Some of these solutions are shown graphically, illustrating both the individual and combined influences of fractional order and noise on selected solutions. These effects appear as alterations in the amplitude and width of the solutions, and as variations in their smoothness.
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(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
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Synchronization of Fractional Delayed Memristive Neural Networks with Jump Mismatches via Event-Based Hybrid Impulsive Controller
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Huiyu Wang, Shutang Liu, Xiang Wu, Jie Sun and Wei Qiao
Fractal Fract. 2024, 8(5), 297; https://doi.org/10.3390/fractalfract8050297 (registering DOI) - 18 May 2024
Abstract
This study investigates the asymptotic synchronization in fractional memristive neural networks of the Riemann–Liouville type, considering mixed time delays and jump mismatches. Addressing the challenges associated with discrepancies in the circuit switching speed and the accuracy of the memristor, this paper introduces an
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This study investigates the asymptotic synchronization in fractional memristive neural networks of the Riemann–Liouville type, considering mixed time delays and jump mismatches. Addressing the challenges associated with discrepancies in the circuit switching speed and the accuracy of the memristor, this paper introduces an enhanced model that effectively navigates these complexities. We propose two novel event-based hybrid impulsive controllers, each characterized by unique triggering conditions. Utilizing advanced techniques in inequality and hybrid impulsive control, we establish the conditions necessary for achieving synchronization through innovative Lyapunov functions. Importantly, the developed controllers are theoretically optimized to minimize control costs, an essential consideration for their practical deployment. Finally, the effectiveness of our proposed approach is demonstrated through two illustrative simulation examples.
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(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
Open AccessArticle
Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints
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Yuanyuan Deng and Yi Zhang
Fractal Fract. 2024, 8(5), 296; https://doi.org/10.3390/fractalfract8050296 (registering DOI) - 18 May 2024
Abstract
This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the
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This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided.
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Open AccessArticle
Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators
by
Hassan M. Serag, Areej A. Almoneef, Mahmoud El-Badawy and Abd-Allah Hyder
Fractal Fract. 2024, 8(5), 295; https://doi.org/10.3390/fractalfract8050295 - 16 May 2024
Abstract
This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which
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This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system’s state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed.
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(This article belongs to the Special Issue Optimal Control Problems for Fractional Differential Equations)
Open AccessArticle
Critical Exponents and Universality for Fractal Time Processes above the Upper Critical Dimensionality
by
Shaolong Zeng, Yangfan Hu, Shijing Tan and Biao Wang
Fractal Fract. 2024, 8(5), 294; https://doi.org/10.3390/fractalfract8050294 - 16 May 2024
Abstract
We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For
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We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For fractal time processes, we not only discover new universality classes with a dimensional constant but also decompose the dangerous irrelevant variables to obtain corrections for critical dynamic behavior and static critical properties. This contrasts with the traditional theory of critical phenomena, which posits that static critical exponents are unrelated to the dynamical processes. Simulations of the Landau–Ginzburg model for fractal time processes and the Ising model with temporal long-range interactions both show good agreement with our set of critical exponents, verifying its universality. The discovery of this new universality class provides a method for examining whether a system is undergoing a fractal time process near the critical point.
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(This article belongs to the Special Issue Fractional Models and Statistical Applications)
Open AccessArticle
Dynamics for a Nonlinear Stochastic Cholera Epidemic Model under Lévy Noise
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Qura Tul Ain, Anwarud Din, Xiaoli Qiang and Zheng Kou
Fractal Fract. 2024, 8(5), 293; https://doi.org/10.3390/fractalfract8050293 - 16 May 2024
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In this study, we develop a comprehensive mathematical model to analyze the dynamics of epidemic cholera, characterized by acute diarrhea due to pathogen overabundance in the human body. The model is first developed from a deterministic point of view, and then it is
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In this study, we develop a comprehensive mathematical model to analyze the dynamics of epidemic cholera, characterized by acute diarrhea due to pathogen overabundance in the human body. The model is first developed from a deterministic point of view, and then it is modified to include the randomness by stochastic differential equations. The study selected Lévy noise above other well-known types of noise, emphasizing its importance in epidemic modeling. Besides presenting a biological justification for the stochastic system, we demonstrate that the equivalent deterministic model exhibits possible equilibria. The introduction is followed by theoretical analysis of the model. Through rigorous analysis, we establish that the stochastic model ensures a unique global solution. Lyapunov function theory is applied to construct necessary conditions, which on average, guarantee the model’s stability for . Our findings suggest the likelihood of eradicating the disease when is below one, a significant insight supported by graphical simulations of the model. Graphical illustrations were generated from simulating the model in order to increase the analytical results’ robustness. This work provides a strong theoretical framework for a thorough comprehension of a range of such diseases. This research not only provides a deeper understanding of cholera dynamics but also offers a robust theoretical framework applicable to a range of similar diseases, alongside a novel approach for constructing Lyapunov functions for nonlinear models with random disturbances.
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Open AccessArticle
Existence Results Related to a Singular Fractional Double-Phase Problem in the Whole Space
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Ramzi Alsaedi
Fractal Fract. 2024, 8(5), 292; https://doi.org/10.3390/fractalfract8050292 - 16 May 2024
Abstract
In this paper, we will study a singular problem involving the fractional - -Laplacian operator in the whole space
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In this paper, we will study a singular problem involving the fractional - -Laplacian operator in the whole space . More precisely, we combine the variational method with monotonicity arguments to prove that the associated functional energy admits a critical point, which is a weak solution for such a problem.
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(This article belongs to the Special Issue Fractional Calculus and Nonlinear Analysis: Theory and Applications)
Open AccessArticle
Fractional-Order Dynamics in Epidemic Disease Modeling with Advanced Perspectives of Fractional Calculus
by
Muhammad Riaz, Zareen A. Khan, Sadique Ahmad and Abdelhamied Ashraf Ateya
Fractal Fract. 2024, 8(5), 291; https://doi.org/10.3390/fractalfract8050291 - 15 May 2024
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Piecewise fractional-order differential operators have received more attention in recent years because they can be used to describe various evolutionary dynamical problems to investigate crossover behaviors. In this manuscript, we use the aforementioned operators to investigate a mathematical model of COVID-19. By utilizing
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Piecewise fractional-order differential operators have received more attention in recent years because they can be used to describe various evolutionary dynamical problems to investigate crossover behaviors. In this manuscript, we use the aforementioned operators to investigate a mathematical model of COVID-19. By utilizing fractional calculus, our approach aims to capture the crossover dynamics of disease spread, considering heterogeneity and transitions between epidemic phases. This research seeks to develop a framework using specialized mathematical techniques, such as the Caputo fractional derivative, with the potential to investigate the crossover dynamical behaviors of the considered epidemic model. The anticipated contribution lies in bridging fractional calculus and epidemiology, offering insights for both theoretical advancements and practical public health interventions. In order to improve our understanding of epidemic dynamics and support, we used MATLAB to simulate numerical results for a visual representation of our findings. For this interpretation, we used various fractional-order values. In addition, we also compare our simulated results with some reported results for infected and death classes to demonstrate the efficiency of our numerical method.
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Open AccessArticle
Exponential H∞ Output Control for Switching Fuzzy Systems via Event-Triggered Mechanism and Logarithmic Quantization
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Jiaojiao Ren, Can Zhao, Jianying Xiao, Renfu Luo and Nanrong He
Fractal Fract. 2024, 8(5), 290; https://doi.org/10.3390/fractalfract8050290 - 15 May 2024
Abstract
This paper investigates the problem of exponential output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT: ) and the upper bound of inter-event intervals (IEIs: T)
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This paper investigates the problem of exponential output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT: ) and the upper bound of inter-event intervals (IEIs: T) satisfy the condition , implying that frequent switching is difficult to achieve, this paper demonstrates that by adopting the mode-dependent event-triggered mechanism (ETM) and a switching law, frequent switching is indeed achieved. Moreover, the question of deriving the normal norm constraint is solved through the ADT method, although only a weighted norm constraint was obtained previously. Additionally, by constructing a controller-mode-dependent Lyapunov function and adopting logarithmic quantizers, the sufficient criteria of exponential output control problem are presented. The validity of established results is demonstrated by a given numerical simulation.
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(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)
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Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces
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Zainab Alsheekhhussain, Ahmed Gamal Ibrahim, M. Mossa Al-Sawalha and Khudhayr A. Rashedi
Fractal Fract. 2024, 8(5), 289; https://doi.org/10.3390/fractalfract8050289 - 13 May 2024
Abstract
The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, -Hilfer, fractional derivative of order with non-instantaneous impulses in Banach spaces
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The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, -Hilfer, fractional derivative of order with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted -Laplace transform, w-weighted -convolution and the measure of non-compactness. Since the operator, the w-weighted -Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results.
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Open AccessArticle
Pore Fractal Characteristics between Marine and Marine–Continental Transitional Black Shales: A Case Study of Niutitang Formation and Longtan Formation
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Shitan Ning, Peng Xia, Fang Hao, Jinqiang Tian, Yong Fu and Ke Wang
Fractal Fract. 2024, 8(5), 288; https://doi.org/10.3390/fractalfract8050288 - 13 May 2024
Abstract
Marine shales from the Niutitang Formation and marine–continental transitional shales from the Longtan Formation are two sets of extremely important hydrocarbon source rocks in South China. In order to quantitatively compare the pore complexity characteristics between marine and marine–continental transitional shales, the shale
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Marine shales from the Niutitang Formation and marine–continental transitional shales from the Longtan Formation are two sets of extremely important hydrocarbon source rocks in South China. In order to quantitatively compare the pore complexity characteristics between marine and marine–continental transitional shales, the shale and kerogen of the Niutitang Formation and the Longtan Formation are taken as our research subjects. Based on organic petrology, geochemistry, and low-temperature gas adsorption analyses, the fractal dimension of their pores is calculated by the Frenkel–Halsey–Hill (FHH) and Sierpinski models, and the influences of total organic carbon (TOC), vitrinite reflectance (Ro), and mineral composition on the pore fractals of the shale and kerogen are discussed. Our results show the following: (1) Marine shale predominantly has wedge-shaped and slit pores, while marine–continental transitional shale has inkpot-shaped and slit pores. (2) Cylindrical pores are common in organic matter of both shale types, with marine shale having a greater gas storage space (CRV) from organic matter pores, while marine–continental transitional shale relies more on inorganic pores, especially interlayer clay mineral pores, for gas storage due to their large specific surface area and high adsorption capacity (CRA). (3) The fractal characteristics of marine and marine–continental transitional shale pores are influenced differently. In marine shale, TOC positively correlates with fractal dimensions, while in marine–continental shale, Ro and clay minerals have a stronger influence. Ro is the primary factor affecting organic matter pore complexity. (4) Our two pore fractal models show that the complexity of the shale in the Longtan Formation surpasses that of the shale in the Niutitang Formation, and type I kerogen has more complex organic matter pores than type III, aiding in evaluating pore connectivity and flow effectiveness in shale reservoirs.
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(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)
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Time-Delay Effects on the Collective Resonant Behavior in Two Coupled Fractional Oscillators with Frequency Fluctuations
by
Minyue He, Huiqi Wang and Lifeng Lin
Fractal Fract. 2024, 8(5), 287; https://doi.org/10.3390/fractalfract8050287 - 11 May 2024
Abstract
In this study, we propose coupled time-delayed fractional oscillators with dichotomous fluctuating frequencies and investigate the collective resonant behavior. Firstly, we obtain the condition of complete synchronization between the average behavior of the two oscillators. Subsequently, we derive the precise analytical expression of
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In this study, we propose coupled time-delayed fractional oscillators with dichotomous fluctuating frequencies and investigate the collective resonant behavior. Firstly, we obtain the condition of complete synchronization between the average behavior of the two oscillators. Subsequently, we derive the precise analytical expression of the output amplitude gain. Based on the analytical results, we observe the collective resonant behavior of the coupled time-delayed system and further study its dependence on various system parameters. The observed results underscore that the coupling strength, fractional order, and time delay play significant roles in controlling the collective resonant behavior by facilitating the occurrence and optimizing the intensity. Finally, numerical simulations are also conducted and verify the accuracy of the analytical results.
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(This article belongs to the Section Mathematical Physics)
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A Novel Technique for Solving the Nonlinear Fractional-Order Smoking Model
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Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran
Fractal Fract. 2024, 8(5), 286; https://doi.org/10.3390/fractalfract8050286 - 10 May 2024
Abstract
In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear
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In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems.
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(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)
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Crop and Weed Segmentation and Fractal Dimension Estimation Using Small Training Data in Heterogeneous Data Environment
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Rehan Akram, Jin Seong Hong, Seung Gu Kim, Haseeb Sultan, Muhammad Usman, Hafiz Ali Hamza Gondal, Muhammad Hamza Tariq, Nadeem Ullah and Kang Ryoung Park
Fractal Fract. 2024, 8(5), 285; https://doi.org/10.3390/fractalfract8050285 - 10 May 2024
Abstract
The segmentation of crops and weeds from camera-captured images is a demanding research area for advancing agricultural and smart farming systems. Previously, the segmentation of crops and weeds was conducted within a homogeneous data environment where training and testing data were from the
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The segmentation of crops and weeds from camera-captured images is a demanding research area for advancing agricultural and smart farming systems. Previously, the segmentation of crops and weeds was conducted within a homogeneous data environment where training and testing data were from the same database. However, in the real-world application of advancing agricultural and smart farming systems, it is often the case of a heterogeneous data environment where a system trained with one database should be used for testing with a different database without additional training. This study pioneers the use of heterogeneous data for crop and weed segmentation, addressing the issue of degraded accuracy. Through adjusting the mean and standard deviation, we minimize the variability in pixel value and contrast, enhancing segmentation robustness. Unlike previous methods relying on extensive training data, our approach achieves real-world applicability with just one training sample for deep learning-based semantic segmentation. Moreover, we seamlessly integrated a method for estimating fractal dimensions into our system, incorporating it as an end-to-end task to provide important information on the distributional characteristics of crops and weeds. We evaluated our framework using the BoniRob dataset and the CWFID. When trained with the BoniRob dataset and tested with the CWFID, we obtained a mean intersection of union (mIoU) of 62% and an F1-score of 75.2%. Furthermore, when trained with the CWFID and tested with the BoniRob dataset, we obtained an mIoU of 63.7% and an F1-score of 74.3%. We confirmed that these values are higher than those obtained by state-of-the-art methods.
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(This article belongs to the Special Issue Fractional Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image Processing Algorithms)
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Screen-Printed Metamaterial Absorber Using Fractal Metal Mesh for Optical Transparency and Flexibility
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Jinwoo Choi, Daecheon Lim and Sungjoon Lim
Fractal Fract. 2024, 8(5), 284; https://doi.org/10.3390/fractalfract8050284 - 9 May 2024
Abstract
In stealth applications, there is a growing emphasis on the development of radar-absorbing structures that are efficient, flexible, and optically transparent. This study proposes a screen-printed metamaterial absorber (MMA) on polyethylene terephthalate (PET) substrates using indium tin oxide (ITO) as the grounding layer,
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In stealth applications, there is a growing emphasis on the development of radar-absorbing structures that are efficient, flexible, and optically transparent. This study proposes a screen-printed metamaterial absorber (MMA) on polyethylene terephthalate (PET) substrates using indium tin oxide (ITO) as the grounding layer, which achieves both optical transparency and flexibility. These materials and methods enhance the overall flexibility and transparency of MMA. To address the limited transparency caused by the silver nanoparticle ink for the top pattern, a metal mesh was incorporated to reduce the area ratio of the printed patterns, thereby enhancing transparency. By incrementing the fractal order of the structure, we optimized the operating frequency to target the X-band, which is most commonly used in radar detection. The proposed MMA demonstrates remarkable performance, with a measured absorption of 91.99% at 8.85 GHz and an average optical transmittance of 46.70% across the visible light spectrum (450 to 700 nm), indicating its potential for applications in transparent windows or drone stealth.
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(This article belongs to the Special Issue Advances in Fractal Antennas: Design, Modeling and Applications)
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Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps
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Sumei Zhang, Haiyang Xiao and Hongquan Yong
Fractal Fract. 2024, 8(5), 283; https://doi.org/10.3390/fractalfract8050283 - 8 May 2024
Abstract
This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log
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This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log return. To obtain the forward characteristic function, we approximate the pricing model with a semimartingale by introducing two small perturbed parameters. Then, we rewrite the forward characteristic function as a conditional expectation of the proportion characteristic function which is expressed in terms of the solution to a classic PDE. With the affine structure of the approximate model, we obtain the solution to the PDE. Based on the derived forward characteristic function and the Fourier transform technique, we develop a pricing algorithm for forward starting options. For comparison, we also develop a simulation scheme for evaluating forward starting options. The numerical results demonstrate that the proposed pricing algorithm is effective. Exhaustive comparative experiments on eight models show that the effects of fractional Brownian motion, mixed-exponential jump, and the second volatility component on forward starting option prices are significant, and especially, the second fractional volatility is necessary to price accurately forward starting options under the framework of fractional Brownian motion.
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(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)
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On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics
by
Fouad Mohammad Salama
Fractal Fract. 2024, 8(5), 282; https://doi.org/10.3390/fractalfract8050282 - 8 May 2024
Abstract
In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling
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In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.
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(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)
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Fractional Scalar Field Cosmology
by
Seyed Meraj Mousavi Rasouli, Samira Cheraghchi and Paulo Moniz
Fractal Fract. 2024, 8(5), 281; https://doi.org/10.3390/fractalfract8050281 - 8 May 2024
Abstract
Considering the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and the Einstein scalar field system as an underlying gravitational model to construct fractional cosmological models has interesting implications in both classical and quantum regimes. Regarding the former, we just review the most fundamental approach to establishing an
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Considering the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and the Einstein scalar field system as an underlying gravitational model to construct fractional cosmological models has interesting implications in both classical and quantum regimes. Regarding the former, we just review the most fundamental approach to establishing an extended cosmological model. We demonstrate that employing new methodologies allows us to obtain exact solutions. Despite the corresponding standard models, we cannot use any arbitrary scalar potentials; instead, it is determined from solving three independent fractional field equations. This article concludes with an overview of a fractional quantum/semi-classical model that provides an inflationary scenario.
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(This article belongs to the Section Mathematical Physics)
Open AccessArticle
Detection of Gate Valve Leaks through the Analysis Fractal Characteristics of Acoustic Signal
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Ayrat Zagretdinov, Shamil Ziganshin, Eugenia Izmailova, Yuri Vankov, Ilya Klyukin and Roman Alexandrov
Fractal Fract. 2024, 8(5), 280; https://doi.org/10.3390/fractalfract8050280 - 8 May 2024
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This paper considers the possibility of using monofractal and multifractal analysis of acoustic signals to detect water leaks through gate valves. Detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MF-DFA) were used. Experimental studies were conducted on a ½-inch nominal diameter wedge
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This paper considers the possibility of using monofractal and multifractal analysis of acoustic signals to detect water leaks through gate valves. Detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MF-DFA) were used. Experimental studies were conducted on a ½-inch nominal diameter wedge valve, which was fitted to a ¾-inch nominal diameter steel pipeline. The water leak was simulated by opening the valve. The resulting leakage rates for different valve opening conditions were 5.3, 10.5, 14, 16.8, and 20 L per minute (L/min). The Hurst exponent for acoustic signals in a hermetically sealed valve is at the same level as a deterministic signal, while the width of the multifractal spectrum closely matches that of a monofractal process. When a leak occurs, turbulent flow pulsations appear, and with small leak sizes, the acoustic signals become anticorrelated with a high degree of multifractality. As the leakage increases, the Hurst exponent also increases and the width of the multifractal spectrum decreases. The main contributor to the multifractal structure of leak signals is small, noise-like fluctuations. The analysis of acoustic signals using the DFA and MF-DFA methods enables determining the extent of water leakage through a non-sealed gate valve. The results of the experimental studies are in agreement with the numerical simulations. Using the Ansys Fluent software (v. 19.2), the frequencies of flow vortices at different positions of gate valve were calculated. The k-ω SST turbulence model was employed for calculations. The calculations were conducted in a transient formulation of the problem. It was found that as the leakage decreases, the areas with a higher turbulence eddy frequency increase. An increase in the frequency of turbulent fluctuations leads to enhanced energy dissipation. Some of the energy from ordered processes is converted into the energy of disordered processes.
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The Optimal Branch Width Convergence Ratio to Maximize the Transport Efficiency of the Combined Electroosmotic and Pressure-Driven Flow within a Fractal Tree-like Convergent Microchannel
by
Dalei Jing and Peng Qi
Fractal Fract. 2024, 8(5), 279; https://doi.org/10.3390/fractalfract8050279 - 7 May 2024
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Building upon the efficient transport capabilities observed in the fractal tree-like convergent structures found in nature, this paper numerically studies the transport process of the combined electroosmotic and pressure-driven flow within a fractal tree-like convergent microchannel (FTCMC) with uniform channel height. The present
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Building upon the efficient transport capabilities observed in the fractal tree-like convergent structures found in nature, this paper numerically studies the transport process of the combined electroosmotic and pressure-driven flow within a fractal tree-like convergent microchannel (FTCMC) with uniform channel height. The present work finds that the flow rate of the combined flow first increases and then decreases with the increasing branch width convergence ratio under the fixed voltage difference and pressure gradient along the FTCMC, which means that there is an optimal branch width convergence ratio to maximize the transport efficiency of the combined flow within the FTCMC. The value of the optimal branch convergence ratio is highly dependent on the ratio of the voltage difference and pressure gradient to drive the combined flow. By adjusting the structural and dimensional parameters of the FTCMC, the dependencies of the optimal branch convergence ratio of the FTCMC on the branching level convergence ratio, the length ratio, the branching number, and the branching level are also investigated. The findings in the present work can be used for the optimization of FTCMC with high transport efficiency for combined electroosmotic and pressure-driven flow.
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