Fractal and Fractional

16 pages, 311 KiB

Open AccessArticle

Novel Estimations of Hadamard-Type Integral Inequalities for Raina’s Fractional Operators

by Merve Coşkun, Çetin Yildiz and Luminiţa-Ioana Cotîrlă

Fractal Fract. 2024, 8(5), 302; https://doi.org/10.3390/fractalfract8050302 - 20 May 2024

Viewed by 1076

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using [...] Read more.

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g.,

s = 1

,

λ = α,

σ (0) = 1,

and

w = 0

. In conclusion, the methodology described in this article is expected to stimulate further research in this area. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)

24 pages, 469 KiB

Open AccessArticle

Time-Varying Function Matrix Projection Synchronization of Caputo Fractional-Order Uncertain Memristive Neural Networks with Multiple Delays via Mixed Open Loop Feedback Control and Impulsive Control

by Hongguang Fan, Yue Rao, Kaibo Shi and Hui Wen

Fractal Fract. 2024, 8(5), 301; https://doi.org/10.3390/fractalfract8050301 - 20 May 2024

Cited by 11 | Viewed by 953

Abstract

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization [...] Read more.

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization (TFMPS), which is a generalized version of traditional matrix projection synchronization, modified projection synchronization, complete synchronization, and anti-synchronization. To achieve the goal of TFMPS, we design a novel mixed controller including the open loop feedback control and impulsive control, which employs the state information in the time-delayed interval and the sampling information at the impulse instants. It has a prominent advantage that impulse intervals are not restricted by time delays. To establish the connection between the error system and the auxiliary system, a generalized fractional-order comparison theorem with time-varying coefficients and impulses is established. Applying the stability theory, the comparison theorem, and the Laplace transform, new synchronization criteria of FOUMNNs are acquired under the mixed impulsive control schemes, and the derived synchronization theorem and corollary can effectively expand the correlative synchronization achievements of fractional-order systems. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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26 pages, 3795 KiB

Open AccessArticle

Augmenting the Stability of Automatic Voltage Regulators through Sophisticated Fractional-Order Controllers

by Emad A. Mohamed, Mokhtar Aly, Waleed Alhosaini and Emad M. Ahmed

Fractal Fract. 2024, 8(5), 300; https://doi.org/10.3390/fractalfract8050300 - 20 May 2024

Cited by 1 | Viewed by 991

Abstract

The transition from traditional to renewable energy sources is a critical issue in current energy-generation systems, which aims to address climate change and the increased demand for energy. This shift, however, imposes additional burdens on control systems to maintain power system stability and [...] Read more.

The transition from traditional to renewable energy sources is a critical issue in current energy-generation systems, which aims to address climate change and the increased demand for energy. This shift, however, imposes additional burdens on control systems to maintain power system stability and quality within predefined limits. Addressing these challenges, this paper proposes an innovative Modified Hybrid Fractional-Order (MHFO) automatic voltage regulator (AVR) equipped with a fractional-order tilt integral and proportional derivative with a filter plus a second-order derivative with a filter FOTI-PDND²N² controller. This advanced controller combines the benefits of a (FOTI) controller, known for enhancing dynamic performance and steady-state response, with a (PDND²N²) controller to improve system robustness and adaptability. The proposed MHFO controller stands out with its nine tunable parameters, providing more extensive control options than the conventional three-parameter PID controller and the five-parameter FOPID controller. Furthermore, a recent optimization approach using a growth optimizer (GO) has been formulated and applied to optimally adjust the MHFO controller’s parameters simultaneously. The performance of the proposed AVR based on the MHFO-GO controller is scrutinized by contrasting it with various established and developed optimization algorithms. The comparative study shows that the AVR based on the MHFO-GO controller surpasses other AVR controllers from the stability, robustness, and dynamic response speed points of view. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications, 2nd Edition)

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21 pages, 1491 KiB

Open AccessArticle

Unveiling the Complexity of HIV Transmission: Integrating Multi-Level Infections via Fractal-Fractional Analysis

by Yasir Nadeem Anjam, Rubayyi Turki Alqahtani, Nadiyah Hussain Alharthi and Saira Tabassum

Fractal Fract. 2024, 8(5), 299; https://doi.org/10.3390/fractalfract8050299 - 20 May 2024

Viewed by 961

Abstract

This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence [...] Read more.

This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence and uniqueness of solutions via fixed-point theory. Ulam-Hyer stability is confirmed through nonlinear functional analysis, accounting for small perturbations. Numerical solutions are obtained using the fractional Adam-Bashforth iterative scheme and corroborated through MATLAB simulations. The results, plotted across various fractional orders and fractal dimensions, are compared with integer orders, revealing trends towards HIV disease-free equilibrium points for infective and recovered populations. Meanwhile, susceptible individuals decrease towards this equilibrium state, indicating stability in HIV exposure. The study emphasizes the critical role of controlling transmission rates to mitigate fatalities, curb HIV transmission, and enhance recovery rates. This proposed strategy offers a competitive advantage, enhancing comprehension of the model’s intricate dynamics. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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14 pages, 3979 KiB

Open AccessArticle

Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach

by Bashayr Almutairi, Muneerah Al Nuwairan and Anwar Aldhafeeri

Fractal Fract. 2024, 8(5), 298; https://doi.org/10.3390/fractalfract8050298 - 18 May 2024

Viewed by 977

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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17 pages, 1266 KiB

Open AccessArticle

Synchronization of Fractional Delayed Memristive Neural Networks with Jump Mismatches via Event-Based Hybrid Impulsive Controller

by Huiyu Wang, Shutang Liu, Xiang Wu, Jie Sun and Wei Qiao

Fractal Fract. 2024, 8(5), 297; https://doi.org/10.3390/fractalfract8050297 - 18 May 2024

Cited by 1 | Viewed by 725

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)

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12 pages, 293 KiB

Open AccessArticle

Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints

by Yuanyuan Deng and Yi Zhang

Fractal Fract. 2024, 8(5), 296; https://doi.org/10.3390/fractalfract8050296 - 18 May 2024

Viewed by 629

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 [...] Read more.

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. Full article

11 pages, 305 KiB

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

Cited by 1 | Viewed by 960

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Optimal Control Problems for Fractional Differential Equations)

10 pages, 1718 KiB

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

Viewed by 1050

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Fractional Models and Statistical Applications)

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31 pages, 3433 KiB

Open AccessArticle

Dynamics for a Nonlinear Stochastic Cholera Epidemic Model under Lévy Noise

by 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

Cited by 1 | Viewed by 1105

Abstract

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 [...] Read more.

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

R_{0}^{s} > 1

. Our findings suggest the likelihood of eradicating the disease when

R^{s}

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. Full article

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10 pages, 265 KiB

Open AccessArticle

Existence Results Related to a Singular Fractional Double-Phase Problem in the Whole Space

by Ramzi Alsaedi

Fractal Fract. 2024, 8(5), 292; https://doi.org/10.3390/fractalfract8050292 - 16 May 2024

Viewed by 891

Abstract

In this paper, we will study a singular problem involving the fractional

(q_{1} (x, .)

-

q_{2} (x, .))

-Laplacian operator in the whole space

R^{N}, (N \geq 2)

[...] Read more.

In this paper, we will study a singular problem involving the fractional

(q_{1} (x, .)

-

q_{2} (x, .))

-Laplacian operator in the whole space

R^{N}, (N \geq 2)

. 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. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Analysis: Theory and Applications)

23 pages, 1416 KiB

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

Viewed by 1590

Abstract

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 [...] Read more.

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. Full article

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22 pages, 835 KiB

Open AccessArticle

Exponential H_∞ Output Control for Switching Fuzzy Systems via Event-Triggered Mechanism and Logarithmic Quantization

by 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

Viewed by 835

Abstract

This paper investigates the problem of exponential

H_{\infty}

output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT:

τ_{a}

) and the upper bound of inter-event intervals (IEIs: T) [...] Read more.

This paper investigates the problem of exponential

H_{\infty}

output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT:

τ_{a}

) and the upper bound of inter-event intervals (IEIs: T) satisfy the condition

τ_{a} \geq \frac{ln μ + (α + β) T}{α} = \frac{ln μ + β T}{α} + T

, 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

L_{2}

norm constraint is solved through the ADT method, although only a weighted

L_{2}

norm constraint was obtained previously. Additionally, by constructing a controller-mode-dependent Lyapunov function and adopting logarithmic quantizers, the sufficient criteria of exponential

H_{\infty}

output control problem are presented. The validity of established results is demonstrated by a given numerical simulation. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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32 pages, 440 KiB

Open AccessArticle

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces

by 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

Cited by 1 | Viewed by 1170

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,

Φ

-Hilfer, fractional derivative of order

μ \in (1, 2)

with non-instantaneous impulses in Banach spaces [...] Read more.

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted,

Φ

-Hilfer, fractional derivative of order

μ \in (1, 2)

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

Φ

-Laplace transform, w-weighted

ψ

-convolution and the measure of non-compactness. Since the operator, the w-weighted

Φ

-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. Full article

29 pages, 13316 KiB

Open AccessArticle

Pore Fractal Characteristics between Marine and Marine–Continental Transitional Black Shales: A Case Study of Niutitang Formation and Longtan Formation

by 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

Cited by 5 | Viewed by 1350

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 [...] Read more.

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 (R_o), 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, R_o and clay minerals have a stronger influence. R_o 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. Full article

(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)

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16 pages, 1090 KiB

Open AccessArticle

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

Viewed by 1039

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 [...] Read more.

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. Full article

(This article belongs to the Section Mathematical Physics)

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21 pages, 786 KiB

Open AccessArticle

A Novel Technique for Solving the Nonlinear Fractional-Order Smoking Model

by 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

Cited by 4 | Viewed by 961

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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29 pages, 10726 KiB

Open AccessArticle

Crop and Weed Segmentation and Fractal Dimension Estimation Using Small Training Data in Heterogeneous Data Environment

by 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

Cited by 1 | Viewed by 1383

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 [...] Read more.

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. Full article

(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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15 pages, 9635 KiB

Open AccessArticle

Screen-Printed Metamaterial Absorber Using Fractal Metal Mesh for Optical Transparency and Flexibility

by Jinwoo Choi, Daecheon Lim and Sungjoon Lim

Fractal Fract. 2024, 8(5), 284; https://doi.org/10.3390/fractalfract8050284 - 9 May 2024

Viewed by 1229

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, [...] Read more.

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. Full article

(This article belongs to the Special Issue Advances in Fractal Antennas: Design, Modeling and Applications)

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13 pages, 2004 KiB

Open AccessArticle

Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps

by Sumei Zhang, Haiyang Xiao and Hongquan Yong

Fractal Fract. 2024, 8(5), 283; https://doi.org/10.3390/fractalfract8050283 - 8 May 2024

Viewed by 923

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)

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27 pages, 987 KiB

Open AccessArticle

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

Cited by 1 | Viewed by 1076

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)

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18 pages, 368 KiB

Open AccessReview

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

Cited by 2 | Viewed by 1147

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 [...] Read more.

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. Full article

(This article belongs to the Section Mathematical Physics)

15 pages, 4276 KiB

Open AccessArticle

Detection of Gate Valve Leaks through the Analysis Fractal Characteristics of Acoustic Signal

by 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

Viewed by 1145

Abstract

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 [...] Read more.

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. Full article

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15 pages, 4073 KiB

Open AccessArticle

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

Cited by 1 | Viewed by 1133

Abstract

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 [...] Read more.

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. Full article

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21 pages, 874 KiB

Open AccessArticle

Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints

by Yuhang Yao, Jiaxin Yuan, Tao Chen, Chen Zhang and Hui Yang

Fractal Fract. 2024, 8(5), 278; https://doi.org/10.3390/fractalfract8050278 - 7 May 2024

Cited by 1 | Viewed by 934

Abstract

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise [...] Read more.

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy. Full article

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17 pages, 4005 KiB

Open AccessArticle

Numerical Study of Time-Fractional Schrödinger Model in One-Dimensional Space Arising in Mathematical Physics

by Muhammad Nadeem and Loredana Florentina Iambor

Fractal Fract. 2024, 8(5), 277; https://doi.org/10.3390/fractalfract8050277 - 7 May 2024

Cited by 2 | Viewed by 820

Abstract

This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the [...] Read more.

This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the Elzaki transform (ET) with the residual power series method (RPSM). This strategy has the advantage of requiring only the premise of limiting at zero for determining the coefficients of the series, and it uses symbolic computation software to perform the least number of calculations. The results obtained through the considered method are in the form of a series solution and converge rapidly. These outcomes closely match the precise results and are discussed through graphical structures to express the physical representation of the considered equation. The results showed that the suggested strategy is a straightforward, suitable, and practical tool for solving and comprehending a wide range of nonlinear physical models. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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25 pages, 3799 KiB

Open AccessArticle

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

by Yasir Nawaz, Muhammad Shoaib Arif, Muavia Mansoor, Kamaleldin Abodayeh and Amani S. Baazeem

Fractal Fract. 2024, 8(5), 276; https://doi.org/10.3390/fractalfract8050276 - 6 May 2024

Cited by 1 | Viewed by 1011

Abstract

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is [...] Read more.

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid’s fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme. Full article

(This article belongs to the Special Issue Heat Transfer and Diffusion Processes in Fractal Domains)

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17 pages, 3351 KiB

Open AccessArticle

Fractal Features of Muscle to Quantify Fatty Infiltration in Aging and Pathology

by Annamaria Zaia, Martina Zannotti, Lucia Losa and Pierluigi Maponi

Fractal Fract. 2024, 8(5), 275; https://doi.org/10.3390/fractalfract8050275 - 6 May 2024

Cited by 1 | Viewed by 1141

Abstract

The physiological loss of muscle mass and strength with aging is referred to as “sarcopenia”, whose combined effect with osteoporosis is a serious threat to the elderly, accounting for decreased mobility and increased risk of falls with consequent fractures. In previous studies, we [...] Read more.

The physiological loss of muscle mass and strength with aging is referred to as “sarcopenia”, whose combined effect with osteoporosis is a serious threat to the elderly, accounting for decreased mobility and increased risk of falls with consequent fractures. In previous studies, we observed a high degree of inter-individual variability in paraspinal muscle fatty infiltration, one of the most relevant indices of muscle wasting. This aspect led us to develop a computerized method to quantitatively characterize muscle fatty infiltration in aging and diseases. Magnetic resonance images of paraspinal muscles from 58 women of different ages (age range of 23–85 years) and physio-pathological status (healthy young, pre-menopause, menopause, and osteoporosis) were used to set up a method based on fractal-derived texture analysis of lean muscle area (contractile muscle) to estimate muscle fatty infiltration. In particular, lacunarity was computed by parameter β from the GBA (gliding box algorithm) curvilinear plot fitted by our hyperbola model function. Succolarity was estimated by parameter µ, for the four main directions through an algorithm implemented with this purpose. The results show that lacunarity, by quantifying muscle fatty infiltration, can discriminate between osteoporosis and healthy aging, while succolarity can separate the other three groups showing similar lacunarity. Therefore, fractal-derived features of contractile muscle, by measuring fatty infiltration, can represent good indices of sarcopenia in aging and disease. Full article

(This article belongs to the Special Issue Biocomplexity and Fractal Analysis: Theory and Applications)

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13 pages, 1062 KiB

Open AccessArticle

Volatility Analysis of Financial Time Series Using the Multifractal Conditional Diffusion Entropy Method

by Maria C. Mariani, William Kubin, Peter K. Asante and Osei K. Tweneboah

Fractal Fract. 2024, 8(5), 274; https://doi.org/10.3390/fractalfract8050274 - 4 May 2024

Viewed by 1135

Abstract

In this article, we introduce the multifractal conditional diffusion entropy method for analyzing the volatility of financial time series. This method utilizes a q-order diffusion entropy based on a q-weighted time lag scale. The technique of conditional diffusion entropy proves valuable [...] Read more.

In this article, we introduce the multifractal conditional diffusion entropy method for analyzing the volatility of financial time series. This method utilizes a q-order diffusion entropy based on a q-weighted time lag scale. The technique of conditional diffusion entropy proves valuable for examining bull and bear behaviors in stock markets across various time scales. Empirical findings from analyzing the Dow Jones Industrial Average (DJI) indicate that employing multi-time lag scales offers greater insight into the complex dynamics of highly fluctuating time series, often characterized by multifractal behavior. A smaller time scale like

t = 2

to

t = 256

coincides more with the state of the DJI index than larger time scales like

t = 256

to

t = 1024

. We observe extreme fluctuations in the conditional diffusion entropy for DJI for a short time lag, while smoother or averaged fluctuations occur over larger time lags. Full article

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13 pages, 12921 KiB

Open AccessArticle

Fractal Evolution Characteristics on the Three-Dimensional Fractures in Coal Induced by CO₂ Phase Transition Fracturing

by Zhen Zhang, Gaofeng Liu, Jia Lin, George Barakos and Ping Chang

Fractal Fract. 2024, 8(5), 273; https://doi.org/10.3390/fractalfract8050273 - 4 May 2024

Cited by 3 | Viewed by 1375

Abstract

To analyze the transformed effect of three-dimensional (3D) fracture in coal by CO₂ phase transition fracturing (CO₂-PTF), the CO₂-PTF experiment under a fracturing pressure of 185 MPa was carried out. Computed Tomography (CT) scanning and fractal theory were [...] Read more.

To analyze the transformed effect of three-dimensional (3D) fracture in coal by CO₂ phase transition fracturing (CO₂-PTF), the CO₂-PTF experiment under a fracturing pressure of 185 MPa was carried out. Computed Tomography (CT) scanning and fractal theory were used to analyze the 3D fracture structure parameters. The fractal evolution characteristics of the 3D fractures in coal induced by CO₂-PTF were analyzed. The results indicate that the CO₂ phase transition fracturing coal has the fracture generation effect and fracture expansion-transformation effect, causing the maximum fracture length, fracture number, fracture volume and fracture surface area to be increased by 71.25%, 161.94%, 3970.88% and 1330.03%. The fractal dimension (D_N) for fracture number increases from 2.3523 to 2.3668, and the fractal dimension (D_V) for fracture volume increases from 2.8440 to 2.9040. The early dynamic high-pressure gas jet stage of CO₂-PTF coal influences the fracture generation effect and promotes the generation of 3D fractures with a length greater than 140 μm. The subsequent quasi-static high-pressure gas stage influences the fracture expansion-transformation effect, which promotes the expansion transformation of 3D fractures with a length of less than 140 μm. The 140 μm is the critical value for the fracture expansion-transformation effect and fracture generation effect. Five indicators are proposed to evaluate the 3D fracture evolution in coal caused by CO₂-PTF, which can provide theoretical and methodological references for the study of fracture evolution characteristics of other unconventional natural gas reservoirs and their reservoir stimulation. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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