Fractal and Fractional

18 pages, 7052 KiB

Open AccessArticle

An Efficient Structure-Preserving Scheme for the Fractional Damped Nonlinear Schrödinger System

by Yao Shi, Xiaozhen Liu and Zhenyu Wang

Fractal Fract. 2025, 9(5), 328; https://doi.org/10.3390/fractalfract9050328 - 21 May 2025

Viewed by 281

This paper introduces a highly accurate and efficient conservative scheme for solving the nonlocal damped Schrödinger system with Riesz fractional derivatives. The proposed approach combines the Fourier spectral method with the Crank–Nicolson time-stepping scheme. To begin, the original equation is reformulated into an [...] Read more.

This paper introduces a highly accurate and efficient conservative scheme for solving the nonlocal damped Schrödinger system with Riesz fractional derivatives. The proposed approach combines the Fourier spectral method with the Crank–Nicolson time-stepping scheme. To begin, the original equation is reformulated into an equivalent system by introducing a new variable that modifies both energy and mass. The Fourier spectral method is employed to achieve high spatial accuracy in this semi-discrete formulation. For time discretization, the Crank–Nicolson scheme is applied, ensuring conservation of the modified energy and mass in the fully discrete system. Numerical experiments validate the scheme’s precision and its ability to preserve conservation properties. Full article

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33 pages, 2459 KiB

Open AccessArticle

Skewed Multifractal Cross-Correlations Between Green Bond Index and Energy Futures Markets: A New Perspective Based on Change Point

by Yun Tian, Zhihui Li, Jue Wang, Xu Wu and Huan Huang

Fractal Fract. 2025, 9(5), 327; https://doi.org/10.3390/fractalfract9050327 - 20 May 2025

Viewed by 336

Abstract

This study is the first to use the Bayesian Estimator of Abrupt Change, Seasonality, and Trend (BEAST) algorithm to detect trend change points in the nexuses between the green bond index (Green Bond) and WTI of crude oil, gasoline, as well as natural [...] Read more.

This study is the first to use the Bayesian Estimator of Abrupt Change, Seasonality, and Trend (BEAST) algorithm to detect trend change points in the nexuses between the green bond index (Green Bond) and WTI of crude oil, gasoline, as well as natural gas futures. The COVID-19 pandemic and the Russia–Ukraine war are identified as common significant trend change points, and the total sample is subsequently divided into three stages based on these points. Utilizing a skewed MF-DCCA method, this study analyzed the skewed multifractal characteristics between the Green Bond and the energy futures across these stages. The results revealed that both the multifractal characteristics and risk levels experienced significant changes across different periods, exhibiting skewed multifractality. Specifically, from the pre-pandemic period to the post-Russia–Ukraine conflict period, the multifractal features and risk of the Green Bond and WTI and Green Bond and Gasoline groups first declined and then increased, while the Green Bond and Natural Gas group displayed an opposite trend, showing an initial increase followed by a decline. A portfolio analysis further indicated that Green Bond provided effective hedging against all three types of energy futures, particularly during crisis periods. Notably, the portfolios constructed using the Mean-MF-DCCA model, which incorporated multifractal features, outperformed those constructed by traditional portfolio models. These findings offered new insights into the dynamic characteristics of the Green Bond and energy futures markets and provided important policy implications for portfolio optimization and risk management strategies. Full article

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15 pages, 2557 KiB

Open AccessArticle

Multifractal Cross-Correlation Analysis of Carbon Emission Markets Between the European Union and China: A Study Based on the Multifractal Detrended Cross-Correlation Analysis and Empirical Mode Decomposition Multifractal Detrended Cross-Correlation Analysis Methods

by Xin Liao, Zheyu Wang and Huimin Tong

Fractal Fract. 2025, 9(5), 326; https://doi.org/10.3390/fractalfract9050326 - 20 May 2025

Viewed by 347

Abstract

Using the multifractal detrended cross-correlation analysis (MF-DCCA) method and the Empirical Mode Decomposition (EMD)-MF-DCCA method, this study quantifies the dynamic interrelation between carbon emission allowance returns in the Chinese and EU markets. The cross-correlation statistics indicate a moderate acceptance of the cross-correlation between [...] Read more.

Using the multifractal detrended cross-correlation analysis (MF-DCCA) method and the Empirical Mode Decomposition (EMD)-MF-DCCA method, this study quantifies the dynamic interrelation between carbon emission allowance returns in the Chinese and EU markets. The cross-correlation statistics indicate a moderate acceptance of the cross-correlation between the two carbon markets. Applying the MF-DCCA and EMD-MF-DCCA methods to the two markets reveals that their cross-correlation exhibits a power-law nature. Moreover, the apparent persistence of the cross-correlation and notable Hurst index show that the cross-correlation between long-term trends of the returns of the Guangdong and EU carbon emission markets exhibits stronger fractality over the long term, whereas the cross-correlation between the short-term fluctuations of the Hubei and EU carbon emission markets demonstrates stronger fractality. Subsequent investigations show that both fat tails and long memory contribute to the various fractals of the cross-correlation between the returns of the Chinese and EU carbon emission markets, especially for the fractals between the Hubei and EU carbon emission markets. Ultimately, the sliding window analysis demonstrates that national policy, trading activity, and other factors can make the observed multiple fractals more sensitive. The aforementioned findings facilitate an understanding of the current state of the Chinese carbon emission market and inform strategies for its future development. Full article

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21 pages, 476 KiB

Open AccessArticle

A New L2 Type Difference Scheme for the Time-Fractional Diffusion Equation

by Cheng-Yu Hu and Fu-Rong Lin

Fractal Fract. 2025, 9(5), 325; https://doi.org/10.3390/fractalfract9050325 - 20 May 2025

Viewed by 431

Abstract

In this paper, a new L2 (NL2) scheme is proposed to approximate the Caputo temporal fractional derivative, leading to a time-stepping scheme for the time-fractional diffusion equation (TFDE). Subsequently, the space derivative of the resulting system is discretized using a specific finite difference [...] Read more.

In this paper, a new L2 (NL2) scheme is proposed to approximate the Caputo temporal fractional derivative, leading to a time-stepping scheme for the time-fractional diffusion equation (TFDE). Subsequently, the space derivative of the resulting system is discretized using a specific finite difference method, yielding a fully discrete system. We then establish the

H^{1}

-norm stability and convergence of the time-stepping scheme on uniform meshes for the TFDE. In particular, we prove that the proposed scheme has

(3 - α)

th-order accuracy, where

α

(

0 < α < 1

) is the order of the time-fractional derivative. Finally, numerical experiments for several test problems are carried out to validate the obtained theoretical results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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20 pages, 386 KiB

Open AccessArticle

Some Fixed Point Results for Novel Contractions with Applications in Fractional Differential Equations for Market Equilibrium and Economic Growth

by Min Wang, Muhammad Din and Mi Zhou

Fractal Fract. 2025, 9(5), 324; https://doi.org/10.3390/fractalfract9050324 - 19 May 2025

Viewed by 381

Abstract

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction [...] Read more.

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction types, including

χ

-contractions, Banach contractions, enriched contractions, Kannan contractions, Bianchini contractions, Zamfirescu contractions, non-expansive mappings, and

(ρ, χ)

-enriched contractions. We establish related fixed point results for the novel contractions in normed spaces endowed with the binary relations preserving key symmetric properties, ensuring consistency and applicability. The Krasnoselskij iteration method is refined to incorporate symmetric constraints, facilitating fixed point identification within these spaces. By appropriately selecting constants in the definition of enriched

(I, ρ, χ)

-contractions, employing a suitable binary relation, or control function

χ \in Θ

, our framework generalizes and extends classical fixed point theorems. Illustrative examples highlight the significance of our findings in reinforcing fixed point conditions and demonstrating their broader applicability. Additionally, this paper explores how these ideas guarantee the stability of the production–consumption markets equilibrium and the economic growth model. Full article

(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)

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18 pages, 320 KiB

Open AccessArticle

Solvability of a Riemann–Liouville-Type Fractional-Impulsive Differential Equation with a Riemann–Stieltjes Integral Boundary Condition

by Keyu Zhang, Donal O’Regan and Jiafa Xu

Fractal Fract. 2025, 9(5), 323; https://doi.org/10.3390/fractalfract9050323 - 19 May 2025

Viewed by 349

Abstract

In this work, we address the solvability of a Riemann–Liouville-type fractional-impulsive integral boundary value problem. Under some conditions on the spectral radius corresponding to the related linear operator, we use fixed-point methods to obtain several existence theorems for our problem. In particular, we [...] Read more.

In this work, we address the solvability of a Riemann–Liouville-type fractional-impulsive integral boundary value problem. Under some conditions on the spectral radius corresponding to the related linear operator, we use fixed-point methods to obtain several existence theorems for our problem. In particular, we obtain the existence of multiple positive solutions via the Avery–Peterson fixed-point theorem. Note that our linear operator depends on the impulsive term and the integral boundary condition. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

29 pages, 67369 KiB

Open AccessArticle

Fractal–Fractional Synergy in Geo-Energy Systems: A Multiscale Framework for Stress Field Characterization and Fracture Network Evolution Modeling

by Qiqiang Ren, Tianhao Gao, Rongtao Jiang, Jin Wang, Mengping Li, Jianwei Feng and He Du

Fractal Fract. 2025, 9(5), 322; https://doi.org/10.3390/fractalfract9050322 - 19 May 2025

Viewed by 677

Abstract

This research introduces an innovative fractal–fractional synergy framework for multiscale analysis of stress field dynamics in geo-energy systems. By integrating fractional calculus with multiscale fractal dimension analysis, we develop a coupled approach examining stress redistribution patterns across different geological scales. The methodology combines [...] Read more.

This research introduces an innovative fractal–fractional synergy framework for multiscale analysis of stress field dynamics in geo-energy systems. By integrating fractional calculus with multiscale fractal dimension analysis, we develop a coupled approach examining stress redistribution patterns across different geological scales. The methodology combines fractal characterization of rock mechanical parameters with fractional-order stress gradient modeling, validated through integrated analysis of core testing, well logging, and seismic inversion data. Our fractal–fractional operators enable simultaneous characterization of stress memory effects and scale-invariant fracture propagation patterns. Key insights reveal the following: (1) Non-monotonic variations in rock mechanical properties (fractal dimension D = 2.31–2.67) correlate with oil–water ratio changes, exhibiting fractional-order transitional behavior. (2) Critical stress thresholds (12.19–25 MPa) for fracture activation follow fractional power-law relationships with fracture orientation deviations. (3) Fracture network evolution demonstrates dual-scale dynamics—microscale tip propagation governed by fractional stress singularities (order α = 0.63–0.78) and macroscale expansion obeying fractal growth patterns (Hurst exponent H = 0.71 ± 0.05). (4) Multiscale modeling reveals anisotropic development with fractal dimension increasing by 18–22% during multi-well fracturing operations. The fractal–fractional formalism successfully resolves the stress-shadow paradox while quantifying water channeling risks through fractional connectivity metrics. This work establishes a novel paradigm for coupled geomechanical–fluid dynamics analysis in complex reservoir systems. Full article

(This article belongs to the Special Issue Efficient Development of Geo-Energy and Carbon Sequestration in Fractal Geo-Systems: New Challenges)

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17 pages, 742 KiB

Open AccessArticle

Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives

by Lamia Loudahi, Amjad Ali, Jing Yuan, Jalil Ahmad, Lamiaa Galal Amin and Yunlan Wei

Fractal Fract. 2025, 9(5), 321; https://doi.org/10.3390/fractalfract9050321 - 19 May 2025

Viewed by 732

Abstract

Water pollution is a significant threat for human health, particularly in developed countries. This study advances the mathematical understanding of WP transmission dynamics by developing a fractional–fractal derivative framework with non-singular kernels and the Mittage–Leffler function, which successfully preserves the non-local behavior of [...] Read more.

Water pollution is a significant threat for human health, particularly in developed countries. This study advances the mathematical understanding of WP transmission dynamics by developing a fractional–fractal derivative framework with non-singular kernels and the Mittage–Leffler function, which successfully preserves the non-local behavior of pollutants. The fractional–fractal derivatives in sense of the Atangana–Baleanu–Caputo formulation inherently captures the non-local and memory-dependent behavior of pollutant diffusion, addressing limitations of classical differential operators. A novel parameter,

γ

, is introduced to represent the recovery rate of water systems through treatment processes, explicitly modeling the bridge between natural purification mechanisms and engineered remediation efforts. Furthermore, this study establishes stability analysis, and the existence and uniqueness of the solution are established through fixed-point theory to ensure the mathematical stability of the system. Moreover, a numerical scheme based on the Newton polynomial is formulated, by obtaining significant simulations of pollution dynamics under various conditions. Graphical results show the effect of important parameters on pollutant evolution, providing useful information about the behavior of the system. Full article

(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus and Mathematical Modelling)

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27 pages, 4478 KiB

Open AccessArticle

Analytical Insight into Some Fractional Nonlinear Dynamical Systems Involving the Caputo Fractional Derivative Operator

by Mashael M. AlBaidani

Fractal Fract. 2025, 9(5), 320; https://doi.org/10.3390/fractalfract9050320 - 19 May 2025

Cited by 3 | Viewed by 574

Abstract

This work explores modern mathematical avenues as part of fractional calculus research. We apply fractional dispersion relations to the fractional wave equation to numerically examine various formulations of the generalized fractional wave equation. The research explores Drinfeld–Sokolov–Wilson and shallow water equations as fundamental [...] Read more.

This work explores modern mathematical avenues as part of fractional calculus research. We apply fractional dispersion relations to the fractional wave equation to numerically examine various formulations of the generalized fractional wave equation. The research explores Drinfeld–Sokolov–Wilson and shallow water equations as fundamental differential equations forming the basis of wave theory studies. This work presents effective methods to obtain the numerical solution of the fractional-order FDSW and FSW coupled system equations. The analysis employs Caputo fractional derivatives during studies of fractional orders. This study develops the new iterative transform technique (NITM) and homotopy perturbation transform method (HPTM) using Elzaki transform (ET) with a new iteration method and a homotopy perturbation method. The proposed techniques generate approximation solutions that adopt an infinite fractional series with fractional order solutions converging towards analytic integer solutions. The proposed method demonstrates its precision through tabular simulations of computed approximations and their absolute error values while representing results with 2D and 3D graphics. The paper presents the physical analysis of solution dynamics across diverse

ϵ

ranges during a suitable time frame. The developed computational techniques yield numerical and graphical output, which are compared to analytic results to verify the solution convergence. The computational algorithms have proven their high accuracy, flexibility, effectiveness, and simplicity in evaluating fractional-order mathematical models. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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21 pages, 18391 KiB

Open AccessArticle

Multifractal Analysis of Geological Data Using a Moving Window Dynamical Approach

by Gil Silva, Fernando Pellon de Miranda, Mateus Michelon, Ana Ovídio, Felipe Venturelli, Letícia Moraes, João Ferreira, João Parêdes, Alexandre Cury and Flávio Barbosa

Fractal Fract. 2025, 9(5), 319; https://doi.org/10.3390/fractalfract9050319 - 16 May 2025

Viewed by 453

Abstract

Fractal dimension has proven to be a valuable tool in the analysis of geological data. For instance, it can be used for assessing the distribution and connectivity of fractures in rocks, which is important for evaluating hydrocarbon storage potential. However, while calculating a [...] Read more.

Fractal dimension has proven to be a valuable tool in the analysis of geological data. For instance, it can be used for assessing the distribution and connectivity of fractures in rocks, which is important for evaluating hydrocarbon storage potential. However, while calculating a single fractal dimension for an entire geological profile provides a general overview, it can obscure local variations. These localized fluctuations, if analyzed, can offer a more detailed and nuanced understanding of the profile’s characteristics. Hence, this study proposes a fractal characterization procedure using a new strategy based on moving windows applied to the analysis domain, enabling the evaluation of data multifractality through the Dynamical Approach Method. Validations for the proposed methodology were performed using controlled artificial data generated from Weierstrass–Mandelbrot functions. Then, the methodology was applied to real geological profile data measuring permeability and porosity in oil wells, revealing the fractal dimensions of these data along the depth of each analyzed case. The results demonstrate that the proposed methodology effectively captures a wide range of fractal dimensions, from high to low, in artificially generated data. Moreover, when applied to geological datasets, it successfully identifies regions exhibiting distinct fractal characteristics, which may contribute to a deeper understanding of reservoir properties and fluid flow dynamics. Full article

(This article belongs to the Special Issue Flow and Transport in Fractal Models of Rock Mechanics)

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46 pages, 921 KiB

Open AccessArticle

Fractional Time-Delayed Differential Equations: Applications in Cosmological Studies

by Bayron Micolta-Riascos, Byron Droguett, Gisel Mattar Marriaga, Genly Leon, Andronikos Paliathanasis, Luis del Campo and Yoelsy Leyva

Fractal Fract. 2025, 9(5), 318; https://doi.org/10.3390/fractalfract9050318 - 16 May 2025

Viewed by 583

Abstract

Fractional differential equations model processes with memory effects, providing a realistic perspective on complex systems. We examine time-delayed differential equations, discussing first-order and fractional Caputo time-delayed differential equations. We derive their characteristic equations and solve them using the Laplace transform. We derive a [...] Read more.

Fractional differential equations model processes with memory effects, providing a realistic perspective on complex systems. We examine time-delayed differential equations, discussing first-order and fractional Caputo time-delayed differential equations. We derive their characteristic equations and solve them using the Laplace transform. We derive a modified evolution equation for the Hubble parameter incorporating a viscosity term modeled as a function of the delayed Hubble parameter within Eckart’s theory. We extend this equation using the last-step method of fractional calculus, resulting in Caputo’s time-delayed fractional differential equation. This equation accounts for the finite response times of cosmic fluids, resulting in a comprehensive model of the Universe’s behavior. We then solve this equation analytically. Due to the complexity of the analytical solution, we also provide a numerical representation. Our solution reaches the de Sitter equilibrium point. Additionally, we present some generalizations. Full article

(This article belongs to the Special Issue Fractional Gravity/Cosmology in Classical and Quantum Regimes, Second Edition)

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21 pages, 3304 KiB

Open AccessArticle

Personalised Fractional-Order Autotuner for the Maintenance Phase of Anaesthesia Using Sine-Tests

by Marcian D. Mihai, Isabela R. Birs, Nicoleta E. Badau, Erwin T. Hegedus, Amani Ynineb and Cristina I. Muresan

Fractal Fract. 2025, 9(5), 317; https://doi.org/10.3390/fractalfract9050317 - 15 May 2025

Viewed by 328

Abstract

The research field of clinical practice has experienced a substantial increase in the integration of information technology and control engineering, which includes the management of medication administration for general anaesthesia. The invasive nature of input signals is the reason why autotuning methods are [...] Read more.

The research field of clinical practice has experienced a substantial increase in the integration of information technology and control engineering, which includes the management of medication administration for general anaesthesia. The invasive nature of input signals is the reason why autotuning methods are not widely used in this research field. This study proposes a non-invasive method using small-amplitude sine tests to estimate patient parameters, which allows the design of a personalised controller using an autotuning principle. The primary objective is to regulate the Bispectral Index through the administration of Propofol during the maintenance phase of anaesthesia, using a personalised fractional-order PID. This work aims to demonstrate the effectiveness of personalised control, which is facilitated by the proposed sine-based method. The closed-loop simulation results demonstrate the efficiency of the proposed approach. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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13 pages, 4696 KiB

Open AccessArticle

Analysis of Noise on Ordinary and Fractional-Order Financial Systems

by Hunida Malaikah and Jawaher Faisal Alabdali

Fractal Fract. 2025, 9(5), 316; https://doi.org/10.3390/fractalfract9050316 - 15 May 2025

Viewed by 382

Abstract

This study investigated the influence of stochastic fluctuations on financial system stability by analyzing both ordinary and fractional-order financial models under noise. The ordinary financial system experiences perturbations due to bounded random disturbances, whereas the fractional-order counterpart models memory-dependent behaviors by incorporating fractional [...] Read more.

This study investigated the influence of stochastic fluctuations on financial system stability by analyzing both ordinary and fractional-order financial models under noise. The ordinary financial system experiences perturbations due to bounded random disturbances, whereas the fractional-order counterpart models memory-dependent behaviors by incorporating fractional Gaussian noise (FGN) characterized by a Hurst parameter that governs long-term correlations. This study used data generated through MATLAB simulations based on standard financial models from the literature. Numerical simulations compared system behavior in deterministic and noisy environments. The results reveal that ordinary systems experience transient fluctuations, quickly returning to a stable state, whereas fractional systems exhibit persistent deviations due to historical dependencies. This highlights the fundamental difference between integer-order and fractional-order derivatives in financial modeling. Our key findings indicate that noise significantly impacts interest rates, investment needs, price indices, and profit margins, with the fractional system displaying higher sensitivity to external shocks. These insights emphasize the necessity of incorporating memory effects in financial modeling to improve accuracy in predicting market behavior. The study further underscores the importance of adaptive monetary policies and risk management strategies to mitigate financial instability. Future research should explore hybrid models combining short-term stability with long-term memory effects for enhanced financial forecasting and stability analysis. Full article

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36 pages, 7456 KiB

Open AccessArticle

Estimation of Fractal Dimensions and Classification of Plant Disease with Complex Backgrounds

by Muhammad Hamza Tariq, Haseeb Sultan, Rehan Akram, Seung Gu Kim, Jung Soo Kim, Muhammad Usman, Hafiz Ali Hamza Gondal, Juwon Seo, Yong Ho Lee and Kang Ryoung Park

Fractal Fract. 2025, 9(5), 315; https://doi.org/10.3390/fractalfract9050315 - 14 May 2025

Viewed by 856

Abstract

Accurate classification of plant disease by farming robot cameras can increase crop yield and reduce unnecessary agricultural chemicals, which is a fundamental task in the field of sustainable and precision agriculture. However, until now, disease classification has mostly been performed by manual methods, [...] Read more.

Accurate classification of plant disease by farming robot cameras can increase crop yield and reduce unnecessary agricultural chemicals, which is a fundamental task in the field of sustainable and precision agriculture. However, until now, disease classification has mostly been performed by manual methods, such as visual inspection, which are labor-intensive and often lead to misclassification of disease types. Therefore, previous studies have proposed disease classification methods based on machine learning or deep learning techniques; however, most did not consider real-world plant images with complex backgrounds and incurred high computational costs. To address these issues, this study proposes a computationally effective residual convolutional attention network (RCA-Net) for the disease classification of plants in field images with complex backgrounds. RCA-Net leverages attention mechanisms and multiscale feature extraction strategies to enhance salient features while reducing background noises. In addition, we introduce fractal dimension estimation to analyze the complexity and irregularity of class activation maps for both healthy plants and their diseases, confirming that our model can extract important features for the correct classification of plant disease. The experiments utilized two publicly available datasets: the sugarcane leaf disease and potato leaf disease datasets. Furthermore, to improve the capability of our proposed system, we performed fractal dimension estimation to evaluate the structural complexity of healthy and diseased leaf patterns. The experimental results show that RCA-Net outperforms state-of-the-art methods with an accuracy of 93.81% on the first dataset and 78.14% on the second dataset. Furthermore, we confirm that our method can be operated on an embedded system for farming robots or mobile devices at fast processing speed (78.7 frames per second). Full article

(This article belongs to the Special Issue Advances in Pattern Recognition—Image and Time Series Analyses—through Fractal Geometry and Complexity Theory)

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30 pages, 399 KiB

Open AccessArticle

Milstein Scheme for a Stochastic Semilinear Subdiffusion Equation Driven by Fractionally Integrated Multiplicative Noise

by Xiaolei Wu and Yubin Yan

Fractal Fract. 2025, 9(5), 314; https://doi.org/10.3390/fractalfract9050314 - 14 May 2025

Cited by 1 | Viewed by 299

Abstract

This paper investigates the strong convergence of a Milstein scheme for a stochastic semilinear subdiffusion equation driven by fractionally integrated multiplicative noise. The existence and uniqueness of the mild solution are established via the Banach fixed point theorem. Temporal and spatial regularity properties [...] Read more.

This paper investigates the strong convergence of a Milstein scheme for a stochastic semilinear subdiffusion equation driven by fractionally integrated multiplicative noise. The existence and uniqueness of the mild solution are established via the Banach fixed point theorem. Temporal and spatial regularity properties of the mild solution are derived using the semigroup approach. For spatial discretization, the standard Galerkin finite element method is employed, while the Grünwald–Letnikov method is used for time discretization. The Milstein scheme is utilized to approximate the multiplicative noise. For sufficiently smooth noise, the proposed scheme achieves the temporal strong convergence order of

O (τ^{α})

,

α \in (0, 1)

. Numerical experiments are presented to verify that the computational results are consistent with the theoretical predictions. Full article

(This article belongs to the Section Numerical and Computational Methods)

44 pages, 1707 KiB

Open AccessArticle

A High-Order Fractional Parallel Scheme for Efficient Eigenvalue Computation

by Mudassir Shams and Bruno Carpentieri

Fractal Fract. 2025, 9(5), 313; https://doi.org/10.3390/fractalfract9050313 - 13 May 2025

Viewed by 353

Abstract

Eigenvalue problems play a fundamental role in many scientific and engineering disciplines, including structural mechanics, quantum physics, and control theory. In this paper, we propose a fast and stable fractional-order parallel algorithm for solving eigenvalue problems. The method is implemented within a parallel [...] Read more.

Eigenvalue problems play a fundamental role in many scientific and engineering disciplines, including structural mechanics, quantum physics, and control theory. In this paper, we propose a fast and stable fractional-order parallel algorithm for solving eigenvalue problems. The method is implemented within a parallel computing framework, allowing simultaneous computations across multiple processors to improve both efficiency and reliability. A theoretical convergence analysis shows that the scheme achieves a local convergence order of

6 κ + 4

, where

κ \in (0, 1]

denotes the Caputo fractional order prescribing the memory depth of the derivative term. Comparative evaluations based on memory utilization, residual error, CPU time, and iteration count demonstrate that the proposed parallel scheme outperforms existing methods in our test cases, exhibiting faster convergence and greater efficiency. These results highlight the method’s robustness and scalability for large-scale eigenvalue computations. Full article

(This article belongs to the Special Issue Numerical Analysis and Iterative Methods for Fractional Differential Equations)

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28 pages, 3168 KiB

Open AccessReview

FDE-Testset: Comparing Matlab© Codes for Solving Fractional Differential Equations of Caputo Type

by Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro and Mikk Vikerpuur

Fractal Fract. 2025, 9(5), 312; https://doi.org/10.3390/fractalfract9050312 - 13 May 2025

Viewed by 538

Abstract

Fractional differential equations (FDEs) have attracted more and more attention in the last years; among them, equations of Caputo type allow for “more natural” initial conditions when the order is greater than one. As a result, many numerical methods have been devised and [...] Read more.

Fractional differential equations (FDEs) have attracted more and more attention in the last years; among them, equations of Caputo type allow for “more natural” initial conditions when the order is greater than one. As a result, many numerical methods have been devised and investigated for approximating their solution: the Matlab© codes of some of them are also available. The aim of this paper is a systematic comparison of such codes on a selected set of test problems. The obtained results are available on the web. Full article

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20 pages, 11532 KiB

Open AccessArticle

Experimental Study of Confining Pressure-Induced Fracture Network for Shale Gas Reservoir Under Triaxial Compression Conditions

by Jinxuan Han, Ming Gao, Yubo Wu, Ali Raza, Pei He, Jianhui Li, Yanjun Lu, Manping Yang and Hongjian Zhu

Fractal Fract. 2025, 9(5), 311; https://doi.org/10.3390/fractalfract9050311 - 13 May 2025

Viewed by 492

Abstract

The experimental study of shale fracture development is very important. As a channel of permeability, a fracture has a great influence on the development of shale gas. This study presents the results of a fracture evaluation in the Silurian Longmaxi Shale using the [...] Read more.

The experimental study of shale fracture development is very important. As a channel of permeability, a fracture has a great influence on the development of shale gas. This study presents the results of a fracture evaluation in the Silurian Longmaxi Shale using the laboratory triaxial compression experiments and CT reconstruction, considering both mechanical properties and fracture network multi-dimensional quantitative characterization. The results indicate that the plastic deformation stage of shale lasts longer under high confining pressure, whereas radial deformation is restricted. Confining pressure has a nice linear connection with both compressive strength and elastic modulus. The 2D fractal dimension of radial and vertical cracks is 1.09–1.28 when the confining pressure is between 5 and 25 MPa. The 3D fractal dimension of the fracture is 2.08–2.16. There is a linear negative correlation at high confining pressure (R² > 0.80) and a weak linear association between the 3D fractal dimension of the fracture and confining pressure at low confining pressure. The fracture angle calculated by the volume weight of multiple main cracks has a linear relationship with the confining pressure (R² > 0.89), and its value is 73.90°–52.76°. The fracture rupture rate and fracture complexity coefficient are linearly negatively correlated with confining pressure (R² > 0.82). The Euler number can well characterize the connectivity of shale fractures, and the two show a strong linear positive correlation (R² = 0.98). We suggest that the bedding plane gap compression, radial deformation limitation, and interlayer effect weakening are efficient mechanisms for the formation of shale fracture networks induced by confining pressure, and that confining pressure plays a significant role in limiting and weakening the development of shale fractures, based on the quantitative characterization results of fractures. Full article

(This article belongs to the Special Issue Flow and Transport in Fractal Models of Rock Mechanics)

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21 pages, 7529 KiB

Open AccessArticle

Multifractal Detrended Fluctuation Analysis Combined with Allen–Cahn Equation for Image Segmentation

by Minzhen Wang, Yanshan Wang, Renkang Xu, Runqiao Peng, Jian Wang and Junseok Kim

Fractal Fract. 2025, 9(5), 310; https://doi.org/10.3390/fractalfract9050310 - 12 May 2025

Viewed by 452

Abstract

This study proposes a novel image segmentation method, MF-DFA combined with the Allen–Cahn equation (MF-AC-DFA). By utilizing the Allen–Cahn equation instead of the least squares method employed in traditional MF-DFA for fitting, the accuracy and robustness of image segmentation are significantly improved. The [...] Read more.

This study proposes a novel image segmentation method, MF-DFA combined with the Allen–Cahn equation (MF-AC-DFA). By utilizing the Allen–Cahn equation instead of the least squares method employed in traditional MF-DFA for fitting, the accuracy and robustness of image segmentation are significantly improved. The article first conducts segmentation experiments under various conditions, including different target shapes, image backgrounds, and resolutions, to verify the feasibility of MF-AC-DFA. It then compares the proposed method with gradient segmentation methods and demonstrates the superiority of MF-AC-DFA. Finally, real-life wire diagrams and transmission tower diagrams are used for segmentation, which shows the application potential of MF-AC-DFA in complex scenes. This method is expected to be applied to the real-time state monitoring and analysis of power facilities, and it is anticipated to improve the safety and reliability of the power grid. Full article

(This article belongs to the Special Issue Fractal and Multifractal Analysis in Environmental, Medical and Technical Fields: Pattern Recognition, Analysis and Processing of Images)

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19 pages, 2581 KiB

Open AccessArticle

Analytical and Dynamical Study of Solitary Waves in a Fractional Magneto-Electro-Elastic System

by Sait San, Beenish and Fehaid Salem Alshammari

Fractal Fract. 2025, 9(5), 309; https://doi.org/10.3390/fractalfract9050309 - 10 May 2025

Cited by 3 | Viewed by 339

Abstract

Magneto-electro-elastic materials, a novel class of smart materials, exhibit remarkable energy conversion properties, making them highly suitable for applications in nanotechnology. This study focuses on various aspects of the fractional nonlinear longitudinal wave equation (FNLWE) that models wave propagation in a magneto-electro-elastic circular [...] Read more.

Magneto-electro-elastic materials, a novel class of smart materials, exhibit remarkable energy conversion properties, making them highly suitable for applications in nanotechnology. This study focuses on various aspects of the fractional nonlinear longitudinal wave equation (FNLWE) that models wave propagation in a magneto-electro-elastic circular rod. Using the direct algebraic method, several new soliton solutions were derived under specific parameter constraints. In addition, Galilean transformation was employed to explore the system’s sensitivity and quasi-periodic dynamics. The study incorporates 2D, 3D, and time-series visualizations as effective tools for analyzing quasi-periodic behavior. The results contribute to a deeper understanding of the nonlinear dynamical features of such systems and demonstrate the robustness of the applied methodologies. This research not only extends existing knowledge of nonlinear wave equations but also introduces a substantial number of new solutions with broad applicability. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)

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12 pages, 267 KiB

Open AccessArticle

Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem

by Keyu Zhang, Tian Wang, Donal O’Regan and Jiafa Xu

Fractal Fract. 2025, 9(5), 308; https://doi.org/10.3390/fractalfract9050308 - 10 May 2025

Viewed by 319

Abstract

In this paper, we study the existence of extremal solutions for a Caputo-type fractional-order initial value problem. By using the monotone iteration technique and the upper–lower solution method, we obtain our existence theorem when the nonlinearity satisfies a reverse-type Lipschitz condition. Note that [...] Read more.

In this paper, we study the existence of extremal solutions for a Caputo-type fractional-order initial value problem. By using the monotone iteration technique and the upper–lower solution method, we obtain our existence theorem when the nonlinearity satisfies a reverse-type Lipschitz condition. Note that our nonlinearity depends on the unknown function and its fractional-order derivative. Full article

(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)

13 pages, 436 KiB

Open AccessArticle

Graph-Theoretic Characterization of Separation Conditions in Self-Affine Iterated Function Systems

by Ming-Qi Bai, Jun Luo and Yi Wu

Fractal Fract. 2025, 9(5), 307; https://doi.org/10.3390/fractalfract9050307 - 8 May 2025

Viewed by 323

Abstract

For a self-affine iterated function system (IFS)

{f_{j}}_{j = 1}^{N}

on

R^{d}

defined by

f_{j} (x) = A^{- 1} (x + d_{j})

, where A is an expansive matrix and [...] Read more.

For a self-affine iterated function system (IFS)

{f_{j}}_{j = 1}^{N}

on

R^{d}

defined by

f_{j} (x) = A^{- 1} (x + d_{j})

, where A is an expansive matrix and

d_{j} \in R^{d}

, we reveal a novel characterization of the open set and weak separation conditions through the bounded degree of the augmented tree induced by the IFS. Furthermore, the augmented tree is shown to be a Gromov hyperbolic graph, and its hyperbolic boundary is Hölder equivalent to the self-affine set generated by the IFS, establishing a canonical association between self-affine IFSs and Gromov hyperbolic graphs. Full article

(This article belongs to the Section Geometry)

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18 pages, 1319 KiB

Open AccessArticle

The Influence of Lévy Noise and Independent Jumps on the Dynamics of a Stochastic COVID-19 Model with Immune Response and Intracellular Transmission

by Yuqin Song, Peijiang Liu and Anwarud Din

Fractal Fract. 2025, 9(5), 306; https://doi.org/10.3390/fractalfract9050306 - 8 May 2025

Viewed by 347

Abstract

The coronavirus (COVID-19) expanded rapidly and affected almost the whole world since December 2019. COVID-19 has an unusual ability to spread quickly through airborne viruses and substances. Taking into account the disease’s natural progression, this study considers that viral spread is unpredictable rather [...] Read more.

The coronavirus (COVID-19) expanded rapidly and affected almost the whole world since December 2019. COVID-19 has an unusual ability to spread quickly through airborne viruses and substances. Taking into account the disease’s natural progression, this study considers that viral spread is unpredictable rather than deterministic. The continuous time Markov chain (CTMC) stochastic model technique has been used to anticipate upcoming states using random variables. The suggested study focuses on a model with five distinct compartments. The first class contains Lévy noise-based infection rates (termed as vulnerable people), while the second class refers to the infectious compartment having similar perturbation incidence as the others. We demonstrate the existence and uniqueness of the positive solution of the model. Subsequently, we define a stochastic threshold as a requisite condition for the extinction and durability of the disease’s mean. By assuming that the threshold value

R_{0}^{D}

is smaller than one, it is demonstrated that the solution trajectories oscillate around the disease-free state (DFS) of the corresponding deterministic model. The solution curves of the SDE model fluctuate in the neighborhood of the endemic state of the base ODE system, when

R_{0}^{P} > 1

elucidates the definitive persistence theory of the suggested model. Ultimately, numerical simulations are provided to confirm our theoretical findings. Moreover, the results indicate that stochastic environmental disturbances might influence the propagation of infectious diseases. Significantly, increased noise levels could hinder the transmission of epidemics within the community. Full article

(This article belongs to the Special Issue Fractional Dynamics in Epidemic Models: Theoretical Approaches and Applications)

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28 pages, 3560 KiB

Open AccessArticle

Solitons, Cnoidal Waves and Nonlinear Effects in Oceanic Shallow Water Waves

by Huanhe Dong, Shengfang Yang, Yong Fang and Mingshuo Liu

Fractal Fract. 2025, 9(5), 305; https://doi.org/10.3390/fractalfract9050305 - 7 May 2025

Viewed by 354

Abstract

Gravity water waves in the shallow-ocean scenario described by generalized Boussinesq–Broer–Kaup–Whitham (gBBKW) equations are discussed. The residual symmetry and Bäcklund transformation associated with the gBBKW equations are systematically constructed. The time and space evolution of wave velocity and height are explored. Additionally, it [...] Read more.

Gravity water waves in the shallow-ocean scenario described by generalized Boussinesq–Broer–Kaup–Whitham (gBBKW) equations are discussed. The residual symmetry and Bäcklund transformation associated with the gBBKW equations are systematically constructed. The time and space evolution of wave velocity and height are explored. Additionally, it is demonstrated that the gBBKW equations are solvable through the consistent Riccati expansion method. Leveraging this property, a novel Bäcklund transformation, solitary wave solution, and soliton–cnoidal wave solution are derived. Furthermore, miscellaneous novel solutions of gBBKW equations are obtained using the modified Sardar sub-equation method. The impact of variations in the diffusion power parameter on wave velocity and height is quantitatively analyzed. The exact solutions of gBBKW equations provide precise description of propagation characteristics for a deeper understanding and the prediction of some ocean wave phenomena. Full article

(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)

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10 pages, 266 KiB

Open AccessArticle

Ulam–Hyers–Rassias Stability of ψ-Hilfer Volterra Integro-Differential Equations of Fractional Order Containing Multiple Variable Delays

by John R. Graef, Osman Tunç and Cemil Tunç

Fractal Fract. 2025, 9(5), 304; https://doi.org/10.3390/fractalfract9050304 - 6 May 2025

Cited by 2 | Viewed by 433

Abstract

The authors consider a nonlinear

ψ

-Hilfer fractional-order Volterra integro-differential equation (

ψ

-Hilfer FrOVIDE) that incorporates N-multiple variable time delays into the equation. By utilizing the

ψ

-Hilfer fractional derivative, they investigate the Ulam–Hyers–Rassias and Ulam–Hyers stability of the equation by [...] Read more.

The authors consider a nonlinear

ψ

-Hilfer fractional-order Volterra integro-differential equation (

ψ

-Hilfer FrOVIDE) that incorporates N-multiple variable time delays into the equation. By utilizing the

ψ

-Hilfer fractional derivative, they investigate the Ulam–Hyers–Rassias and Ulam–Hyers stability of the equation by using fixed-point methods. Their results improve existing ones both with and without delays by extending them to nonlinear

ψ

-Hilfer FrOVIDEs that incorporate N-multiple variable time delays. Full article

(This article belongs to the Special Issue Advances in Fractional Operators: Mathematical Perspectives and Multidisciplinary Applications)

15 pages, 516 KiB

Open AccessArticle

Remarks on the Relationship Between Fractal Dimensions and Convergence Speed

by Jiaqi Qiu and Yongshun Liang

Fractal Fract. 2025, 9(5), 303; https://doi.org/10.3390/fractalfract9050303 - 6 May 2025

Viewed by 471

Abstract

This paper conducts an in-depth investigation into the fundamental relationship between the fractal dimensions and convergence properties of mathematical sequences. By concentrating on three representative classes of sequences, namely, the factorial-decay, logarithmic-decay, and factorial–exponential types, a comprehensive framework is established to link their [...] Read more.

This paper conducts an in-depth investigation into the fundamental relationship between the fractal dimensions and convergence properties of mathematical sequences. By concentrating on three representative classes of sequences, namely, the factorial-decay, logarithmic-decay, and factorial–exponential types, a comprehensive framework is established to link their geometric characteristics with asymptotic behavior. This study makes two significant contributions to the field of fractal analysis. Firstly, a unified methodology is developed for the calculation of multiple fractal dimensions, including the Box, Hausdorff, Packing, and Assouad dimensions, of discrete sequences. This methodology reveals how these dimensional quantities jointly describe the structures of sequences, providing a more comprehensive understanding of their geometric properties. Secondly, it is demonstrated that different fractal dimensions play distinct yet complementary roles in regulating convergence rates. Specifically, the Box dimension determines the global convergence properties of sequences, while the Assouad dimension characterizes the local constraints on the speed of convergence. The theoretical results presented herein offer novel insights into the inherent connection between geometric complexity and analytical behavior within sequence spaces. These findings have immediate and far-reaching implications for various applications that demand precise control over convergence properties, such as numerical algorithm design and signal processing. Notably, the identification of dimension-based convergence criteria provides practical and effective tools for the analysis of sequence behavior in both pure mathematical research and applied fields. Full article

(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)

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17 pages, 317 KiB

Open AccessArticle

An Exploration of the Qualitative Analysis of the Generalized Pantograph Equation with the q-Hilfer Fractional Derivative

by R. Vivek, K. Kanagarajan, D. Vivek, T. D. Alharbi and E. M. Elsayed

Fractal Fract. 2025, 9(5), 302; https://doi.org/10.3390/fractalfract9050302 - 6 May 2025

Viewed by 354

Abstract

This manuscript tries to show that there is only one solution to the problem of the q-Hilfer fractional generalized pantograph differential equations with a nonlocal condition, and it does so by employing a particular technique known as Schaefer’s fixed point theorem and the [...] Read more.

This manuscript tries to show that there is only one solution to the problem of the q-Hilfer fractional generalized pantograph differential equations with a nonlocal condition, and it does so by employing a particular technique known as Schaefer’s fixed point theorem and the Banach contraction principle. Then, we verify that the Ulam-type stability is valid. To illustrate the results, an example is provided. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 3rd Edition)

18 pages, 1838 KiB

Open AccessArticle

On Solving Modified Time Caputo Fractional Kawahara Equations in the Framework of Hilbert Algebras Using the Laplace Residual Power Series Method

by Faten H. Damag and Amin Saif

Fractal Fract. 2025, 9(5), 301; https://doi.org/10.3390/fractalfract9050301 - 6 May 2025

Viewed by 327

Abstract

In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series method, combined with the Laplace [...] Read more.

In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series method, combined with the Laplace transform, to introduce a new effective technique called the Laplace Residual Power Series Method (LRPSM). This method is applied to derive the coefficients of the series solution for MTCFKEs in the context of Hilbert algebras. In real Hilbert algebras, we obtain approximate solutions for MTCFKEs under both exact and approximate initial conditions. We present both graphical and numerical results of the approximate analytical solutions to demonstrate the capability, efficiency, and reliability of the LRPSM. Furthermore, we compare our results with solutions obtained using the homotopy analysis method and the natural transform decomposition method. Full article

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8 pages, 799 KiB

Open AccessArticle

Optical Solutions of the Nonlinear Kodama Equation with the M-Truncated Derivative via the Extended (G^′/G)-Expansion Method

by Zhao Li

Fractal Fract. 2025, 9(5), 300; https://doi.org/10.3390/fractalfract9050300 - 5 May 2025

Cited by 7 | Viewed by 397

Abstract

The main purpose of this article is to study the optical solutions of the nonlinear Kodama equation with the M-truncated derivative by using the extended

(G^{'} / G)

-expansion method. Firstly, the nonlinear Kodama equation with the M-truncated derivative is [...] Read more.

The main purpose of this article is to study the optical solutions of the nonlinear Kodama equation with the M-truncated derivative by using the extended

(G^{'} / G)

-expansion method. Firstly, the nonlinear Kodama equation with the M-truncated derivative is transformed into a nonlinear ordinary differential equation based on the principle of homogeneous equilibrium and the traveling wave transformation. Secondly, the optical solutions of the nonlinear Kodama equation with the M-truncated derivative are constructed by using the extended

(G^{'} / G)

-expansion method. Finally, three-dimensional, two-dimensional, and contour maps of partial solutions are obtained by using Matlab R2023b mathematical software. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)

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16 pages, 2958 KiB

Open AccessArticle

Fractional Uncertain Forecasting of the Impact of Groundwater Over-Exploitation on Temperature in the Largest Groundwater Depression Cone

by Xiangyue Ren, Liyuan Ren and Lifeng Wu

Fractal Fract. 2025, 9(5), 299; https://doi.org/10.3390/fractalfract9050299 - 5 May 2025

Viewed by 428

Abstract

China currently faces critical climatic conditions, with persistent global warming trends and extreme heat waves across the northern hemisphere. To explore the predictive trajectory of regional extreme high temperature influenced by groundwater over-exploitation, the SGMC(1,N) was established. Additionally, the SGMC(1,N) model was validated [...] Read more.

China currently faces critical climatic conditions, with persistent global warming trends and extreme heat waves across the northern hemisphere. To explore the predictive trajectory of regional extreme high temperature influenced by groundwater over-exploitation, the SGMC(1,N) was established. Additionally, the SGMC(1,N) model was validated using 2019–2023 observational data from the world’s largest groundwater depression cone. The results demonstrate superior performance, with the model achieving a MAPE of 1.97% compared to benchmark models. Scenario simulations with annual groundwater reduction rates (−15%, −20%, −25%) successfully project extreme heat evolution for 2024–2028. When the decline rate of annual groundwater over-exploitation is set at −20%, a 6.66 °C temperature reduction from baseline by 2028 is projected. Stable decline trends emerge when GOE reduction exceeds 20%. To mitigate regional extreme heat, implementing phased groundwater extraction quotas and total extraction cap regulations is recommended. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Grey Models, 2nd Edition)

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Fractal Fract., Volume 9, Issue 5 (May 2025) – 62 articles

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