Fractal and Fractional

22 pages, 10576 KiB

Open AccessArticle

Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm

by Qianqian Fan, Qiumei Liu, Yiming Chen, Yuhuan Cui, Jingguo Qu and Lei Wang

Fractal Fract. 2025, 9(7), 477; https://doi.org/10.3390/fractalfract9070477 - 21 Jul 2025

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This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the [...] Read more.

This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the Fractional Kelvin–Voigt (FKV) model with the Caputo fractional derivative and the theory of nonlocal elasticity. Then, the shifted Legendre polynomial is used to approximate the displacement function, and the governing equations are transformed into algebraic equations to facilitate the numerical solution in the time domain. Moreover, the systematic convergence analysis is carried out to verify the convergence of the ternary displacement function and its fractional derivatives in the equation, ensuring the rigor of the mathematical model. Finally, a dimensionless numerical example is given to verify the feasibility of the proposed algorithm, and the effects of material parameters on plate displacement are analyzed for double-layer plates with different materials. Full article

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13 pages, 9670 KiB

Open AccessArticle

Exact Solitary Wave Solutions and Sensitivity Analysis of the Fractional (3+1)D KdV–ZK Equation

by Asif Khan, Fehaid Salem Alshammari, Sadia Yasin and Beenish

Fractal Fract. 2025, 9(7), 476; https://doi.org/10.3390/fractalfract9070476 - 21 Jul 2025

Viewed by 143

Abstract

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a [...] Read more.

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a case study, which describes some intricate phenomena of wave behavior in plasma physics and fluid dynamics. With the implementation of SSEM, we yield new solitary wave solutions and explicitly examine the role of the fractional-order parameter in the dynamics of the solutions. In addition, the sensitivity analysis of the results is conducted in the Galilean transformation in order to ensure that the obtained results are valid and have physical significance. Besides expanding the toolbox of analytical methods to address high-dimensional nonlinear FDEs, the proposed method helps to better understand how fractional-order dynamics affect the nonlinear wave phenomenon. The results are compared to known methods and a discussion about their possible applications and limitations is given. The results show the effectiveness and flexibility of SSEM along with JMRLD in forming new categories of exact solutions to nonlinear fractional models. Full article

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

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20 pages, 1485 KiB

Open AccessArticle

The Continued Fraction Structure in Physical Fractal Theory

by Ruiheng Jiang, Tianyi Zhou and Yajun Yin

Fractal Fract. 2025, 9(7), 475; https://doi.org/10.3390/fractalfract9070475 - 21 Jul 2025

Viewed by 77

Abstract

The objective of this study is to reveal the intrinsic connection between fractal operators in physical fractal spaces and continued fractions. The specific contributions include: (1) reviewing fundamental concepts of continued fractions and physical fractal theory; (2) establishing algebraic structure consistency between continued [...] Read more.

The objective of this study is to reveal the intrinsic connection between fractal operators in physical fractal spaces and continued fractions. The specific contributions include: (1) reviewing fundamental concepts of continued fractions and physical fractal theory; (2) establishing algebraic structure consistency between continued fractions and fractal operators through the medium of generation mappings; (3) discussing the convergence of fractal operators by employing theory from continued fraction analysis; and (4) confirming the correspondence between fixed points of infinite continued fractions and algebraic equations governing fractal operators. Full article

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3 pages, 165 KiB

Open AccessEditorial

Fractal and Fractional Analysis in Biomedical Sciences and Engineering

by Andjelija Ž. Ilić

Fractal Fract. 2025, 9(7), 474; https://doi.org/10.3390/fractalfract9070474 - 21 Jul 2025

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Abstract

The visual appearance of fractal patterns in biology has been noted from the early days of research in this field and, henceforth, has been studied for a long time [...] Full article

(This article belongs to the Special Issue Fractal and Fractional Analysis in Biomedical Sciences and Engineering)

16 pages, 1993 KiB

Open AccessArticle

A Fractional Derivative Insight into Full-Stage Creep Behavior in Deep Coal

by Shuai Yang, Hongchen Song, Hongwei Zhou, Senlin Xie, Lei Zhang and Wentao Zhou

Fractal Fract. 2025, 9(7), 473; https://doi.org/10.3390/fractalfract9070473 - 21 Jul 2025

Viewed by 144

Abstract

The time-dependent creep behavior of coal is essential for assessing long-term structural stability and operational safety in deep coal mining. Therefore, this work develops a full-stage creep constitutive model. By integrating fractional calculus theory with statistical damage mechanics, a nonlinear fractional-order (FO) damage [...] Read more.

The time-dependent creep behavior of coal is essential for assessing long-term structural stability and operational safety in deep coal mining. Therefore, this work develops a full-stage creep constitutive model. By integrating fractional calculus theory with statistical damage mechanics, a nonlinear fractional-order (FO) damage creep model is constructed through serial connection of elastic, viscous, viscoelastic, and viscoelastic–plastic components. Based on this model, both one-dimensional and three-dimensional (3D) fractional creep damage constitutive equations are acquired. Model parameters are identified using experimental data from deep coal samples in the mining area. The result curves of the improved model coincide with experimental data points, accurately describing the deceleration creep stage (DCS), steady-state creep stage (SCS), and accelerated creep stage (ACS). Furthermore, a sensitivity analysis elucidates the impact of model parameters on coal creep behavior, thereby confirming the model’s robustness and applicability. Consequently, the proposed model offers a solid theoretical basis for evaluating the sustained stability of deep coal mining and has great application potential in deep underground engineering. Full article

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23 pages, 406 KiB

Open AccessArticle

Periodically Kicked Rotator with Power-Law Memory: Exact Solution and Discrete Maps

by Vasily E. Tarasov

Fractal Fract. 2025, 9(7), 472; https://doi.org/10.3390/fractalfract9070472 - 21 Jul 2025

Viewed by 249

Abstract

This article discusses the transformation of a continuous-time model of the fractional system into a discrete-time model of the fractional system. For the continuous-time model, the exact solution of the nonlinear equation with fractional derivatives (FDs) that has the form of the damped [...] Read more.

This article discusses the transformation of a continuous-time model of the fractional system into a discrete-time model of the fractional system. For the continuous-time model, the exact solution of the nonlinear equation with fractional derivatives (FDs) that has the form of the damped rotator type with power non-locality in time is obtained.This equation with two FDs and periodic kicks is solved in the general case for the arbitrary orders of FDs without any approximations. A three-stage method for solving a nonlinear equation with two FDs and deriving discrete maps with memory (DMMs) is proposed. The exact solutions of the nonlinear equation with two FDs are obtained for arbitrary values of the orders of these derivatives. In this article, the orders of two FDs are not related to each other, unlike in previous works. The exact solution of nonlinear equation with two FDs of different orders and periodic kicks are proposed. Using this exact solution, we derive DMMs that describe a kicked damped rotator with power-law non-localities in time. For the discrete-time model, these damped DMMs are described by the exact solution of nonlinear equations with FDs at discrete time points as the functions of all past discrete moments of time. An example of the application, the exact solution and DMMs are proposed for the economic growth model with two-parameter power-law memory and price kicks. It should be emphasized that the manuscript proposes exact analytical solutions to nonlinear equations with FDs, which are derived without any approximations. Therefore, it does not require any numerical proofs, justifications, or numerical validation. The proposed method gives exact analytical solutions, where approximations are not used at all. Full article

(This article belongs to the Special Issue Continuous/Discrete-Time Fractional Systems: Modelling, Design and Estimation)

27 pages, 3887 KiB

Open AccessArticle

Methodologies for Improved Optimisation of the Derivative Order and Neural Network Parameters in Neural FDE Models

by Cecília Coelho, M. Fernanda P. Costa, Oliver Niggemann and Luís L. Ferrás

Fractal Fract. 2025, 9(7), 471; https://doi.org/10.3390/fractalfract9070471 - 20 Jul 2025

Viewed by 106

Abstract

This work presents and compares different methodologies for the joint optimisation of the fractional derivative order and the parameters of the right-hand-side neural network in Neural Fractional Differential Equation models. The proposed strategies aim to tackle the training difficulties typically encountered when learning [...] Read more.

This work presents and compares different methodologies for the joint optimisation of the fractional derivative order and the parameters of the right-hand-side neural network in Neural Fractional Differential Equation models. The proposed strategies aim to tackle the training difficulties typically encountered when learning the fractional order

α

together with the network weights. One approach is based on regulating the gradient magnitude of the loss function with respect to

α

, encouraging more stable and effective updates. Another strategy introduces an online pre-training scheme, where the network parameters are initially optimised over progressively longer time intervals, while

α

is updated more conservatively using the full time trajectory. The study focuses only on a foundational setting with one-dimensional problems, and numerical experiments demonstrate that the proposed techniques improve both training stability and accuracy. Nonetheless, the issue of non-uniqueness in the optimal derivative order remains, particularly in less well-posed scenarios, suggesting the need for further research in data-driven modelling of fractional-order systems. Full article

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

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18 pages, 451 KiB

Open AccessArticle

Distinctive LMI Formulations for Admissibility and Stabilization Algorithms of Singular Fractional-Order Systems with Order Less than One

by Xinhai Wang, Xuefeng Zhang, Qing-Guo Wang and Driss Boutat

Fractal Fract. 2025, 9(7), 470; https://doi.org/10.3390/fractalfract9070470 - 19 Jul 2025

Viewed by 125

Abstract

This paper presents three novel sufficient and necessary conditions for the admissibility of singular fractional-order systems (FOSs), a stabilization criterion, and a solution algorithm. The strict linear matrix inequality (LMI) stability criterion for integer-order systems is generalized to singular FOSs by using column-full [...] Read more.

This paper presents three novel sufficient and necessary conditions for the admissibility of singular fractional-order systems (FOSs), a stabilization criterion, and a solution algorithm. The strict linear matrix inequality (LMI) stability criterion for integer-order systems is generalized to singular FOSs by using column-full rank matrices. This admissibility criterion does not involve complex variables and is different from all previous results, filling a gap in this area. Based on the LMIs in the generalized condition, the improved criterion utilizes a variable substitution technique to reduce the number of matrix variables to be solved from one pair to one, reflecting the admissibility more essentially. This improved result simplifies the programming process compared to the traditional approach that requires two matrix variables. To complete the state feedback controller design, the system matrices in the generalized admissibility criterion are decoupled, but bilinear constraints still occur in the stabilization criterion. For this case, where a feasible solution cannot be found using the MATLAB LMI toolbox, a branch-and-bound algorithm (BBA) is designed to solve it. Finally, the validity of these criteria and the BBA is verified by three examples, including a real circuit model. Full article

(This article belongs to the Special Issue Fractional Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image Processing Algorithms, Second Edition)

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28 pages, 404 KiB

Open AccessArticle

Normalized Ground States for Mixed Fractional Schrödinger Equations with Combined Local and Nonlocal Nonlinearities

by Jie Yang and Haibo Chen

Fractal Fract. 2025, 9(7), 469; https://doi.org/10.3390/fractalfract9070469 - 18 Jul 2025

Viewed by 165

Abstract

This paper studies the existence, regularity, and properties of normalized ground state solutions for the mixed fractional Schrödinger equations. For subcritical cases, we establish the boundedness and Sobolev regularity of solutions, derive Pohozaev identities, and prove the existence of radial, decreasing ground states, [...] Read more.

This paper studies the existence, regularity, and properties of normalized ground state solutions for the mixed fractional Schrödinger equations. For subcritical cases, we establish the boundedness and Sobolev regularity of solutions, derive Pohozaev identities, and prove the existence of radial, decreasing ground states, while showing nonexistence in the

L^{2}

-critical case. For

L^{2}

-supercritical exponents, we identify parameter regimes where ground states exist, characterized by a negative Lagrange multiplier. The analysis combines variational methods, scaling techniques, and the careful study of fibering maps to address challenges posed by competing nonlinearities and nonlocal interactions. Full article

(This article belongs to the Special Issue Variational Problems and Fractional Differential Equations)

23 pages, 765 KiB

Open AccessArticle

Inverse Problem for a Time-Dependent Source in Distributed-Order Time-Space Fractional Diffusion Equations

by Yushan Li and Huimin Wang

Fractal Fract. 2025, 9(7), 468; https://doi.org/10.3390/fractalfract9070468 - 18 Jul 2025

Viewed by 220

Abstract

This paper investigates the problem of identifying a time-dependent source term in distributed-order time-space fractional diffusion equations (FDEs) based on boundary observation data. Firstly, the existence, uniqueness, and regularity of the solution to the direct problem are proved. Using the regularity of the [...] Read more.

This paper investigates the problem of identifying a time-dependent source term in distributed-order time-space fractional diffusion equations (FDEs) based on boundary observation data. Firstly, the existence, uniqueness, and regularity of the solution to the direct problem are proved. Using the regularity of the solution and a Gronwall inequality with a weakly singular kernel, the uniqueness and stability estimates of the solution to the inverse problem are obtained. Subsequently, the inverse source problem is transformed into a minimization problem of a functional using the Tikhonov regularization method, and an approximate solution is obtained by the conjugate gradient method. Numerical experiments confirm that the method provides both accurate and robust results. Full article

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26 pages, 11154 KiB

Open AccessArticle

The Pore Structure and Fractal Characteristics of Upper Paleozoic Coal-Bearing Shale Reservoirs in the Yangquan Block, Qinshui Basin

by Jinqing Zhang, Xianqing Li, Xueqing Zhang, Xiaoyan Zou, Yunfeng Yang and Shujuan Kang

Fractal Fract. 2025, 9(7), 467; https://doi.org/10.3390/fractalfract9070467 - 18 Jul 2025

Viewed by 230

Abstract

The investigation of the pore structure and fractal characteristics of coal-bearing shale is critical for unraveling reservoir heterogeneity, storage-seepage capacity, and gas occurrence mechanisms. In this study, 12 representative Upper Paleozoic coal-bearing shale samples from the Yangquan Block of the Qinshui Basin were [...] Read more.

The investigation of the pore structure and fractal characteristics of coal-bearing shale is critical for unraveling reservoir heterogeneity, storage-seepage capacity, and gas occurrence mechanisms. In this study, 12 representative Upper Paleozoic coal-bearing shale samples from the Yangquan Block of the Qinshui Basin were systematically analyzed through field emission scanning electron microscopy (FE-SEM), high-pressure mercury intrusion, and gas adsorption experiments to characterize pore structures and calculate multi-scale fractal dimensions (D₁–D₅). Key findings reveal that reservoir pores are predominantly composed of macropores generated by brittle fracturing and interlayer pores within clay minerals, with residual organic pores exhibiting low proportions. Macropores dominate the total pore volume, while mesopores primarily contribute to the specific surface area. Fractal dimension D₁ shows a significant positive correlation with clay mineral content, highlighting the role of diagenetic modification in enhancing the complexity of interlayer pores. D₂ is strongly correlated with the quartz content, indicating that brittle fracturing serves as a key driver of macropore network complexity. Fractal dimensions D₃–D₅ further unveil the synergistic control of tectonic activity and dissolution on the spatial distribution of pore-fracture systems. Notably, during the overmature stage, the collapse of organic pores suppresses mesopore complexity, whereas inorganic diagenetic processes (e.g., quartz cementation and tectonic fracturing) significantly amplify the heterogeneity of macropores and fractures. These findings provide multi-scale fractal theoretical insights for evaluating coal-bearing shale gas reservoirs and offer actionable recommendations for optimizing the exploration and development of Upper Paleozoic coal-bearing shale gas resources in the Yangquan Block of the Qinshui Basin. Full article

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

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22 pages, 320 KiB

Open AccessArticle

A New Caputo Fractional Differential Equation with Infinite-Point Boundary Conditions: Positive Solutions

by Jing Ren, Zijuan Du and Chengbo Zhai

Fractal Fract. 2025, 9(7), 466; https://doi.org/10.3390/fractalfract9070466 - 18 Jul 2025

Viewed by 137

Abstract

This paper mainly studies a different infinite-point Caputo fractional differential equation, whose nonlinear term may be singular. Under some conditions, we first use spectral analysis and fixed-point index theorem to explore the existence of positive solutions of the equation, and then use Banach [...] Read more.

This paper mainly studies a different infinite-point Caputo fractional differential equation, whose nonlinear term may be singular. Under some conditions, we first use spectral analysis and fixed-point index theorem to explore the existence of positive solutions of the equation, and then use Banach fixed-point theorem to prove the uniqueness of positive solutions. Finally, an interesting example is used to explain the main result. Full article

(This article belongs to the Section General Mathematics, Analysis)

22 pages, 14847 KiB

Open AccessArticle

Formation Control of Underactuated AUVs Using a Fractional-Order Sliding Mode Observer

by Long He, Mengting Xie, Ya Zhang, Shizhong Li, Bo Li, Zehui Yuan and Chenrui Bai

Fractal Fract. 2025, 9(7), 465; https://doi.org/10.3390/fractalfract9070465 - 18 Jul 2025

Viewed by 209

Abstract

This paper proposes a control method that combines a fractional-order sliding mode observer and a cooperative control strategy to address the problem of path-following for underactuated autonomous underwater vehicles (AUVs) in complex marine environments. First, a fractional-order sliding mode observer is designed, combining [...] Read more.

This paper proposes a control method that combines a fractional-order sliding mode observer and a cooperative control strategy to address the problem of path-following for underactuated autonomous underwater vehicles (AUVs) in complex marine environments. First, a fractional-order sliding mode observer is designed, combining fractional calculus and double-power convergence laws to enhance the estimation accuracy of high-frequency disturbances. An adaptive gain mechanism is introduced to avoid dependence on the upper bound of disturbances. Second, a formation cooperative control strategy based on path parameter coordination is proposed. By setting independent reference points for each AUV and exchanging path parameters, formation consistency is achieved with low communication overhead. For the followers’ speed control problem, an error-based expected speed adjustment mechanism is introduced, and a hyperbolic tangent function is used to replace the traditional arctangent function to improve the response speed of the system. Numerical simulation results show that this control method performs well in terms of path-following accuracy, formation maintenance capability, and disturbance suppression, verifying its effectiveness and robustness in complex marine environments. Full article

(This article belongs to the Special Issue Fractional Control Systems and Estimation: Control, State Observers and Parameter Estimation)

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5 pages, 161 KiB

Open AccessEditorial

Closing Editorial for Applications of Fractional-Order Systems in Automatic Control

by German A. Munoz-Hernandez and Jose Fermi Guerrero-Castellanos

Fractal Fract. 2025, 9(7), 464; https://doi.org/10.3390/fractalfract9070464 - 17 Jul 2025

Viewed by 163

Abstract

Fractional-order systems have been applied in very diverse areas of science and engineering [...] Full article

(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)

22 pages, 3655 KiB

Open AccessArticle

Analytical Description of Three-Dimensional Fractal Surface Wear Process Based on Multi-Stage Contact Theory

by Wang Zhang, Ling Li, Yang Liu, Jingjing Wang, Yao Li and Guozhang Chen

Fractal Fract. 2025, 9(7), 463; https://doi.org/10.3390/fractalfract9070463 - 16 Jul 2025

Viewed by 205

Abstract

Accurately revealing the sliding wear mechanisms of mechanical surfaces is crucial for enhancing the performance of mechanical surfaces. This study reveals the mechanism of stage transitions in three-dimensional surface wear processes from a microscopic contact perspective. Firstly, according to the fractal theory, a [...] Read more.

Accurately revealing the sliding wear mechanisms of mechanical surfaces is crucial for enhancing the performance of mechanical surfaces. This study reveals the mechanism of stage transitions in three-dimensional surface wear processes from a microscopic contact perspective. Firstly, according to the fractal theory, a mathematical model for the critical area scale controlling debris particle formation is established. Secondly, incorporating the contact area scale, a mathematical expression for the wear coefficient of surfaces, is proposed based on the multi-stage contact theory. Finally, the influences of fractal parameters on critical contact load, wear rate, and wear coefficient are systematically examined. The experimental findings substantiate that the proposed wear model exhibits an explicit deterministic formulation and demonstrates high predictive accuracy for the wear rate. Full article

(This article belongs to the Special Issue Multi-Scale and Fractal Research on Advanced Manufacturing and Tribology)

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22 pages, 436 KiB

Open AccessArticle

Fractional Boundary Value Problems with Parameter-Dependent and Asymptotic Conditions

by Kateryna Marynets and Dona Pantova

Fractal Fract. 2025, 9(7), 462; https://doi.org/10.3390/fractalfract9070462 - 16 Jul 2025

Viewed by 150

Abstract

We study a nonlinear fractional differential equation, defined on a finite and infinite interval. In the finite interval setting, we attach initial conditions and parameter-dependent boundary conditions to the problem. We apply a dichotomy approach, coupled with the numerical-analytic method, to analyze the [...] Read more.

We study a nonlinear fractional differential equation, defined on a finite and infinite interval. In the finite interval setting, we attach initial conditions and parameter-dependent boundary conditions to the problem. We apply a dichotomy approach, coupled with the numerical-analytic method, to analyze the problem and to construct a sequence of approximations. Additionally, we study the existence of bounded solutions in the case when the fractional differential equation is defined on the half-axis and is subject to asymptotic conditions. Our theoretical results are applied to the Arctic gyre equation in the fractional setting on a finite interval. Full article

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19 pages, 670 KiB

Open AccessArticle

Solutions to Variable-Order Fractional BVPs with Multipoint Data in W^s,p Spaces

by Zineb Bellabes, Kadda Maazouz, Naima Boussekkine and Rosana Rodríguez-López

Fractal Fract. 2025, 9(7), 461; https://doi.org/10.3390/fractalfract9070461 - 15 Jul 2025

Viewed by 198

Abstract

This study explores the existence of positive solutions within a Sobolev space for a boundary value problem that involves Riemann–Liouville fractional derivatives of variable order. The analysis utilizes the method of upper and lower solutions in combination with the Schauder fixed-point theorem. To [...] Read more.

This study explores the existence of positive solutions within a Sobolev space for a boundary value problem that involves Riemann–Liouville fractional derivatives of variable order. The analysis utilizes the method of upper and lower solutions in combination with the Schauder fixed-point theorem. To illustrate the theoretical findings, a numerical example is included. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)

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39 pages, 7470 KiB

Open AccessArticle

Estimation of Fractal Dimension and Semantic Segmentation of Motion-Blurred Images by Knowledge Distillation in Autonomous Vehicle

by Seong In Jeong, Min Su Jeong and Kang Ryoung Park

Fractal Fract. 2025, 9(7), 460; https://doi.org/10.3390/fractalfract9070460 - 15 Jul 2025

Viewed by 313

Abstract

Research on semantic segmentation for remote sensing road scenes advanced significantly, driven by autonomous driving technology. However, motion blur from camera or subject movements hampers segmentation performance. To address this issue, we propose a knowledge distillation-based semantic segmentation network (KDS-Net) that is robust [...] Read more.

Research on semantic segmentation for remote sensing road scenes advanced significantly, driven by autonomous driving technology. However, motion blur from camera or subject movements hampers segmentation performance. To address this issue, we propose a knowledge distillation-based semantic segmentation network (KDS-Net) that is robust to motion blur, eliminating the need for image restoration networks. KDS-Net leverages innovative knowledge distillation techniques and edge-enhanced segmentation loss to refine edge regions and improve segmentation precision across various receptive fields. To enhance the interpretability of segmentation quality under motion blur, we incorporate fractal dimension estimation to quantify the geometric complexity of class-specific regions, allowing for a structural assessment of predictions generated by the proposed knowledge distillation framework for autonomous driving. Experiments on well-known motion-blurred remote sensing road scene datasets (CamVid and KITTI) demonstrate mean IoU scores of 72.42% and 59.29%, respectively, surpassing state-of-the-art methods. Additionally, the lightweight KDS-Net (21.44 M parameters) enables real-time edge computing, mitigating data privacy concerns and communication overheads in internet of vehicles scenarios. 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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17 pages, 1795 KiB

Open AccessArticle

A Double-Parameter Regularization Scheme for the Backward Diffusion Problem with a Time-Fractional Derivative

by Qun Chen and Zewen Wang

Fractal Fract. 2025, 9(7), 459; https://doi.org/10.3390/fractalfract9070459 - 14 Jul 2025

Viewed by 176

Abstract

In this paper, we investigate the regularization of the backward problem for a diffusion process with a time-fractional derivative. We propose a novel double-parameter regularization scheme that integrates the quasi-reversibility method for the governing equation with the quasi-boundary method. Theoretical analysis establishes the [...] Read more.

In this paper, we investigate the regularization of the backward problem for a diffusion process with a time-fractional derivative. We propose a novel double-parameter regularization scheme that integrates the quasi-reversibility method for the governing equation with the quasi-boundary method. Theoretical analysis establishes the regularity and the convergence analysis of the regularized solution, along with a convergence rate under an a-priori regularization parameter choice rule in the general-dimensional case. Finally, numerical experiments validate the effectiveness of the proposed scheme. Full article

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43 pages, 511 KiB

Open AccessArticle

Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs

by Waqar Afzal, Mujahid Abbas, Jorge E. Macías-Díaz, Armando Gallegos and Yahya Almalki

Fractal Fract. 2025, 9(7), 458; https://doi.org/10.3390/fractalfract9070458 - 14 Jul 2025

Viewed by 173

Abstract

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces

L^{p (\cdot)}

. To the best of our knowledge, the boundedness of such operators has not [...] Read more.

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces

L^{p (\cdot)}

. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function

p (\cdot)

. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter

λ = 0

. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

(This article belongs to the Special Issue Advances in Fractional Integral Inequalities: Theory and Applications)

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19 pages, 4953 KiB

Open AccessArticle

Modeling Fractals in the Setting of Graphical Fuzzy Cone Metric Spaces

by Ilyas Khan, Fahim Ud Din, Luminiţa-Ioana Cotîrlă and Daniel Breaz

Fractal Fract. 2025, 9(7), 457; https://doi.org/10.3390/fractalfract9070457 - 13 Jul 2025

Viewed by 194

Abstract

This study introduces a new metric structure called the Graphical Fuzzy Cone Metric Space (GFCMS) and explores its essential properties in detail. We examine its topological aspects in detail and introduce the notion of Hausdorff distance within this setting—an advancement not previously explored [...] Read more.

This study introduces a new metric structure called the Graphical Fuzzy Cone Metric Space (GFCMS) and explores its essential properties in detail. We examine its topological aspects in detail and introduce the notion of Hausdorff distance within this setting—an advancement not previously explored in any graphical structure. Furthermore, a fixed-point result is proven within the framework of GFCMS, accompanied by examples that demonstrate the applicability of the theoretical results. As a significant application, we construct fractals within GFCMS, marking the first instance of fractal generation in a graphical structure. This pioneering work opens new avenues for research in graph theory, fuzzy metric spaces, topology, and fractal geometry, with promising implications for diverse scientific and computational domains. Full article

(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)

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26 pages, 1906 KiB

Open AccessArticle

The Thermoelastic Component of the Photoacoustic Response in a 3D-Printed Polyamide Coated with Pigment Dye: A Two-Layer Model Incorporating Fractional Heat Conduction Theories

by Marica N. Popovic, Slobodanka P. Galovic, Ervin K. Lenzi and Aloisi Somer

Fractal Fract. 2025, 9(7), 456; https://doi.org/10.3390/fractalfract9070456 - 12 Jul 2025

Viewed by 173

Abstract

This study presents a theoretical model for the thermoelastic response in transmission-mode photoacoustic systems that feature a two-layer structure. The model incorporates volumetric optical absorption in both layers and is based on classical heat conduction theory, hyperbolic generalized heat conduction theory, and fractional [...] Read more.

This study presents a theoretical model for the thermoelastic response in transmission-mode photoacoustic systems that feature a two-layer structure. The model incorporates volumetric optical absorption in both layers and is based on classical heat conduction theory, hyperbolic generalized heat conduction theory, and fractional heat conduction models including inertial memory in Generalizations of the Cattaneo Equation (GCEI, GCEII, and GCEIII). To validate the model, comparisons were made with the existing literature models. Using the proposed model, the thermoelastic photoacoustic response of a two-layer system composed of a 3D-printed porous polyamide (PA12) substrate coated with a thin, highly absorptive protective dye layer is analyzed. We obtain that the thickness and thermal conduction in properties of the coating are very important in influencing the thermoelastic component and should not be overlooked. Furthermore, the thermoelastic component is affected by the selected fractional model—whether it is subdiffusion or superdiffusion—along with the value of the order of the fractional derivative, as well as the optical absorption coefficient of the layer being investigated. Additionally, it is concluded that the phase has a greater impact than the amplitude when selecting the appropriate theoretical heat conduction model. Full article

(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)

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22 pages, 2326 KiB

Open AccessArticle

A Structure-Preserving Finite Difference Scheme for the Nonlinear Space Fractional Sine-Gordon Equation with Damping Based on the T-SAV Approach

by Penglin Jiang and Yu Li

Fractal Fract. 2025, 9(7), 455; https://doi.org/10.3390/fractalfract9070455 - 11 Jul 2025

Viewed by 264

Abstract

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly [...] Read more.

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter

ε

on error dynamics is investigated. Full article

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

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20 pages, 4263 KiB

Open AccessArticle

Quantitative Fractal Analysis of Fracture Mechanics and Damage Evolution in Recycled Aggregate Concrete Beams: Investigation of Dosage-Dependent Mechanical Response Under Incremental Load

by Xiu-Cheng Zhang and Xue-Fei Chen

Fractal Fract. 2025, 9(7), 454; https://doi.org/10.3390/fractalfract9070454 - 11 Jul 2025

Viewed by 210

Abstract

This study investigated the fracture behavior of concrete beams with recycled coarse aggregate (RCA) and recycled fine aggregate (RFA) using the box-counting method to measure crack fractal dimensions under load. Beams with RCA showed higher fractal dimensions due to RCA’s lower elastic moduli [...] Read more.

This study investigated the fracture behavior of concrete beams with recycled coarse aggregate (RCA) and recycled fine aggregate (RFA) using the box-counting method to measure crack fractal dimensions under load. Beams with RCA showed higher fractal dimensions due to RCA’s lower elastic moduli and compressive strengths, resulting in reduced deformation resistance, ductility, and more late-stage crack propagation. A direct proportional relationship existed between RCA/RFA replacement ratios and crack fractal dimensions. Second-order and third-order polynomial trend surface-fitting techniques were applied to examine the complex relationships among RFA/RCA dosage, applied load, and crack fractal dimension. The results indicated that the RFA dosage had a negative quadratic influence, while load had a positive linear effect, with dosage impact increasing with load. A second-order functional relationship was found between mid-span deflection and crack fractal dimension, reflecting nonlinear behavior consistent with concrete mechanics. This study enhances the understanding of recycled aggregate concrete beam fracture behavior, with the crack fractal dimension serving as a valuable quantitative indicator for damage state and crack complexity assessment. These findings are crucial for engineering design and application, enabling better evaluation of structural performance under various conditions. Full article

(This article belongs to the Section Engineering)

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23 pages, 1107 KiB

Open AccessArticle

Mathematical and Physical Analysis of the Fractional Dynamical Model

by Mohammed Ahmed Alomair and Haitham Qawaqneh

Fractal Fract. 2025, 9(7), 453; https://doi.org/10.3390/fractalfract9070453 - 11 Jul 2025

Viewed by 164

Abstract

This paper consists of various kinds of wave solitons to the mathematical model known as the truncated M-fractional FitzHugh–Nagumo model. This model explains the transmission of the electromechanical pulses in nerves. Through the application of the modified extended tanh function technique and the [...] Read more.

This paper consists of various kinds of wave solitons to the mathematical model known as the truncated M-fractional FitzHugh–Nagumo model. This model explains the transmission of the electromechanical pulses in nerves. Through the application of the modified extended tanh function technique and the modified

(G^{'} / G^{2})

-expansion technique, we are able to achieve the series of exact solitons. The results differ from the current solutions because of the fractional derivative. These solutions could be helpful in the telecommunication and bioscience domains. Contour plots, in two and three dimensions, are used to describe the results. Stability analysis is used to check the stability of the obtained solutions. Moreover, the stationary solutions of the focusing equation are studied through modulation instability. Future research on the focused model in question will benefit from the findings. The techniques used are simple and effective. Full article

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

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23 pages, 1065 KiB

Open AccessArticle

Modeling and Neural Network Approximation of Asymptotic Behavior for Delta Fractional Difference Equations with Mittag-Leffler Kernels

by Pshtiwan Othman Mohammed, Muteb R. Alharthi, Majeed Ahmad Yousif, Alina Alb Lupas and Shrooq Mohammed Azzo

Fractal Fract. 2025, 9(7), 452; https://doi.org/10.3390/fractalfract9070452 - 9 Jul 2025

Viewed by 252

Abstract

The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order

0.5

subject to a given initial condition. We establish [...] Read more.

The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order

0.5

subject to a given initial condition. We establish the existence of a unique solution expressed via a Mittag-Leffler-type function. The delta-asymptotic behavior of the solution is examined, and its convergence properties are rigorously analyzed. Numerical experiments are conducted to illustrate the qualitative features of the solution. Furthermore, a neural network-based approximation is employed to validate and compare with the analytical results, confirming the accuracy, stability, and sensitivity of the proposed method. Full article

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

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14 pages, 11562 KiB

Open AccessArticle

An Eighth-Order Numerical Method for Spatial Variable-Coefficient Time-Fractional Convection–Diffusion–Reaction Equations

by Yuelong Feng, Xindong Zhang and Leilei Wei

Fractal Fract. 2025, 9(7), 451; https://doi.org/10.3390/fractalfract9070451 - 9 Jul 2025

Viewed by 203

Abstract

In this paper, we propose a high-order compact difference scheme for a class of time-fractional convection–diffusion–reaction equations (CDREs) with variable coefficients. Using the Lagrange polynomial interpolation formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish [...] Read more.

In this paper, we propose a high-order compact difference scheme for a class of time-fractional convection–diffusion–reaction equations (CDREs) with variable coefficients. Using the Lagrange polynomial interpolation formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish an unconditionally stable compact difference method. The stability and convergence properties of the method are rigorously analyzed using the Fourier method. The convergence order of our discrete scheme is

O (τ^{4 - α} + h^{8})

, where

τ

and h represent the time step size and space step size, respectively. This work contributes to providing a better understanding of the dependability of the method by thoroughly examining convergence and conducting an error analysis. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research. Full article

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

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17 pages, 3679 KiB

Open AccessArticle

Binary-Classification Physical Fractal Models in Different Coal Structures

by Guangui Zou, Yuyan Che, Tailang Zhao, Yajun Yin, Suping Peng and Jiasheng She

Fractal Fract. 2025, 9(7), 450; https://doi.org/10.3390/fractalfract9070450 - 8 Jul 2025

Viewed by 192

Abstract

Existing theoretical models of wave-induced flow face challenges in coal applications due to the scarcity of experimental data in the seismic-frequency band. Additionally, traditional viscoelastic combination models exhibit inherent limitations in accurately capturing the attenuation characteristics of rocks. To overcome these constraints, we [...] Read more.

Existing theoretical models of wave-induced flow face challenges in coal applications due to the scarcity of experimental data in the seismic-frequency band. Additionally, traditional viscoelastic combination models exhibit inherent limitations in accurately capturing the attenuation characteristics of rocks. To overcome these constraints, we propose a novel binary classification physical fractal model, which provides a more robust framework for analyzing wave dispersion and attenuation in complex coal. The fractal cell was regarded as an element to re-establish the viscoelastic constitutive equation. In the new constitutive equation, three key fractional orders,

α

,

β

, and

γ

, emerged. Among them,

α

mainly affects the attenuation at low frequencies;

β

controls the attenuation in the middle-frequency band; and

γ

dominates the attenuation in the tail-frequency band. After fitting with the measured attenuation data of partially saturated coal samples under variable confining pressures and variable temperature conditions, the results show that this model can effectively represent the attenuation characteristics of elastic wave propagation in coals with different coal structures. It provides a new theoretical model and analysis ideas for the study of elastic wave attenuation in tectonic coals and is of great significance for an in-depth understanding of the physical properties of coals and related geophysical prospecting. Full article

(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)

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25 pages, 3106 KiB

Open AccessArticle

Multifractal-Aware Convolutional Attention Synergistic Network for Carbon Market Price Forecasting

by Liran Wei, Mingzhu Tang, Na Li, Jingwen Deng, Xinpeng Zhou and Haijun Hu

Fractal Fract. 2025, 9(7), 449; https://doi.org/10.3390/fractalfract9070449 - 7 Jul 2025

Viewed by 315

Abstract

Accurate carbon market price prediction is crucial for promoting a low-carbon economy and sustainable engineering. Traditional models often face challenges in effectively capturing the multifractality inherent in carbon market prices. Inspired by the self-similarity and scale invariance inherent in fractal structures, this study [...] Read more.

Accurate carbon market price prediction is crucial for promoting a low-carbon economy and sustainable engineering. Traditional models often face challenges in effectively capturing the multifractality inherent in carbon market prices. Inspired by the self-similarity and scale invariance inherent in fractal structures, this study proposes a novel multifractal-aware model, MF-Transformer-DEC, for carbon market price prediction. The multi-scale convolution (MSC) module employs multi-layer dilated convolutions constrained by shared convolution kernel weights to construct a scale-invariant convolutional network. By projecting and reconstructing time series data within a multi-scale fractal space, MSC enhances the model’s ability to adapt to complex nonlinear fluctuations while significantly suppressing noise interference. The fractal attention (FA) module calculates similarity matrices within a multi-scale feature space through multi-head attention, adaptively integrating multifractal market dynamics and implicit associations. The dynamic error correction (DEC) module models error commonality through variational autoencoder (VAE), and uncertainty-guided dynamic weighting achieves robust error correction. The proposed model achieved an average R² of 0.9777 and 0.9942 for 7-step ahead predictions on the Shanghai and Guangdong carbon price datasets, respectively. This study pioneers the interdisciplinary integration of fractal theory and artificial intelligence methods for complex engineering analysis, enhancing the accuracy of carbon market price prediction. The proposed technical pathway of “multi-scale deconstruction and similarity mining” offers a valuable reference for AI-driven fractal modeling. Full article

(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)

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26 pages, 18598 KiB

Open AccessArticle

Fractal Feature of Manufactured Sand Ultra-High-Performance Concrete (UHPC) Based on MIP

by Xinlin Wang, Tinghong Pan, Yang Yang, Rongqing Qi, Dian Guan, Kaihe Dong, Run-Sheng Lin and Rongxin Guo

Fractal Fract. 2025, 9(7), 448; https://doi.org/10.3390/fractalfract9070448 - 5 Jul 2025

Viewed by 398

Abstract

To alleviate environmental pressures, manufactured sand (MS) are increasingly being used in the production of ultra-high-performance concrete (UHPC) due to their consistent supply and environmental benefits. However, manufactured sand properties are critically influenced by processing and production techniques, resulting in substantial variations in [...] Read more.

To alleviate environmental pressures, manufactured sand (MS) are increasingly being used in the production of ultra-high-performance concrete (UHPC) due to their consistent supply and environmental benefits. However, manufactured sand properties are critically influenced by processing and production techniques, resulting in substantial variations in fundamental characteristics that directly impact UHPC matrix pore structure and ultimately compromise performance. Traditional testing methods inadequately characterize UHPC’s pore structure, necessitating multifractal theory implementation to enhance pore structural interpretation capabilities. In this study, UHPC specimens were fabricated with five types of MS exhibiting distinct properties and at varying water to binder (w/b) ratios. The flowability, mechanical strength, and pore structure of the specimens were systematically evaluated. Additionally, multifractal analysis was conducted on each specimen group using mercury intrusion porosimetry (MIP) data to characterize pore complexity. SM-type sands have a more uniform distribution of pores of different scales, better pore structure and matrix homogeneity due to their finer particles, moderate stone powder content, and better cleanliness. Both excessively high and low stone powder content, as well as low cleanliness, will lead to pore aggregation and poor closure, degrading the pore structure. Full article

(This article belongs to the Special Issue Fractal and Fractional in Construction Materials)

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Fractal Fract., Volume 9, Issue 7 (July 2025) – 81 articles

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