Fractal and Fractional

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 (registering DOI) - 13 May 2025

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)

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

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

Viewed by 63

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 60

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 120

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 107

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 50

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

Viewed by 158

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 152

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 102

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 115

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

Viewed by 138

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 186

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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18 pages, 1218 KiB

Open AccessArticle

Modification to an Auxiliary Function Method for Solving Space-Fractional Stochastic Regularized Long-Wave Equation

by Muneerah Al Nuwairan and Adel Elmandouh

Fractal Fract. 2025, 9(5), 298; https://doi.org/10.3390/fractalfract9050298 - 4 May 2025

Viewed by 140

Abstract

This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function [...] Read more.

This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function method with the complete discriminant polynomial system. This method proves to be effective in discovering precise solutions for stochastic fractional partial differential equations (SFPDEs), including special cases. Applying this method to the SFRLWE yields new exact solutions, offering fresh insights. We investigated how noise affects stochastic solutions and discovered that more intense noise can result in flatter surfaces. We note that multiplicative noise can stabilize the solution, and we show how fractional derivatives influence the dynamics of noise. We found that the noise strength and fractional derivative affect the width, amplitude, and smoothness of the obtained solutions. Additionally, we conclude that multiplicative noise impacts and stabilizes the behavior of SFRLWE solutions. Full article

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19 pages, 440 KiB

Open AccessArticle

Finite-Time Synchronization and Practical Synchronization for Caputo Fractional-Order Fuzzy Cellular Neural Networks with Transmission Delays and Uncertainties via Information Feedback

by Hongguang Fan, Hui Wen, Kaibo Shi and Anran Zhou

Fractal Fract. 2025, 9(5), 297; https://doi.org/10.3390/fractalfract9050297 - 2 May 2025

Viewed by 271

Abstract

This article considers a class of Caputo fractional-order fuzzy cellular neural networks (CFOFCNNs) with transmission delays and uncertain perturbations. In particular, nonlinear activations and fuzzy operators AND and OR are investigated in the drive-response neural networks (NNs). To achieve practical finite-time (PFT) synchronization [...] Read more.

This article considers a class of Caputo fractional-order fuzzy cellular neural networks (CFOFCNNs) with transmission delays and uncertain perturbations. In particular, nonlinear activations and fuzzy operators AND and OR are investigated in the drive-response neural networks (NNs). To achieve practical finite-time (PFT) synchronization and finite-time (FT) synchronization of the studied systems, we design new nonlinear controllers including four feedback terms in this paper, and each carries a different role in the control process. Integrating different comparison principles and nonlinear feedback schemes, straightforward synchronization criteria of the CFOFCNNs are derived. Unlike existing works, a significant finding is that adjusting the feedback coefficients and parameters can enable synchronization switching. Namely, changing one of the feedback terms from positive to negative can cause PFT synchronization to switch to FT synchronization via adjusted control parameters, making our control methods applicable to different scenarios. The settling time depends explicitly on feedback coefficients, initial conditions, and fractional order. Full article

(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)

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31 pages, 476 KiB

Open AccessArticle

Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients

by Zhaoqiang Yang and Chenglong Xu

Fractal Fract. 2025, 9(5), 296; https://doi.org/10.3390/fractalfract9050296 - 1 May 2025

Viewed by 214

Abstract

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using [...] Read more.

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the

L^{2}

sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the

L^{2}

sense. Full article

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

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14 pages, 336 KiB

Open AccessArticle

The Existence and Stability of Integral Fractional Differential Equations

by Rahman Ullah Khan and Ioan-Lucian Popa

Fractal Fract. 2025, 9(5), 295; https://doi.org/10.3390/fractalfract9050295 - 1 May 2025

Viewed by 292

Abstract

The main goal of this research is to study integro-fractional differential equations and simulate their dynamic behavior using ABC-fractional derivatives. We investigate the Hyers–Ulam stability of the proposed system and further expand the prerequisites for the existence and uniqueness of the solutions. The [...] Read more.

The main goal of this research is to study integro-fractional differential equations and simulate their dynamic behavior using ABC-fractional derivatives. We investigate the Hyers–Ulam stability of the proposed system and further expand the prerequisites for the existence and uniqueness of the solutions. The Schauder fixed-point theorem and the Banach contraction principle are employed to obtain the results. Finally, we present an example to demonstrate the practical application of our theoretical conclusions. Full article

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

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19 pages, 10310 KiB

Open AccessArticle

Nonlinear Dynamic Characteristics of Single-Point Suspension Isolation System of Maglev Vehicle Based on Fractional-Order Nonlinear Nishimura Model

by Minghe Qu, Lianchun Wang, Shijie Gu, Peichang Yu, Qicai Li, Danfeng Zhou and Jie Li

Fractal Fract. 2025, 9(5), 294; https://doi.org/10.3390/fractalfract9050294 - 1 May 2025

Viewed by 225

Abstract

Base excitation sources significantly impact vehicle-body vibrations in maglev systems, with the dynamic performance of the suspension system playing a crucial role in mitigating these effects. The second-series suspension system of a maglev vehicle typically employs an air spring, which has a great [...] Read more.

Base excitation sources significantly impact vehicle-body vibrations in maglev systems, with the dynamic performance of the suspension system playing a crucial role in mitigating these effects. The second-series suspension system of a maglev vehicle typically employs an air spring, which has a great impact on the stability of maglev vehicle operation. Considering that the suspension system has certain dynamic characteristics under the foundation excitation, the present study proposes the fractional-order nonlinear Nishimura model to describe the memory-restoring force characteristics of the air spring. The fractional-order derivative term is made equivalent to a term in the form of trigonometric function, the steady-state response of the system is solved by the harmonic balance method, and the results are compared with a variety of other methods. The influence of the foundation excitation source on the dynamic behavior of the vibration isolation system is discussed significantly. The variation law of the jump phenomenon and the diversity of periodic motion of the multi-value amplitude curve are summarized. The numerical simulation also revealed the presence of multi-periodic motion in the system when variations occurred in the gap of the suspension system. Combined with the cell mapping algorithm, the distribution law of different attractors on the attraction domain of periodic motion is discussed, and the rule of the transition of periodic motion stability with different fundamental excitation amplitudes is summarized with the Lyapunov exponent. Full article

(This article belongs to the Special Issue New Advances and Applications of Fractional Oscillate System)

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21 pages, 7083 KiB

Open AccessArticle

Pore Structure Evolution Characteristics and Damage Mechanism of Sandstone Subjected to Freeze–Thaw Cycle Treatment: Insights from Low-Field Nuclear Magnetic Resonance Testing and Fractal Theory

by Xin Xiong, Feng Gao, Jielin Li, Keping Zhou and Chengye Yang

Fractal Fract. 2025, 9(5), 293; https://doi.org/10.3390/fractalfract9050293 - 1 May 2025

Viewed by 276

Abstract

To investigate the pore structure evolution characteristics and damage mechanism of sandstone subjected to treatment with freeze–thaw cycles, quantitative analyses were conducted on the longitudinal wave velocity (LWV) and T₂ spectrum of sandstone before and after 10, 20, 30, and 40 freeze–thaw [...] Read more.

To investigate the pore structure evolution characteristics and damage mechanism of sandstone subjected to treatment with freeze–thaw cycles, quantitative analyses were conducted on the longitudinal wave velocity (LWV) and T₂ spectrum of sandstone before and after 10, 20, 30, and 40 freeze–thaw cycles, using longitudinal wave velocity testing, low-field nuclear magnetic resonance (NMR) testing, and fractal theory. The results show that, with the increase in the number of freeze–thaw cycles, the LWV of sandstone gradually decreases, the amplitude of the saturated T₂ spectrum gradually increases, the amplitude of the centrifugal T₂ spectrum gradually decreases, the total porosity and effective porosity increase, and the residual porosity decreases. After undergoing freeze–thaw cycles, sandstone exhibits obvious fractal characteristics in both the total porosity NMR fractal dimension and the effective porosity NMR fractal dimension, and the growth rates of both decrease exponentially with the increase in the number of freeze–thaw cycles. The magnitude of the fractal dimensions reflects the complexity of the pore structure, with smaller fractal dimensions indicating better pore connectivity. In summary, the damage evolution mechanism of sandstone under freeze–thaw cycles is characterized by the gradual expansion and interconnection of internal closed micro-pores (cracks), along with increased total porosity and effective porosity, leading to enhanced freeze–thaw damage. Full article

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

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26 pages, 3452 KiB

Open AccessArticle

Exploring Multifractal Asymmetric Detrended Cross-Correlation Behavior in Semiconductor Stocks

by Werner Kristjanpoller

Fractal Fract. 2025, 9(5), 292; https://doi.org/10.3390/fractalfract9050292 - 1 May 2025

Viewed by 301

Abstract

This study investigates the multifractal behavior of four leading semiconductor stocks—Intel (INTC), Advanced Micro Devices (AMD), Nvidia (NVDA), and Broadcom (AVGO)—in relation to key financial assets, including the Dow Jones Industrial Average (DJI), the Euro–U.S. Dollar exchange rate (EUR), gold (XAU), crude oil [...] Read more.

This study investigates the multifractal behavior of four leading semiconductor stocks—Intel (INTC), Advanced Micro Devices (AMD), Nvidia (NVDA), and Broadcom (AVGO)—in relation to key financial assets, including the Dow Jones Industrial Average (DJI), the Euro–U.S. Dollar exchange rate (EUR), gold (XAU), crude oil (WTI), and Bitcoin (BTC), using Multifractal Asymmetric Detrended Cross-Correlation Analysis (MF-ADCCA). The analysis is based on daily price return time series from January 2015 to January 2025. Results reveal consistent evidence of multifractality across all asset pairs, with the generalized Hurst exponent exhibiting significant variability, indicative of complex and nonlinear stock price dynamics. Among the semiconductor stocks, NVDA and AVGO exhibit the highest levels of multifractal cross-correlation, particularly with DJI, WTI, and BTC, while AMD consistently shows the lowest, suggesting comparatively more stable behavior. Notably, cross-correlation Hurst exponents with BTC are the highest, reaching approximately 0.54 for NVDA and AMD. Conversely, pairs with EUR display long-term negative correlations, with exponents around 0.46 across all semiconductor stocks. Multifractal spectrum analysis highlights that NVDA and AVGO exhibit broader and more pronounced multifractal characteristics, largely driven by higher fluctuation intensities. Asymmetric cross-correlation analysis reveals that stocks paired with DJI show greater persistence during market downturns, whereas those paired with XAU demonstrate stronger persistence during upward trends. Analysis of multifractality sources using surrogate time series confirms the influence of fat-tailed distributions and temporal linear correlations in most asset pairs, with the exception of WTI, which shows less complex behavior. Overall, the findings underscore the utility of multifractal asymmetric cross-correlation analysis in capturing the intricate dynamics of semiconductor stocks. This approach provides valuable insights for investors and portfolio managers by accounting for the multifaceted and asset-dependent nature of stock behavior under varying market conditions. Full article

(This article belongs to the Special Issue Fractal and Multifractal Analysis in Econometric Models and Empirical Finance)

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17 pages, 792 KiB

Open AccessArticle

The Application of Abstract Algebra in Operational Calculus

by Ruiheng Jiang, Tianyi Zhou and Yajun Yin

Fractal Fract. 2025, 9(5), 291; https://doi.org/10.3390/fractalfract9050291 - 30 Apr 2025

Viewed by 172

Abstract

This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of Mikusiński has been revised. We restate the concept of Mikusiński’s operator field as a convolutional field and [...] Read more.

This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of Mikusiński has been revised. We restate the concept of Mikusiński’s operator field as a convolutional field and define a new concept of operator field with the differential operator p as the core, effectively overcoming the confusion between the concepts of operators and functions, which represents the limitation of traditional theory. In addition, the classical Laplace transform is integrated into our theory in a homomorphic form, revealing the principle that the Laplace transform is compatible with operational calculus theory. Full article

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21 pages, 4797 KiB

Open AccessArticle

Multifractal Characterization of Pore Heterogeneity and Water Distribution in Medium- and High-Rank Coals via Nuclear Magnetic Resonance

by Huan Liu, Shasha Zhang, Yu Qiao, Danfeng Xie and Long Chang

Fractal Fract. 2025, 9(5), 290; https://doi.org/10.3390/fractalfract9050290 - 28 Apr 2025

Viewed by 168

Abstract

Comprehensive assessment of pore structure and multiphase water distribution is critical to the flow and transport process in coalbed methane (CBM) reservoirs. In this study, nuclear magnetic resonance (NMR) and multifractal analysis were integrated to quantify the multiscale heterogeneity of nine medium- and [...] Read more.

Comprehensive assessment of pore structure and multiphase water distribution is critical to the flow and transport process in coalbed methane (CBM) reservoirs. In this study, nuclear magnetic resonance (NMR) and multifractal analysis were integrated to quantify the multiscale heterogeneity of nine medium- and high-rank coals under water-saturated and dry conditions. By applying the box-counting method to transverse relaxation time (T₂) spectra, multifractal parameters were derived to characterize pore heterogeneity and residual water distribution. The influencing factors of pore heterogeneity were also discussed. The results show that pore structures in high-rank coals (HCs) exhibit a broader multifractal spectrum and stronger rightward spectrum than those of medium-rank coals, reflecting micropore-dominated heterogeneity and the complexity induced by aromatization in HCs. The vitrinite content enhances micropore development, increasing the heterogeneity and complexity of pore structure and residual water distribution. Inertinite content shows opposite trends compared to vitrinite content for the effect on pore structure and water distribution. Volatile yield reflects coal metamorphism and thermal maturity, which inversely correlates with pore heterogeneity and complexity. Residual water mainly distributes to adsorption pores and pore throats, shortening T₂ relaxation (bound water effect) and reducing spectral asymmetry. The equivalence of the multifractal dimension and singularity spectrum validates their joint utility in characterizing pore structure. Minerals enhance pore connectivity but suppress complexity, while moisture and ash contents show negligible impacts. These findings provide a theoretical reference for CBM exploration, especially in optimizing fluid transportation and CBM production strategies and identifying CBM sweet spots. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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15 pages, 463 KiB

Open AccessArticle

On Extended Numerical Discretization Technique of Fractional Models with Caputo-Type Derivatives

by Reem Allogmany and S. S. Alzahrani

Fractal Fract. 2025, 9(5), 289; https://doi.org/10.3390/fractalfract9050289 - 28 Apr 2025

Viewed by 147

Abstract

In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found [...] Read more.

In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found that the method produces solutions that are identical to the exact solution and approximate series solutions. The ENDT is its ability to proficiently handle complex systems governed by fractional differential equations while preserving memory and hereditary characteristics. Its simplicity, accuracy, and flexibility render it an effective instrument for replicating real-world phenomena in physics and biology. The ENDT method offers accuracy, stability, and efficiency compared to traditional methods. It effectively handles challenges in complex systems, supports any fractional order, is simple to implement, improves computing efficiency with sophisticated methodologies, and applies it to epidemic predictions and biological simulations. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

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19 pages, 619 KiB

Open AccessArticle

Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems

by Xing Zhang and Mengmeng Li

Fractal Fract. 2025, 9(5), 288; https://doi.org/10.3390/fractalfract9050288 - 28 Apr 2025

Viewed by 178

Abstract

Fractional multi-delay differential systems, as an important mathematical model, can effectively describe viscoelastic materials and non-local delay responses in ecosystem population dynamics. It has a wide and profound application background in interdisciplinary fields such as physics, biomedicine, and intelligent control. Through literature review [...] Read more.

Fractional multi-delay differential systems, as an important mathematical model, can effectively describe viscoelastic materials and non-local delay responses in ecosystem population dynamics. It has a wide and profound application background in interdisciplinary fields such as physics, biomedicine, and intelligent control. Through literature review and classification, it is evident that for the fractional multi-delay differential system, the existence and uniqueness of the solution and Ulam-Hyers stability (UHS), Ulam-Hyers-Rassias stability (UHRS) of the fractional multi-delay differential system are rarely studied by using the multi-delayed perturbation of two parameter Mittag-Leffler typematrix function. In this paper, we first establish the existence and uniqueness of the solution for the Riemann-Liouville fractional multi-delay differential system on finite intervals using the Banach and Schauder fixed point theorems. Second, we demonstrate the existence and uniqueness of the solution for the system on the unbounded intervals in the weighted function space. Furthermore, we investigate UHS and UHRS for the nonlinear fractional multi-delay differential system in unbounded intervals. Finally, numerical examples are provided to validate the key theoretical results. Full article

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23 pages, 2348 KiB

Open AccessArticle

Chaotic Analysis and Wave Photon Dynamics of Fractional Whitham–Broer–Kaup Model with β Derivative

by Muhammad Idrees Afridi, Theodoros E. Karakasidis and Abdullah Alhushaybari

Fractal Fract. 2025, 9(5), 287; https://doi.org/10.3390/fractalfract9050287 - 27 Apr 2025

Viewed by 145

Abstract

This study uses a conformable derivative of order

β

to investigate a fractional Whitham–Broer–Kaup (

FWBK

) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation

EMSSE

approach is applied [...] Read more.

This study uses a conformable derivative of order

β

to investigate a fractional Whitham–Broer–Kaup (

FWBK

) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation

EMSSE

approach is applied to achieve precise analytical solutions, demonstrating its effectiveness in resolving complex wave photons. Bright, solitary, trigonometric, dark, and plane waves are among the various wave dynamics that may be effectively and precisely determined using the

FWBK

model. Furthermore, the study explores the chaotic behaviour of both perturbed and unperturbed systems, revealing illumination on their dynamic characteristics. By demonstrating its validity in examining wave propagation in nonlinear fractional systems, the effectiveness and reliability of the suggested method in fractional modelling are confirmed through thorough investigation. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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18 pages, 552 KiB

Open AccessArticle

Error Analysis of the L1 Scheme on a Modified Graded Mesh for a Caputo–Hadamard Fractional Diffusion Equation

by Dan Liu, Libin Liu, Hongbin Chen and Xiongfa Mai

Fractal Fract. 2025, 9(5), 286; https://doi.org/10.3390/fractalfract9050286 - 27 Apr 2025

Viewed by 131

Abstract

The

L 1

scheme on a modified graded mesh is proposed to solve a Caputo–Hadamard fractional diffusion equation with order

α \in (0, 1)

. Firstly, an improved graded mesh frame is innovatively constructed, and its mathematical properties are verified. [...] Read more.

The

L 1

scheme on a modified graded mesh is proposed to solve a Caputo–Hadamard fractional diffusion equation with order

α \in (0, 1)

. Firstly, an improved graded mesh frame is innovatively constructed, and its mathematical properties are verified. Subsequently, a new truncation error bound for the L1 discretisation format of Caputo–Hadamard fractional-order derivatives is established by means of a Taylor cosine expansion of the integral form, and a second-order central difference method is used to achieve high-precision discretisation of spatial derivatives. Furthermore, a rigorous analysis of stability and convergence under the maximum norm is conducted, with special attention devoted to validating that the

L 1

approximation scheme manifests an optimal convergence order of

2 - α

when deployed on the modified graded mesh. Finally, the theoretical results are substantiated through a series of numerical experiments, which validate their accuracy and applicability. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)

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11 pages, 712 KiB

Open AccessArticle

Qualitative Analysis and Traveling Wave Solutions of a (3 + 1)- Dimensional Generalized Nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt System

by Zhao Li and Ejaz Hussain

Fractal Fract. 2025, 9(5), 285; https://doi.org/10.3390/fractalfract9050285 - 27 Apr 2025

Cited by 1 | Viewed by 182

Abstract

This article investigates the qualitative analysis and traveling wave solutions of a (3 + 1)-dimensional generalized nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt system. This equation is commonly used to simulate nonlinear wave problems in the fields of fluid mechanics, plasma physics, and nonlinear optics, as well as [...] Read more.

This article investigates the qualitative analysis and traveling wave solutions of a (3 + 1)-dimensional generalized nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt system. This equation is commonly used to simulate nonlinear wave problems in the fields of fluid mechanics, plasma physics, and nonlinear optics, as well as to transform nonlinear partial differential equations into nonlinear ordinary differential equations through wave transformations. Based on the analysis of planar dynamical systems, a nonlinear ordinary differential equation is transformed into a two-dimensional dynamical system, and the qualitative behavior of the two-dimensional dynamical system and its periodic disturbance system is studied. A two-dimensional phase portrait, three-dimensional phase portrait, sensitivity analysis diagrams, Poincaré section diagrams, and Lyapunov exponent diagrams are provided to illustrate the dynamic behavior of two-dimensional dynamical systems with disturbances. The traveling wave solution of a Konopelchenko-Dubrovsky-Kaup-Kupershmidt system is studied based on the complete discriminant system method, and its three-dimensional, two-dimensional graphs and contour plots are plotted. These works can provide a deeper understanding of the dynamic behavior of Konopelchenko-Dubrovsky-Kaup-Kupershmidt systems and the propagation process of waves. Full article

(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)

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40 pages, 3324 KiB

Open AccessArticle

Numerical Analysis of a Fractional Cauchy Problem for the Laplace Equation in an Annular Circular Region

by José Julio Conde Mones, Julio Andrés Acevedo Vázquez, Eduardo Hernández Montero, María Monserrat Morín Castillo, Carlos Arturo Hernández Gracidas and José Jacobo Oliveros Oliveros

Fractal Fract. 2025, 9(5), 284; https://doi.org/10.3390/fractalfract9050284 - 27 Apr 2025

Viewed by 259

Abstract

The Cauchy problem for the Laplace equation in an annular bounded region consists of finding a harmonic function from the Dirichlet and Neumann data known on the exterior boundary. This work considers a fractional boundary condition instead of the Dirichlet condition in a [...] Read more.

The Cauchy problem for the Laplace equation in an annular bounded region consists of finding a harmonic function from the Dirichlet and Neumann data known on the exterior boundary. This work considers a fractional boundary condition instead of the Dirichlet condition in a circular annular region. We found the solution to the fractional boundary problem using circular harmonics. Then, the Tikhonov regularization is used to handle the numerical instability of the fractional Cauchy problem. The regularization parameter was chosen using the L-curve method, Morozov’s discrepancy principle, and the Tikhonov criterion. From numerical tests, we found that the series expansion of the solution to the Cauchy problem can be truncated in

N = 20

,

N = 25

, or

N = 30

for smooth functions. For other functions, such as absolute value and the jump function, we have to choose other values of N. Thus, we found a stable method for finding the solution to the problem studied. To illustrate the proposed method, we elaborate on synthetic examples and MATLAB 2021 programs to implement it. The numerical results show the feasibility of the proposed stable algorithm. In almost all cases, the L-curve method gives better results than the Tikhonov Criterion and Morozov’s discrepancy principle. In all cases, the regularization using the L-curve method gives better results than without regularization. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)

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15 pages, 2574 KiB

Open AccessArticle

The Effect of Organic Acid Modification on the Pore Structure and Fractal Features of 1/3 Coking Coal

by Jiafeng Fan and Feng Cai

Fractal Fract. 2025, 9(5), 283; https://doi.org/10.3390/fractalfract9050283 - 26 Apr 2025

Viewed by 130

Abstract

The acidification modification of coal seams is a significant technical measure for transforming coalbed methane reservoirs and enhancing the permeability of coal seams, thereby improving the extractability of coalbed methane. However, the acids currently used in fracturing fluids are predominantly inorganic acids, which [...] Read more.

The acidification modification of coal seams is a significant technical measure for transforming coalbed methane reservoirs and enhancing the permeability of coal seams, thereby improving the extractability of coalbed methane. However, the acids currently used in fracturing fluids are predominantly inorganic acids, which are highly corrosive and can contaminate groundwater reservoirs. In contrast, organic acids are not only significantly less corrosive than inorganic acids but also readily bind with the coal matrix. Some organic acids even exhibit complexing and flocculating effects, thus avoiding groundwater contamination. This study focuses on the 1/3 coking coal from the Guqiao Coal Mine of Huainan Mining Group Co., Ltd., in China. It systematically investigates the fractal characteristics and chemical structure of coal samples before and after pore modification using four organic acids (acetic acid, glycolic acid, oxalic acid, and citric acid) and compares their effects with those of hydrochloric acid solutions at the same concentration. Following treatment with organic acids, the coal samples exhibit an increase in surface fractal dimension, a reduction in spatial fractal dimension, a decline in micropore volume proportion, and a rise in the proportions of transitional and mesopore volumes, and the structure of the hydroxyl group and oxygen-containing functional group decreased. This indicates that treating coal samples with organic acids enhances their pore structure and chemical structure. A comparative analysis reveals that hydrochloric acid is more effective than acetic acid in modifying coal pores, while oxalic acid and citric acid outperform hydrochloric acid, and citric acid shows the best results. The findings provide essential theoretical support for organic acidification modification technology in coalbed methane reservoirs and hydraulic fracturing techniques for coalbed methane extraction. Full article

(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering)

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20 pages, 7934 KiB

Open AccessArticle

Dynamical Behavior Analysis of Generalized Chen–Lee–Liu Equation via the Riemann–Hilbert Approach

by Wenxia Chen, Chaosheng Zhang and Lixin Tian

Fractal Fract. 2025, 9(5), 282; https://doi.org/10.3390/fractalfract9050282 - 26 Apr 2025

Viewed by 152

Abstract

In this paper, we investigate the dynamics of the generalized Chen–Lee–Liu (gCLL) equation utilizing the Riemann–Hilbert method to derive its

N

-soliton solution. We incorporate a self-steepening term and a Kerr nonlinear term to characterize nonlinear light propagation in optical fibers based on [...] Read more.

In this paper, we investigate the dynamics of the generalized Chen–Lee–Liu (gCLL) equation utilizing the Riemann–Hilbert method to derive its

N

-soliton solution. We incorporate a self-steepening term and a Kerr nonlinear term to characterize nonlinear light propagation in optical fibers based on the Chen–Lee–Liu (CLL) equation more accurately. The Riemann–Hilbert problem is addressed through spectral analysis derived from the Lax pair formulation, resulting in an

N

-soliton solution for a reflectance-less system. We also present explicit formulas for solutions involving one and two solitons, thereby providing theoretical support for stable long-distance signal transmission in optical fiber communication. Furthermore, by adjusting parameters and conducting comparative analyses, we generate three-dimensional soliton images that warrant further exploration. The stability of soliton solutions in optical fibers offers novel insights into the intricate propagation behavior of light pulses, and it is crucial for maintaining the integrity of communication signals. 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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Fractal Fract., Volume 9, Issue 5 (May 2025) – 45 articles

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