Fractal and Fractional

Studying mining fracture development is vital for geotechnical and mining engineering and geological disaster prevention. This research assesses crack effects on rock mass stress equilibrium during coal mining, potentially causing geological disasters such as land subsidence and landslides. Using fractal geometry theory, the present study investigates the development of horizontal and vertical mining cracks, revealing their propagation patterns. The fractal dimension generally increases as the propulsion distance increases; however, fluctuations vary from 250 to 287.5 m, forming a wavering line chart. The proportion of mining fracture area relative to mining space area increases with greater propulsion distance, indicating expanded upward mining space due to separation layers. The horizontal distribution of mining cracks persists, while the vertical distribution decreases, suggesting ground subsidence results from upward transmission. The fastest increase in fractal dimension occurs at 87.5–100 m. At 250 m, it peaks at 1.4136, indicating complex crack structures. During propulsion, the fractal dimension decreases due to upward mining space expansion through overlying rock layer collapse, forming new cracks. The proportion of mining crack area to mining space area increases gradually throughout the mining process. The present study presents a simulation model for crack identification, noting limitations in identifying tiny cracks.

Jinsui Wu

Dongyu Xie

Sihai Yi

Shangxian Yin

Dezhi Hu

Yuanyuan Li

Yun Wang

Fractal Study of the Development Law of Mining Cracks

矿井开采裂隙发展规律的分形研究

Wind energy is a clean, sustainable, and renewable source. It is receiving a large amount of attention from governments and energy companies worldwide as it plays a significant role as an alternative source of energy in reducing carbon emissions. However, due to long-term operation in reduced and difficult weather conditions, wind turbine blades are always seriously damaged. Hence, damage detection in blade structure is essential to evaluate its operational condition and ensure its structural integrity and safety. We aim to use fractal, entropy, and chaos concepts as descriptors for the diagnosis of wind turbine blade condition. They are, respectively, estimated by the correlation dimension, approximate entropy, and the Lyapunov exponent. Formal statistical tests are performed to check how they are different across wind turbine blade conditions. The experimental results follow. First, the correlation dimension is not able to distinguish between all conditions of wind turbine blades. Second, approximate entropy is suitable to distinguish between healthy and erosion conditions and between healthy and mass imbalance conditions. Third, chaos is not a discriminative feature to distinguish between wind turbine blade conditions. Fourth, wind turbine blades with either erosion or mass imbalance exhibit less irregularity in their respective signals than healthy wind turbine blades.

Salim Lahmiri

Wind Turbine Blade Fault Diagnosis: Approximate Entropy as a Tool to Detect Erosion and Mass Imbalance

风力涡轮机叶片故障诊断：近似熵作为检测侵蚀和质量不平衡的工具

This paper proposes an iterative algorithm for the search for common fixed points of two mappings. The properties of approximation and convergence of the method are analyzed in the context of Banach spaces. In particular, this article provides sufficient conditions for the strong convergence of the sequence generated by the iterative scheme to a common fixed point of two operators. The method is illustrated with some examples of application. The procedure is used to approach a common solution of two Fredholm integral equations of the second kind. In the second part of the article, the existence of a fractal function coming from two different Read–Bajraktarević operators is proved. Afterwards, a study of the approximation of fixed points of a fractal convolution of operators is performed, in the framework of Lebesgue or Bochner spaces.

María A. Navascués

An Iterative Method for the Approximation of Common Fixed Points of Two Mappings: Application to Fractal Functions

一种求解两个映射公共不动点近似的迭代方法：应用于分形函数

Fractal dimension is a characteristic parameter used to measure the complexity and irregularity of geometric shapes and patterns. It is applied in architecture to explore complexity and irregularity and to assess the aesthetic preferences in architectural design. Office building facade design pattern, as an observation unit, has a positive connection with the aesthetic value. This study aims to evaluate facade design styles in terms of two aesthetic qualities, visual complexity and visual diversity, via applying fractal dimension to three design styles of office building facades in Erbil City. The study uses a combination of qualitative and quantitative evaluations to achieve this goal. It employs box-counting analysis through the ImageJ plugin to FracLac and the mathematical perplexity equation to evaluate visual complexity and diversity. The results indicate that the neo-classical office facade style, with a visual complexity value of 1.7008 and visual diversity of 21.27, presents an elevated level of aesthetics similar to the saccadic pattern facade. This study concluded that a neo-classical architectural style for office building facades is the most aesthetically preferable. Modern facade design is considered a secondary architectural style aimed at achieving aesthetic value. Ultimately, the high-tech style is the least attractive facade style. This study contributes to avoiding designs of unattractive office building facades due to a lack of architectural design vocabulary while avoiding overly complex designs that prove visually upsetting for viewers.

Lana Abubakr Ali

Faris Ali Mustafa

Fractal Dimensional Analysis of Building Facades: The Case of Office Buildings in Erbil City

建筑立面分形维数分析：以埃尔比勒市办公建筑为例

Networks are designed to communicate, operate, and allocate tasks to respective commodities. Operating supercomputers became challenging, which was handled by the network design commonly known as hypercube, denoted by Qn. In a recent study, the hypercube networks were insufficient to hold supercomputers’ parallel processors. Thus, variants of hypercubes were discovered to produce an alternative to the hypercube. A new variant of the hypercube, the fractal cubic network, can be used as the best alternative in the case of hypercubes. Our research investigates that the fractal cubic network is a rooted product of two graphs. We try to determine its domination and resolving domination parameters, which could be applied to resource location and broadcasting-related problems.

Savari Prabhu

Arumugam Krishnan Arulmozhi

M. Arulperumjothi

Certain Domination Parameters and Their Resolving Versions of Fractal Cubic Networks

分形立方网络的某些支配参数及其解析版本

This study focuses on optical twin-core couplers, which facilitate light transmission between two closely aligned optical fibers. These couplers operate based on the principle of coupling, allowing signals in one core to interact with those in the other. The Kerr effect, which describes how a material’s refractive index changes in response to the intensity of light, induces the nonlinear behavior essential for generating solitons—self-sustaining wave packets that preserve their shape and speed. In our research, we employ fractional derivatives to investigate how fractional-order variations influence wave propagation and soliton dynamics. By utilizing the modified extended mapping method (MEMM), we derive solitary wave solutions for the equations governing the behavior of optical twin-core couplers under Kerr nonlinearity. This methodology produces novel fractional traveling wave solutions, including dark, bright, singular, and combined bright–dark solitons, as well as hyperbolic, Jacobi elliptic function (JEF), periodic, and singular periodic solutions. To enhance understanding, we present physical interpretations through contour plots and include both 2D and 3D graphical representations of the results.

Noorah Mshary

Hamdy M. Ahmed

Wafaa B. Rabie

Fractional Solitons in Optical Twin-Core Couplers with Kerr Law Nonlinearity and Local M-Derivative Using Modified Extended Mapping Method

分数阶孤子在具有Kerr律非线性和局部M导数的改进扩展映射方法中的光学双芯耦合器

In this paper, we study a system of nonlinear tempered fractional differential equations with multi-point coupled boundary conditions. By applying the properties of Green’s function and the operator and combining the method of matrix analysis, we obtain the corresponding Lyapunov inequalities under two Banach spaces. And, we have compared two Lyapunov inequalities under certain conditions. An example is given to verify our results.

Hailong Ma

Hongyu Li

Lyapunov Inequalities for Systems of Tempered Fractional Differential Equations with Multi-Point Coupled Boundary Conditions via a Fix Point Approach

通过定点方法研究多点耦合边界条件下缓变分数阶微分方程系统的Lyapunov不等式

This paper deals with the exact wave results of the (1+1)-dimensional nonlinear compound Korteweg–De Vries and Burgers (KdVB) equation with a truncated M-fractional derivative. This model represents the generalization of Korteweg–De Vries-modified Korteweg–De Vries and Burgers equations. We obtained periodic, combo singular, dark–bright, and other wave results with the use of the extended sinh-Gordon equation expansion (EShGEE) and modified (G′/G2)-expansion techniques. The use of the effective fractional derivative makes our results much better than the existing results. The obtained solutions are useful as well as applicable in various fields, including mathematical physics, plasma physics, ocean engineering, optics, etc. The obtained solutions are demonstrated by 2D, 3D, and contour plots. The achieved results will be fruitful for future research on this equation. Stability analysis is used to check that the results are precise as well as exact. Modulation instability (MI) analysis is performed to find stable steady-state solutions of the abovementioned model. In the end, it is concluded that the methods used are easy and reliable.

Abdulrahman Alomair

Abdulaziz Saud Al Naim

Mustafa Bayram

Qualitative Analysis and Novel Exact Soliton Solutions to the Compound Korteweg–De Vries–Burgers Equation

复合Korteweg-De Vries-Burgers方程的定性分析与新型精确孤子解

The novelty of this work is the development of a new fractional boundary element model based on the Caputo derivative to investigate anomalous thermal stress effects on cement-based materials. To obtain the BEM integral equations for the proposed formulation, we employ the weighted residuals technique, with the anisotropic fundamental solution serving as the weighting function in the anomalous heat governing equation. The Caputo fractional derivative was employed as an integrand for the domain integral of the proposed formulation. The time step selection is less dependent on the time derivative order. This allows the approach to overcome the non-locality of the fractional operators. The key benefit provided by the suggested formulation is the ability to analyze situations with tiny values of the fractional time derivative. The current BEM methodology proves that it is a useful tool for solving fractional calculus problems.

Mohamed Abdelsabour Fahmy

Roqia Abdullah A. Jeli

A New Fractional Boundary Element Model for Anomalous Thermal Stress Effects on Cement-Based Materials

一种新的分数阶边界元模型用于水泥基材料异常热应力效应

This study demonstrates that the Grünwald–Letnikov fractional proportional–integral–derivative (GPID) controller outperforms traditional PID controllers in adaptive cruise control systems, while conventional PID controllers struggle with nonlinearities, dynamic uncertainties, and stability, the GPID enhances robustness and provides more precise control across various driving conditions. Simulation results show that the GPID improves the accuracy, reducing errors better than the PID controller. Additionally, the GPID maintains a more consistent speed and reaches the target speed faster, demonstrating superior speed control. The GPID’s performance across different fractional orders highlights its adaptability to changing road conditions, which is crucial for ensuring safety and comfort. By leveraging fractional calculus, the GPID also improves acceleration and deceleration profiles. These findings emphasize the GPID’s potential to revolutionize adaptive cruise control, significantly enhancing driving performance and comfort. Numerical results obtained in α=0.99 from the GPID controller have shown better accuracy and speed consistency, adapting to road conditions for improved safety and comfort. The GPID also demonstrated faster stabilization of speed at 60 km/h with smaller errors and reduced the error to 0.59 km/h at 50 s compared to 0.78 km/h for the PID.

Diaa Eldin Elgezouli

Hassan Eltayeb

Mohamed A. Abdoon

Novel GPID: Grünwald–Letnikov Fractional PID for Enhanced Adaptive Cruise Control

新型GPID：用于增强自适应巡航控制的Grünwald-Letnikov分数阶PID

Abbreviation	Fractal Fract
Journal Impact	3.48
Quartiles(Global)	MATHEMATICS, INTERDISCIPLINARY APPLICATIONS(Q1)

ISSN	2504-3110
h-index	33

Publication Information	Publisher: Multidisciplinary Digital Publishing Institute (MDPI)，Publishing cycle: ，Journal Type: journal，Open Access Journals: Yes
Basic data	Year of publication: 2017，Proportion of original research papers: 99.19%，Self Citation Rate:19.40%， Gold OA Rate: 99.74%
Journal Fees	$Details
Average review cycle	网友分享经验：11 Weeks

Fractal and Fractional

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