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 study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation R, proposing a novel definition for a generalized ℷC-type Suzuki R-contraction specific to these spaces. By doing so, we pave the way for a range of relation-theoretical common fixed-point theorems, highlighting the versatility of our approach. To illustrate the practical relevance of our findings, we present a compelling example. Ultimately, this work aims to enrich the existing academic discourse and stimulate further research and practical applications within the field.

Abdurrahman Büyükkaya

Dilek Kesik

Ülkü Yeşil

Mahpeyker Öztürk

New Studies for Dynamic Programming and Fractional Differential Equations in Partial Modular b-Metric Spaces

部分模b-度量空间中动态规划与分数阶微分方程的新研究

For a complex parameter c outside the unit disk and an integer n≥2, we examine the n-ary collinear fractal E(c,n), defined as the attractor of the iterated function system {fk:C⟶C}k=1n, where fk(z):=1+n−2k+c−1z. We investigate some topological features of the connectedness locus Mn defined as the set of those c for which E(c,n) is connected. In particular, we provide a detailed answer to an open question posed by Calegari, Koch, and Walker in 2017. We also extend and refine the technique of the “covering property” by Solomyak and Xu to any n≥2. We use it to show that a nontrivial portion of Mn is regular closed. When n≥21, we enhance this result by showing that in fact, the whole Mn∖R lies within the closure of its interior, thus proving that the generalized Bandt’s conjecture is true.

Bernat Espigule

David Juher

Joan Saldaña

Collinear Fractals and Bandt’s Conjecture

共线分形与班德特猜想

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the Ψ-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. Sufficient conditions for controllability are established using the notion of measure of noncompactness (MNC) and the Mönch fixed-point theorem. An example is given to illustrate the abstract findings.

Abdur Raheem

Fahad M. Alamrani

Javed Akhtar

Adel Alatawi

Esmail Alshaban

Areefa Khatoon

Faizan Ahmad Khan

Study on Controllability for Ψ-Hilfer Fractional Stochastic Differential Equations

关于Ψ-Hilfer分数阶随机微分方程可控性的研究

In this paper, we aim to solve the issue of pattern formation mechanisms in a spatiotemporally discrete activator–inhibitor model that incorporates self- and cross-diffusions. We seek to identify the conditions that lead to the emergence of complex patterns and to elucidate the principles governing the dynamic behaviors that result in these patterns. We first construct a corresponding coupled map lattice (CML) model based on the continuous activator–inhibitor reaction–diffusion system. In the reaction stage, we examine the existence, uniqueness, and stability of the homogeneous stationary state and specify the parametric conditions for realizing these properties. Furthermore, by applying the center manifold theorem, we perform a flip bifurcation analysis and confirm that the model is capable of undergoing flip bifurcation. In the diffusion stage, we focus on the analysis of Turing bifurcation and determine the parameter conditions for the emergence of Turing instability. Through numerical simulations, we validate and explain the results of our theoretical analysis. Our study reveals various Turing instability mechanisms by coupling Turing and flip bifurcations, which include pure-self-diffusion-Turing instability, pure-cross-diffusion-Turing instability, flip-self-diffusion-Turing instability, flip-cross-diffusion-Turing instability, and chaos-self-diffusion-Turing instability mechanisms. Under different mechanisms, we illustrate the corresponding Turing patterns and discover a rich variety of pattern types such as labyrinthine, mosaic, alternating mosaic, colorful mottled grid patterns with winding and twisted bands, and patterns with dense patches and twisted bands nested together. Our research provides a theoretical framework and numerical support for understanding the complex dynamical behaviors and pattern formations in activator–inhibitor models with self- and cross-diffusions.

You Li

Ying Sun

Jingyu Luo

Jiayi Pang

Bingjie Liu

Pattern Formation Mechanisms of Spatiotemporally Discrete Activator–Inhibitor Model with Self- and Cross-Diffusions

时空离散激活因子-抑制因子模型的自扩散与交叉扩散模式形成机制

A high-order variable-step numerical scheme is formulated for the space–time-fractional Cahn–Hilliard equation, employing the variable-step fractional BDF2 formula. The unique solvability and mass conservation at the discretization setting are established. Subject to the constraint of time-step ratios, i.e., 0.4159≤rk≤4.660, a careful analysis based on the discrete gradient structure of the fractional BDF2 formula reveals that the proposed scheme adheres to the energy dissipation law. Remarkably, the modified energy exhibits asymptotic compatibility with that of the classical Cahn–Hilliard equation. Moreover, the modified energy dissipation law of the resulting scheme for the space–time-fractional Cahn–Hilliard equation aligns asymptotically with that of the variable-step BDF2 scheme for its classical counterpart. Finally, a few numerical experiments combined with the adaptive method are presented, which confirm the accuracy and efficiency of the proposed scheme.

Xuan Zhao

Bingqing Hu

Hong Sun

Energy Dissipation Law of the Temporal Variable-Step Fractional BDF2 Scheme for Space–Time-Fractional Cahn–Hilliard Equation

时空分数阶Cahn-Hilliard方程的时间变步长分数阶BDF2格式的能量耗散定律

This paper investigates the dynamical behavior of a discrete fractional-order modified Leslie–Gower model with a Michaelis–Menten-type harvesting mechanism and a Holling-II functional response. We analyze the existence and stability of the nonnegative equilibrium points. For the interior equilibrium points, we study the conditions for period-doubling and Neimark–Sacker bifurcations using the center manifold theorem and bifurcation theory. To control the chaos arising from these bifurcations, two chaos control strategies are proposed. Numerical simulations are performed to validate the theoretical results. The findings provide valuable insights into the sustainable management and conservation of ecological systems.

Yao Shi

Xiaozhen Liu

Zhenyu Wang

Bifurcation Analysis and Chaos Control of a Discrete Fractional-Order Modified Leslie–Gower Model with Nonlinear Harvesting Effects

离散分数阶改进Leslie-Gower模型的分岔分析与混沌控制及非线性捕捞效应

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the φ−Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra R and commutative Banach algebra of the collection of continuous functions in R.

Faten H. Damag

Amin Saif

Adem Kiliçman

φ−Hilfer Fractional Cauchy Problems with Almost Sectorial and Lie Bracket Operators in Banach Algebras

φ−Hilfer 分数阶柯西问题在 Banach 代数中的几乎扇形和李括号算子

Two rheological Burgers–Faraday models and rheological dynamical systems were created by using two new rheological models: Kelvin–Voigt–Faraday fractional-type model and Maxwell–Faraday fractional-type model. The Burgers–Faraday models described in the paper are new models that examine the dynamical behavior of materials with coupled fields: mechanical stress and strain and the electric field of polarization through the Faraday element. The analysis of the constitutive relation of the fractional order for Burgers–Faraday models is given. Two Burgers–Faraday fractional-type dynamical systems were created under certain approximations. Both rheological Burgers-Faraday dynamic systems have two internal degrees of freedom, which are introduced into the system by each standard light Burgers-Faraday bonding element. It is shown that the sequence of bonding elements in the structure of the standard light Burgers-Faraday bonding element changes the dynamic properties of the rheological dynamic system, so that in one case the system behaves as a fractional-type oscillator, while in the other case, it exhibits a creeping or pulsating behavior under the influence of an external periodic force. These models of rheological dynamic systems can be used to model new natural and synthetic biomaterials that possess both viscoelastic/viscoplastic and piezoelectric properties and have dynamical properties of stress relaxation.

Katica R. (Stevanović) Hedrih

Andjelka N. Hedrih

Rheological Burgers–Faraday Models and Rheological Dynamical Systems with Fractional Derivatives and Their Application in Biomechanics

Burgers-Faraday流变模型与分数阶导数流变动力系统及其在生物力学中的应用

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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