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Calculation of the Relaxation Modulus in the Andrade Model by Using the Laplace Transform 利用拉普拉斯变换计算安德拉德模型中的松弛模量
Pub Date : 2024-07-26 DOI: 10.3390/fractalfract8080439
Juan Luis González-Santander, Giorgio Spada, Francesco Mainardi, Alexander Apelblat
In the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model Gαt for the case of rational parameter α=m/n∈(0,1) in terms of Mittag–Leffler functions from its Laplace transform G˜αs. It turns out that the expression obtained can be rewritten in terms of Rabotnov functions. Moreover, for the original parameter α=1/3 in the Andrade model, we obtain an expression in terms of Miller-Ross functions. The asymptotic behaviours of Gαt for t→0+ and t→+∞ are also derived applying the Tauberian theorem. The analytical results obtained have been numerically checked by solving the Volterra integral equation satisfied by Gαt by using a successive approximation approach, as well as computing the inverse Laplace transform of G˜αs by using Talbot’s method.
在线性粘弹性理论的框架内,我们从安德拉德模型 Gαs 的拉普拉斯变换中推导出了有理参数 α=m/n∈(0,1)情况下安德拉德模型 Gαt 中松弛模量的分析表达式,该表达式用 Mittag-Leffler 函数表示。结果发现,所得到的表达式可以用拉博特诺夫函数重写。此外,对于安德拉德模型中的原始参数 α=1/3,我们可以得到米勒-罗斯函数的表达式。我们还应用陶伯定理推导出了 Gαt 在 t→0+ 和 t→+∞ 时的渐近行为。通过使用逐次逼近法求解 Gαt 满足的 Volterra 积分方程,以及使用塔尔博特方法计算 G˜αs 的反拉普拉斯变换,对得到的分析结果进行了数值检验。
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引用次数: 0
Existence of Solutions for Caputo Sequential Fractional Differential Inclusions with Nonlocal Generalized Riemann–Liouville Boundary Conditions 具有非局部广义黎曼-刘维尔边界条件的卡普托序列分微分方程的解的存在性
Pub Date : 2024-07-26 DOI: 10.3390/fractalfract8080441
M. Manigandan, Saravanan Shanmugam, Mohamed Rhaima, Elango Sekar
In this study, we explore the existence and uniqueness of solutions for a boundary value problem defined by coupled sequential fractional differential inclusions. This investigation is augmented by the introduction of a novel set of generalized Riemann–Liouville boundary conditions. Utilizing Carathéodory functions and Lipschitz mappings, we establish existence results for these nonlocal boundary conditions. Utilizing fixed-point theorems designed for multi-valued maps, we obtain significant existence results for the problem, considering both convex and non-convex values. The derived results are clearly demonstrated with an illustrative example. Numerical examples are provided to validate the theoretical conclusions, contributing to a deeper understanding of fractional-order boundary value problems.
在本研究中,我们探讨了由耦合序列分数微分夹杂定义的边界值问题的解的存在性和唯一性。通过引入一组新颖的广义黎曼-刘维尔边界条件,这一研究得到了加强。利用 Carathéodory 函数和 Lipschitz 映射,我们建立了这些非局部边界条件的存在性结果。利用专为多值映射设计的定点定理,考虑到凸值和非凸值,我们获得了问题的重要存在性结果。我们通过一个示例清楚地演示了得出的结果。我们还提供了数值示例来验证理论结论,有助于加深对分数阶边界值问题的理解。
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引用次数: 0
Morphological Features of Mathematical and Real-World Fractals: A Survey 数学和现实世界分形的形态特征:概览
Pub Date : 2024-07-26 DOI: 10.3390/fractalfract8080440
M. Patiño-Ortiz, J. Patiño-Ortiz, M. Martínez-Cruz, Fernando René Esquivel-Patiño, A. Balankin
The aim of this review paper is to survey the fractal morphology of scale-invariant patterns. We are particularly focusing on the scale and conformal invariance, as well as on the fractal non-uniformity (multifractality), inhomogeneity (lacunarity), and anisotropy (succolarity). We argue that these features can be properly quantified by the following six adimensional numbers: the fractal (e.g., similarity, box-counting, or Assouad) dimension, conformal dimension, degree of multifractal non-uniformity, coefficient of multifractal asymmetry, index of lacunarity, and index of fractal anisotropy. The difference between morphological properties of mathematical and real-world fractals is especially outlined in this review paper.
本综述旨在研究尺度不变模式的分形形态。我们尤其关注尺度不变性和共形不变性,以及分形的不均匀性(多分形)、不均匀性(裂隙性)和各向异性(琥珀性)。我们认为,这些特征可以用以下六个维度的数字来适当量化:分形(如相似性、盒数或阿苏阿德)维度、保形维度、多分形不均匀度、多分形不对称系数、裂隙度指数和分形各向异性指数。这篇综述论文特别概述了数学分形与现实世界分形的形态特性之间的区别。
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引用次数: 0
An Application of Multiple Erdélyi–Kober Fractional Integral Operators to Establish New Inequalities Involving a General Class of Functions 应用多重埃代利-科贝尔分式积分算子建立涉及一般函数类别的新不等式
Pub Date : 2024-07-25 DOI: 10.3390/fractalfract8080438
Asifa Tassaddiq, R. Srivastava, Rabab Alharbi, R. Kasmani, Sania Qureshi
This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in the finite interval a≤t≤x, the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results.
本研究旨在利用多重埃尔德利-科贝尔(E-K)分式积分算子,发展广义分式积分不等式。利用一组在有限区间 a≤t≤x 内 j,(j∈N) 正连续且衰减的函数,Fox-H 函数参与建立新颖的分式积分不等式。由于 Fox-H 函数是最一般的特殊函数,因此所得到的不等式与现有文献相比具有足够的广泛性和重要性,从而产生了新颖独特的结果。
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引用次数: 0
Evolution of Pore Structure and Fractal Characteristics in Red Sandstone under Cyclic Impact Loading 循环冲击载荷下红色砂岩孔隙结构和分形特征的演变
Pub Date : 2024-07-24 DOI: 10.3390/fractalfract8080437
Huanhuan Qiao, Peng Wang, Zhen Jiang, Yao Liu, Guanglin Tian, Bokun Zhao
Fatigue damage can occur in surface rock engineering due to various factors, including earthquakes, blasting, and impacts. The underlying cause for the variations in physical and mechanical properties of the rock resulting from impact loading is the alteration in the internal pore structure. To investigate the evolution characteristics of the pore structure under impact fatigue damage, red sandstone subjected to cyclic impact compression by split Hopkinson pressure bar (SHPB) was analyzed using nuclear magnetic resonance (NMR) technology. The parameters describing the evolution of pore structure were obtained and quantified using fractal methods. The development of the pore structure in rocks subjected to cyclic impact was quantitatively analyzed, and two fractal evolution models based on pore size and pore connectivity were constructed. The results indicate that with an increasing number of impact loading cycles, the porosity of the red sandstone gradually increases, the T2 cutoff (T2c) value decreases, the most probable gray value of magnetic resonance imaging (MRI) increases, the pores’ connectivity is enhanced, and the fractal dimension decreases gradually. Moreover, the pore distribution space tends to transition from three-dimensional to two-dimensional, suggesting the expansion of dominant pores into clusters, forming microfractures or even macroscopic fissures. The findings provide valuable insights into the impact fatigue characteristics of rocks from a microscopic perspective and contribute to the evaluation of time-varying stability and the assessment of progressive damage in rock engineering.
由于地震、爆破和撞击等各种因素,表层岩石工程可能会出现疲劳破坏。冲击荷载导致岩石物理和机械性能变化的根本原因是内部孔隙结构的改变。为了研究冲击疲劳破坏下孔隙结构的演变特征,使用核磁共振(NMR)技术分析了受到分体式霍普金森压力棒(SHPB)循环冲击压缩的红砂岩。利用分形方法获得并量化了描述孔隙结构演变的参数。定量分析了受到循环冲击的岩石中孔隙结构的发展,并构建了基于孔隙大小和孔隙连通性的两种分形演化模型。结果表明,随着冲击载荷循环次数的增加,红砂岩的孔隙率逐渐增大,T2截止值(T2c)减小,磁共振成像(MRI)最可能灰度值增大,孔隙连通性增强,分形维数逐渐减小。此外,孔隙分布空间趋于从三维过渡到二维,表明优势孔隙扩展成团,形成微裂缝甚至宏观裂缝。研究结果为从微观角度了解岩石的冲击疲劳特性提供了宝贵的见解,有助于岩石工程中的时变稳定性评估和渐进损伤评估。
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引用次数: 0
Semi-Regular Continued Fractions with Fast-Growing Partial Quotients 具有快速增长部分商的半整式连续分数
Pub Date : 2024-07-24 DOI: 10.3390/fractalfract8080436
Sh. Kadyrov, A. Kazin, F. Mashurov
In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field.
在数论中,连续分数是必不可少的工具,因为它们提供了实数的独特表示,并提供了有关实数特征的信息。人们对规则连续分数进行了深入研究,但对半规则连续分数的研究较少,因为半规则连续分数是由正负1交替产生的。在本研究中,我们通过维度理论的视角研究半规则连续分数的结构和特征。我们证明了一个关于数集的豪斯多夫维度的主要结果,这些数集的部分商的增加速度快于给定的速度。此外,我们还通过数值分析来说明正则连续分数与半正则连续分数之间的差异,为这一领域未来的潜在发展方向提供启示。
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引用次数: 0
Analytic Solution of the Time-Fractional Partial Differential Equation Using a Multi-G-Laplace Transform Method 用多G-拉普拉斯变换法解析时分数偏微分方程
Pub Date : 2024-07-23 DOI: 10.3390/fractalfract8080435
Hassan Eltayeb
In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) are proved. The solution of the system of time-fractional partial differential equations is addressed using a new analytical method. This technique is a combination of the multi-G-Laplace transform and decomposition methods (MGLTDM). Moreover, we discuss the convergence of this method. Two examples are provided to check the applicability and efficiency of our technique.
在最近的几项研究中,许多研究人员都展示了分数微积分在生成大量线性和非线性偏微分方程特定解方面的优势。在这项研究工作中,证明了与 G 双拉普拉斯变换 (DGLT) 有关的不同定理。时分数偏微分方程系统的求解采用了一种新的分析方法。该技术是多 G 拉普拉斯变换和分解方法(MGLTDM)的结合。此外,我们还讨论了该方法的收敛性。我们提供了两个例子来检验我们技术的适用性和效率。
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引用次数: 0
A Stable Forward Modeling Approach in Heterogeneous Attenuating Media Using Reapplied Hilbert Transform 使用重应用希尔伯特变换的异质衰减介质稳定前向建模方法
Pub Date : 2024-07-22 DOI: 10.3390/fractalfract8070434
Songmei Deng, Shaolin Shi, Hongwei Liu
In the field of geological exploration and wave propagation theory, particularly in heterogeneous attenuating media, the stability of numerical simulations is a significant challenge for implementing effective attenuation compensation strategies. Consequently, the development and optimization of algorithms and techniques that can mitigate these numerical instabilities are critical for ensuring the accuracy and practicality of attenuation compensation methods. This is essential to reveal subsurface structure information accurately and enhance the reliability of geological interpretation. We present a method for stable forward modeling in strongly attenuating media by reapplying the Hilbert transform to eliminate increasing negative frequency components. We derived and validated new constant-Q wave equation (CWE) formulations and a stable solving method. Our study reveals that the original CWE equations, when utilizing the analytic signal, regenerate and amplify negative frequencies, leading to instability. Implementing our method maintains high accuracy between analytical and numerical solutions. The application of our approach to the Chimney Model, compared with results from the acoustic wave equation, confirms the reliability and effectiveness of the proposed equations and method.
在地质勘探和波传播理论领域,特别是在异质衰减介质中,数值模拟的稳定性是实施有效衰减补偿策略的重大挑战。因此,开发和优化能够缓解这些数值不稳定性的算法和技术,对于确保衰减补偿方法的准确性和实用性至关重要。这对于准确揭示地下结构信息和提高地质解释的可靠性至关重要。我们提出了一种在强衰减介质中进行稳定正演建模的方法,通过重新应用希尔伯特变换来消除不断增加的负频率成分。我们推导并验证了新的常Q波方程(CWE)公式和稳定求解方法。我们的研究发现,利用解析信号时,原始的 CWE 方程会重新生成并放大负频率,从而导致不稳定性。采用我们的方法,可以在分析和数值解之间保持高精度。将我们的方法应用于烟囱模型,并与声波方程的结果进行比较,证实了所提出方程和方法的可靠性和有效性。
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引用次数: 0
Numerical Performance of the Fractional Direct Spreading Cholera Disease Model: An Artificial Neural Network Approach 分式直接传播霍乱疾病模型的数值性能:人工神经网络方法
Pub Date : 2024-07-22 DOI: 10.3390/fractalfract8070432
Saadia Malik
The current investigation examines the numerical performance of the fractional-order endemic disease model based on the direct spreading of cholera by applying the neuro-computing Bayesian regularization (BR) neural network process. The purpose is to present the numerical solutions of the fractional-order model, which provides more precise solutions as compared to the integer-order one. Real values based on the parameters can be obtained and one can achieve better results by utilizing these values. The mathematical form of the fractional direct spreading cholera disease is categorized as susceptible, infected, treatment, and recovered, which represents a nonlinear model. The construction of the dataset is performed through the implicit Runge–Kutta method, which is used to lessen the mean square error by taking 74% of the data for training, while 8% is used for both validation and testing. Twenty-two neurons and the log-sigmoid fitness function in the hidden layer are used in the stochastic neural network process. The optimization of BR is performed in order to solve the direct spreading cholera disease problem. The accuracy of the stochastic process is authenticated through the valuation of the outputs, whereas the negligible calculated absolute error values demonstrate the approach’s correctness. Furthermore, the statistical operator performance establishes the reliability of the proposed scheme.
本次研究通过应用神经计算贝叶斯正则化(BR)神经网络过程,检验了基于霍乱直接传播的分数阶地方病模型的数值性能。目的是给出分数阶模型的数值解,与整数阶模型相比,分数阶模型提供了更精确的解。可以获得基于参数的实数值,利用这些数值可以获得更好的结果。分数直接传播霍乱疾病的数学形式分为易感、感染、治疗和康复,这代表了一种非线性模型。数据集的构建采用隐式 Runge-Kutta 方法,通过将 74% 的数据用于训练,8% 的数据用于验证和测试来减少均方误差。在随机神经网络过程中,隐层使用了 22 个神经元和对数拟合函数。为了解决霍乱疾病的直接传播问题,对 BR 进行了优化。随机过程的准确性通过对输出的估价得到验证,而计算出的绝对误差值可以忽略不计,这证明了该方法的正确性。此外,统计算子的性能也证明了建议方案的可靠性。
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引用次数: 0
Securing Bipartite Nonlinear Fractional-Order Multi-Agent Systems against False Data Injection Attacks (FDIAs) Considering Hostile Environment 考虑到敌对环境,确保双部分非线性分数阶多代理系统免受虚假数据注入攻击(FDIAs)的安全
Pub Date : 2024-07-22 DOI: 10.3390/fractalfract8070430
Hanen Louati, Saadia Rehman, Farhat Imtiaz, Nafisa A. Albasheir, A. Y. Al-Rezami, Mohammed M. A. Almazah, A. U. K. Niazi
This study investigated the stability of bipartite nonlinear fractional-order multi-agent systems (FOMASs) in the presence of false data injection attacks (FDIAs) in a hostile environment. To tackle this problem we used signed graph theory, the Razumikhin methodology, and the Lyapunov function method. The main focus of our proposed work is to provide a method of stability for FOMASs against FDIAs. The technique of Razumikhin improves the Lyapunov-based stability analysis by supporting the handling of the intricacies of fractional-order dynamics. Moreover, utilizing signed graph theory, we analyzed both hostile and cooperative interactions between agents within the MASs. We determined the system stability requirements to ensure robustness against erroneous data injections through comprehensive theoretical investigation. We present numerical examples to illustrate the robustness and efficiency of our proposed technique.
本研究探讨了敌对环境中存在虚假数据注入攻击(FDIAs)时双方非线性分数阶多代理系统(FOMASs)的稳定性问题。为了解决这个问题,我们使用了符号图论、拉祖米欣方法和李亚普诺夫函数法。我们提出的工作重点是提供一种针对 FDIA 的 FOMAS 稳定性方法。Razumikhin 技术通过支持处理复杂的分数阶动力学,改进了基于 Lyapunov 的稳定性分析。此外,我们还利用符号图理论分析了 MAS 中代理之间的敌对与合作互动。通过全面的理论研究,我们确定了系统稳定性要求,以确保对错误数据注入的鲁棒性。我们列举了一些数值示例,以说明我们提出的技术的鲁棒性和效率。
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引用次数: 0
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Fractal and Fractional
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