Fractal and Fractional

24 pages, 4059 KiB

Open AccessArticle

Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation

by Sheng Zhang, Hongwei Li and Bo Xu

Fractal Fract. 2024, 8(9), 520; https://doi.org/10.3390/fractalfract8090520 (registering DOI) - 31 Aug 2024

Abstract

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order [...] Read more.

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

24 pages, 1057 KiB

Open AccessArticle

Synchronization for Delayed Fractional-Order Memristive Neural Networks Based on Intermittent-Hold Control with Application in Secure Communication

by Xueqi Yao, Jingxi Shi, Shouming Zhong and Yuanhua Du

Fractal Fract. 2024, 8(9), 519; https://doi.org/10.3390/fractalfract8090519 - 30 Aug 2024

Viewed by 143

Abstract

This article investigates the dynamic behaviors of delayed fractional-order memristive fuzzy cellular neural networks via the Lyapunov method. To address the delay terms of fractional-order systems, a novel lemma is provided to make the solutions of the systems exponentially stable. Furthermore, two new [...] Read more.

This article investigates the dynamic behaviors of delayed fractional-order memristive fuzzy cellular neural networks via the Lyapunov method. To address the delay terms of fractional-order systems, a novel lemma is provided to make the solutions of the systems exponentially stable. Furthermore, two new intermittent-hold controllers are designed to improve the robustness of the system and reduce the cost of the controller. One intermittent-hold controller is based on the feedback control strategy, while the other one integrates an adaptive control strategy. Moreover, two crucial theorems are derived from the proposed lemma and controllers, guaranteeing the exponential synchronization between drive and response systems. Finally, the superior performance of the controllers in achieving exponential synchronization is demonstrated through simulations. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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32 pages, 505 KiB

Open AccessArticle

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓ_q(·)(M_p(·),v(·)) with Variable Exponents

by Waqar Afzal, Mujahid Abbas, Daniel Breaz and Luminiţa-Ioana Cotîrlă

Fractal Fract. 2024, 8(9), 518; https://doi.org/10.3390/fractalfract8090518 - 30 Aug 2024

Viewed by 154

Abstract

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is [...] Read more.

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

23 pages, 1471 KiB

Open AccessArticle

Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension

by Lihua Zhang, Zitong Zheng, Bo Shen, Gangwei Wang and Zhenli Wang

Fractal Fract. 2024, 8(9), 517; https://doi.org/10.3390/fractalfract8090517 - 30 Aug 2024

Viewed by 135

Abstract

Abstract: We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative [...] Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

12 pages, 325 KiB

Open AccessArticle

L^p(L^q)-Maximal Regularity for Damped Equations in a Cylindrical Domain

by Edgardo Alvarez, Stiven Díaz and Carlos Lizama

Fractal Fract. 2024, 8(9), 516; https://doi.org/10.3390/fractalfract8090516 - 30 Aug 2024

Viewed by 220

Abstract

We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are [...] Read more.

We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are still valid for fractional powers of the negative Laplacian operator. We base our methods on the theory of operator-valued Fourier multipliers on vector-valued Lebesgue spaces of periodic functions. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

18 pages, 750 KiB

Open AccessArticle

Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons

by Aly R. Seadway, Asghar Ali, Ahmet Bekir and Adem C. Cevikel

Fractal Fract. 2024, 8(9), 515; https://doi.org/10.3390/fractalfract8090515 - 30 Aug 2024

Viewed by 232

Abstract

We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves [...] Read more.

We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves traveling in both directions, the zero-mass assumption, which is required for many scientific applications, is not required by the KP-B equation. In several applications in engineering and physics, taking these features into account allows researchers to acquire more precise conclusions, particularly in studies pertaining to the dynamics of water waves. The foremost purpose of this manuscript is to establish diverse solutions in the form of exponential, trigonometric, hyperbolic, and rational functions of the (3+1)-dimensional fractional (KP-B) via the application of four analytical methods. This KP-B model has fruitful applications in fluid dynamics and plasma physics. Additionally, in order to better explain the potential and physical behavior of the equation, the relevant models of the findings are visually indicated, and 2-dimensional (2D) and 3-dimensional (3D) graphics are drawn. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

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16 pages, 3378 KiB

Open AccessArticle

Multifractal Analysis of Neuronal Morphology in the Human Dorsal Striatum: Age-Related Changes and Spatial Differences

by Zorana Nedeljković, Bojana Krstonošić, Nebojša Milošević, Olivera Stanojlović, Dragan Hrnčić and Nemanja Rajković

Fractal Fract. 2024, 8(9), 514; https://doi.org/10.3390/fractalfract8090514 - 30 Aug 2024

Viewed by 194

Abstract

Multifractal analysis offers a sophisticated method to examine the complex morphology of neurons, which traditionally have been analyzed using monofractal techniques. This study investigates the multifractal properties of two-dimensional neuron projections from the human dorsal striatum, focusing on potential morphological changes related to [...] Read more.

Multifractal analysis offers a sophisticated method to examine the complex morphology of neurons, which traditionally have been analyzed using monofractal techniques. This study investigates the multifractal properties of two-dimensional neuron projections from the human dorsal striatum, focusing on potential morphological changes related to aging and differences based on spatial origin within the nucleus. Using multifractal spectra, we analyzed various parameters, including generalized dimensions and Hölder exponents, to characterize the neurons’ morphology. Despite the detailed analysis, no significant correlation was found between neuronal morphology and age. However, clear morphological differences were observed between neurons from the caudate nucleus and the putamen. Neurons from the putamen displayed higher morphological complexity and greater local homogeneity, while those from the caudate nucleus exhibited more scaling laws and higher local heterogeneity. These findings suggest that while age may not significantly impact neuronal morphology in the dorsal striatum, the spatial origin within this brain region plays a crucial role in determining neuronal structure. Further studies with larger samples are recommended to confirm these findings and to explore the full potential of multifractal analysis in neuronal morphology research. Full article

(This article belongs to the Special Issue Advances in Pattern Recognition—Image and Time Series Analyses—through Fractal Geometry and Complexity Theory)

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9 pages, 340 KiB

Open AccessBrief Report

Modeling Double Stochastic Opinion Dynamics with Fractional Inflow of New Opinions

by Vygintas Gontis

Fractal Fract. 2024, 8(9), 513; https://doi.org/10.3390/fractalfract8090513 - 29 Aug 2024

Viewed by 172

Abstract

Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential distribution, closely aligned with the second [...] Read more.

Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential distribution, closely aligned with the second form of the Pareto distribution. We elucidate this distinctive power-law statistical property through the lens of agent heterogeneity in trading activity and asset possession. Our study introduces a novel modeling approach that combines fractional Lévy stable motion for limit order inflow with this power-law distribution for cancellation times, significantly enhancing the prediction of order imbalances. This model not only addresses gaps in current financial market modeling but also extends to broader contexts such as opinion dynamics in social systems, capturing the finite lifespan of opinions. Characterized by stationary increments and a departure from self-similarity, our model provides a unique framework for exploring long-range dependencies in time series. This work paves the way for more precise financial market analyses and offers new insights into the dynamic nature of opinion formation in social systems. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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15 pages, 523 KiB

Open AccessArticle

Stability Analysis Study of Time-Fractional Nonlinear Modified Kawahara Equation Based on the Homotopy Perturbation Sadik Transform

by Zhihua Chen, Saeed Kosari, Jana Shafi and Mohammad Hossein Derakhshan

Fractal Fract. 2024, 8(9), 512; https://doi.org/10.3390/fractalfract8090512 - 29 Aug 2024

Viewed by 181

Abstract

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the frame of the Caputo–Prabhakar (CP) operator. The nonlinear [...] Read more.

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the frame of the Caputo–Prabhakar (CP) operator. The nonlinear terms are handled with the assistance of the homotopy polynomials. The stability analysis of the implemented method is studied by using S-stable mapping and the Banach contraction principle. Also, we use the fixed-point method to determine the existence and uniqueness of solutions in the given suggested model. Finally, some numerical simulations are illustrated to display the accuracy and efficiency of the present numerical method. Moreover, numerical behaviors are captured to validate the reliability and efficiency of the scheme. Full article

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33 pages, 7565 KiB

Open AccessArticle

Enhancing Medical Image Quality Using Fractional Order Denoising Integrated with Transfer Learning

by Abirami Annadurai, Vidhushavarshini Sureshkumar, Dhayanithi Jaganathan and Seshathiri Dhanasekaran

Fractal Fract. 2024, 8(9), 511; https://doi.org/10.3390/fractalfract8090511 - 29 Aug 2024

Viewed by 200

Abstract

In medical imaging, noise can significantly obscure critical details, complicating diagnosis and treatment. Traditional denoising techniques often struggle to maintain a balance between noise reduction and detail preservation. To address this challenge, we propose an “Efficient Transfer-Learning-Based Fractional Order Image Denoising Approach in [...] Read more.

In medical imaging, noise can significantly obscure critical details, complicating diagnosis and treatment. Traditional denoising techniques often struggle to maintain a balance between noise reduction and detail preservation. To address this challenge, we propose an “Efficient Transfer-Learning-Based Fractional Order Image Denoising Approach in Medical Image Analysis (ETLFOD)” method. Our approach uniquely integrates transfer learning with fractional order techniques, leveraging pre-trained models such as DenseNet121 to adapt to the specific needs of medical image denoising. This method enhances denoising performance while preserving essential image details. The ETLFOD model has demonstrated superior performance compared to state-of-the-art (SOTA) techniques. For instance, our DenseNet121 model achieved an accuracy of 98.01%, precision of 98%, and recall of 98%, significantly outperforming traditional denoising methods. Specific results include a 95% accuracy, 98% precision, 99% recall, and 96% F1-score for MRI brain datasets, and an 88% accuracy, 91% precision, 95% recall, and 88% F1-score for COVID-19 lung data. X-ray pneumonia results in the lung CT dataset showed a 92% accuracy, 97% precision, 98% recall, and 93% F1-score. It is important to note that while we report performance metrics in this paper, the primary evaluation of our approach is based on the comparison of original noisy images with the denoised outputs, ensuring a focus on image quality enhancement rather than classification performance. Full article

(This article belongs to the Section Optimization, Big Data, and AI/ML)

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22 pages, 344 KiB

Open AccessArticle

Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions

by Brahim Tellab, Abdelkader Amara, Mohammed El-Hadi Mezabia, Khaled Zennir and Loay Alkhalifa

Fractal Fract. 2024, 8(9), 510; https://doi.org/10.3390/fractalfract8090510 - 29 Aug 2024

Viewed by 410

Abstract

This research is concerned with the existence and uniqueness of solutions for a coupled system of

Ψ

–Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known [...] Read more.

This research is concerned with the existence and uniqueness of solutions for a coupled system of

Ψ

–Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known fixed-point theorems. By employing these mathematical tools, we demonstrate the existence and uniqueness of solutions for the proposed system. Additionally, to illustrate the practical implications of our findings, we provide several examples that showcase the main results obtained in this study. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

9 pages, 778 KiB

Open AccessCommunication

Applications of Mittag–Leffler Functions on a Subclass of Meromorphic Functions Influenced by the Definition of a Non-Newtonian Derivative

by Daniel Breaz, Kadhavoor R. Karthikeyan and Gangadharan Murugusundaramoorthy

Fractal Fract. 2024, 8(9), 509; https://doi.org/10.3390/fractalfract8090509 - 29 Aug 2024

Viewed by 226

Abstract

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, [...] Read more.

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

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21 pages, 346 KiB

Open AccessArticle

Extensions of Bicomplex Hypergeometric Functions and Riemann–Liouville Fractional Calculus in Bicomplex Numbers

by Ahmed Bakhet, Mohamed Fathi, Mohammed Zakarya, Ghada AlNemer and Mohammed A. Saleem

Fractal Fract. 2024, 8(9), 508; https://doi.org/10.3390/fractalfract8090508 - 28 Aug 2024

Viewed by 375

Abstract

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville [...] Read more.

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

19 pages, 348 KiB

Open AccessArticle

Polynomial Decay of the Energy of Solutions of the Timoshenko System with Two Boundary Fractional Dissipations

by Suleman Alfalqi, Hamid Khiar, Ahmed Bchatnia and Abderrahmane Beniani

Fractal Fract. 2024, 8(9), 507; https://doi.org/10.3390/fractalfract8090507 - 28 Aug 2024

Viewed by 236

Abstract

In this study, we examine Timoshenko systems with boundary conditions featuring two types of fractional dissipations. By applying semigroup theory, we demonstrate the existence and uniqueness of solutions. Our analysis shows that while the system exhibits strong stability, it does not achieve uniform [...] Read more.

In this study, we examine Timoshenko systems with boundary conditions featuring two types of fractional dissipations. By applying semigroup theory, we demonstrate the existence and uniqueness of solutions. Our analysis shows that while the system exhibits strong stability, it does not achieve uniform stability. Consequently, we derive a polynomial decay rate for the system. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

20 pages, 1043 KiB

Open AccessArticle

Fuzzy Adaptive Approaches for Robust Containment Control in Nonlinear Multi-Agent Systems under False Data Injection Attacks

by Ammar Alsinai, Mohammed M. Ali Al-Shamiri, Waqar Ul Hassan, Saadia Rehman and Azmat Ullah Khan Niazi

Fractal Fract. 2024, 8(9), 506; https://doi.org/10.3390/fractalfract8090506 - 28 Aug 2024

Viewed by 230

Abstract

This study addresses the problem of fractional-order nonlinear containment control of heterogeneous multi-agent systems within a leader–follower framework, focusing on the impact of False Data Injection (FDI) attacks. By employing adaptive mechanisms and fuzzy logic, the suggested method enhances system resilience, ensuring reliable [...] Read more.

This study addresses the problem of fractional-order nonlinear containment control of heterogeneous multi-agent systems within a leader–follower framework, focusing on the impact of False Data Injection (FDI) attacks. By employing adaptive mechanisms and fuzzy logic, the suggested method enhances system resilience, ensuring reliable coordination and stability even in the presence of deceptive disturbances. To deal with these uncertainties, our controller makes use of interval type-II (IT2) fuzzy sets, and we create matrix equalities and inequalities to account for the asymmetry of Laplace matrices. Also, we use the Lyapunov functions for the stability analysis of our system. Lastly, we explain the numerical simulations for the effectiveness of our theoretical results, and these simulated examples are used to verify the effectiveness of our approach and designed model. Full article

(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control, 2nd Edition)

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17 pages, 5441 KiB

Open AccessArticle

Analysis of Impact Crushing Characteristics of Steel Fiber Reinforced Recycled Aggregate Concrete Based on Fractal Theory

by Xianggang Zhang, Yanan Zhu, Junbo Wang, Gaoqiang Zhou and Yajun Huang

Fractal Fract. 2024, 8(9), 505; https://doi.org/10.3390/fractalfract8090505 - 27 Aug 2024

Viewed by 352

Abstract

The fractal theory can effectively describe the complexity and multi-scale of concrete under impact load and provide a scientific basis for evaluating concrete’s impact resistance. Therefore, based on the fractal theory, this study carried out the fragmentation size analysis by weighing the quality [...] Read more.

The fractal theory can effectively describe the complexity and multi-scale of concrete under impact load and provide a scientific basis for evaluating concrete’s impact resistance. Therefore, based on the fractal theory, this study carried out the fragmentation size analysis by weighing the quality of SFRRAC fragments, disclosed the distribution characteristics of impact fragmentation size of steel fiber reinforced recycled aggregate concrete (SFRRAC) specimens under different recycled coarse aggregate (RCA) replacement ratio, different steel fiber (SF) contents and different impact pressures. The results indicate that the fractal dimension can describe the degree of fragmentation of the specimen. The greater the fractal dimension, the more the amount of fragmentation of the specimen subjected to impact load, the lesser the fragmentation size, and the greater the degree of fragmentation. Under the impact load, the fractal dimension of SFRRAC is between 1.36 and 2.28. As the impact pressure increases, the energy consumption increases, and the fractal dimension decreases. With the growth in replacement ratio, the fractal dimension gradually increases, and the energy consumption is negatively correlated with the fractal dimension. Along with the growth of SF content, the energy consumption gradually increases, and the fractal dimension continuously decreases. A new metric angle is provided to explore the inherent law between the impact-crushing characteristics of SFRRAC and the dynamic load, thereby offering foundational support for the application of SFRRAC in practical engineering. Full article

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10 pages, 2349 KiB

Open AccessArticle

Multifractal Characteristics of Gain Structures: A Universal Law of Polycrystalline Strain-Hardening Behaviors

by Maoqing Fu, Jiapeng Chen, Zhaowen Huang, Bin Chen, Yangfan Hu and Biao Wang

Fractal Fract. 2024, 8(9), 504; https://doi.org/10.3390/fractalfract8090504 - 27 Aug 2024

Viewed by 239

Abstract

The quantitative relationship between material microstructures, such as grain distributions, and the nonlinear strain-hardening behaviors of polycrystalline metals has not yet been completely understood. This study finds that the grain correlation dimension of polycrystals D is universally equal to the reciprocal of the [...] Read more.

The quantitative relationship between material microstructures, such as grain distributions, and the nonlinear strain-hardening behaviors of polycrystalline metals has not yet been completely understood. This study finds that the grain correlation dimension of polycrystals D is universally equal to the reciprocal of the strain-hardening exponent by experimental research and fractal geometry analysis. From a geometric perspective, the correlation dimension of grains is consistent with that of the equivalent plastic strain field, which represents the correlation dimension of the material manifold. According to the definition of the Hausdorff measure and Ludwik constitutive model, the strain-hardening exponent represents the exponent derived from the Dth root of the measure relationship. This universal law indicates that the strain-hardening behaviors are fractal geometrized and that the strain-hardening exponent represents a geometrical parameter reflecting the multifractal characteristics of grain structures. This conclusion can enhance the comprehension of the relationship between microstructure and mechanical properties of materials and highlights the importance of designing materials with non-uniform grain distributions to achieve desired hardening properties. Full article

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21 pages, 8041 KiB

Open AccessArticle

A Novel Contact Resistance Model for the Spherical–Planar Joint Interface Based on Three Dimensional Fractal Theory

by Qi An, Weikun Wang, Min Huang, Shuangfu Suo, Yue Liu and Shuai Wang

Fractal Fract. 2024, 8(9), 503; https://doi.org/10.3390/fractalfract8090503 - 26 Aug 2024

Viewed by 330

Abstract

In order to obtain the contact resistance of relay contacts more accurately, a novel contact resistance model for the spherical–planar joint interface is constructed based on the three-dimensional fractal theory. In this model, three-dimensional fractal theory is adopted to generate a rough surface [...] Read more.

In order to obtain the contact resistance of relay contacts more accurately, a novel contact resistance model for the spherical–planar joint interface is constructed based on the three-dimensional fractal theory. In this model, three-dimensional fractal theory is adopted to generate a rough surface at microscopic scale. Then, using contact mechanics theory, the deformation mechanism of asperities on rough surfaces is explored. Combined with the distribution of asperities, a contact resistance model for the planar joint interface is established. Furthermore, by introducing the surface contact coefficient, cross-scale coupling between the macro-geometric configuration and micro-surface topography is achieved, and a contact resistance model for the spherical–planar joint interface is constructed. After that, experiments are conducted to verify the accuracy of the proposed model, and the maximum relative error of the proposed model is 8.44%. Ultimately, combining numerical simulation analysis, the patterns of variation in contact resistance influenced by factors such as macroscopic configuration and microscopic topography are discussed, thereby revealing the influence mechanism of the contact resistance for the spherical–planar joint interface. The proposed model provides a solid theoretical foundation for the optimization of relay contact structures and improvements in manufacturing processes, which is of great significance for ensuring the safe and stable operation of power systems and electronic equipment. Full article

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12 pages, 292 KiB

Open AccessArticle

The Ulam Stability of High-Order Variable-Order φ-Hilfer Fractional Implicit Integro-Differential Equations

by Peiguang Wang, Bing Han and Junyan Bao

Fractal Fract. 2024, 8(9), 502; https://doi.org/10.3390/fractalfract8090502 - 26 Aug 2024

Viewed by 323

Abstract

This study investigates the initial value problem of high-order variable-order

φ

-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional [...] Read more.

This study investigates the initial value problem of high-order variable-order

φ

-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions. Full article

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

13 pages, 341 KiB

Open AccessArticle

Applications of Caputo-Type Fractional Derivatives for Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation

by Kholood M. Alsager, Gangadharan Murugusundaramoorthy, Adriana Catas and Sheza M. El-Deeb

Fractal Fract. 2024, 8(9), 501; https://doi.org/10.3390/fractalfract8090501 - 26 Aug 2024

Viewed by 292

Abstract

In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two Taylor–Maclaurin coefficients, [...] Read more.

In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two Taylor–Maclaurin coefficients,

| a_{2} |

and

| a_{3} |

. Furthermore, the famous Fekete–Szegö inequality is obtained for the newly defined subclasses of bi-univalent functions. Several consequences of our results are pointed out which are new and not yet discussed in association with bounded boundary rotation. Some improved results when compared with those already available in the literature are also stated as corollaries. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

13 pages, 320 KiB

Open AccessBrief Report

Fractional Differential Equations with Impulsive Effects

by Michal Fečkan, Marius-F. Danca and Guanrong Chen

Fractal Fract. 2024, 8(9), 500; https://doi.org/10.3390/fractalfract8090500 - 26 Aug 2024

Viewed by 297

Abstract

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically [...] Read more.

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically changing lower limits. Then, the impulsive effects are modeled for fractional differential equations regarding the nonlinearities rather than the initial value conditions. The proposed impulsive model differs from common discontinuous and nonsmooth dynamical systems. Full article

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

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17 pages, 410 KiB

Open AccessArticle

Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2

by Anurag Shukla, Sumati Kumari Panda, Velusamy Vijayakumar, Kamalendra Kumar and Kothandabani Thilagavathi

Fractal Fract. 2024, 8(9), 499; https://doi.org/10.3390/fractalfract8090499 - 24 Aug 2024

Viewed by 349

Abstract

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order

1 < q < 2

. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects [...] Read more.

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order

1 < q < 2

. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

21 pages, 1723 KiB

Open AccessArticle

Exploring Solitons Solutions of a (3+1)-Dimensional Fractional mKdV-ZK Equation

by Amjad E. Hamza, Osman Osman, Muhammad Umair Sarwar, Khaled Aldwoah, Hicham Saber and Manel Hleili

Fractal Fract. 2024, 8(9), 498; https://doi.org/10.3390/fractalfract8090498 - 24 Aug 2024

Viewed by 361

Abstract

This study presents the application of the

ϕ^{6}

model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep [...] Read more.

This study presents the application of the

ϕ^{6}

model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep insights into wave behavior. Clear graphics demonstrate that the solutions are greatly affected by changes in the fractional order, deepening our understanding and revealing the hidden dynamics of wave propagation. The considered equation has several applications in fluid dynamics, plasma physics, and nonlinear optics. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

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12 pages, 1090 KiB

Open AccessArticle

Dynamics of the Traveling Wave Solutions of Fractional Date–Jimbo–Kashiwara–Miwa Equation via Riccati–Bernoulli Sub-ODE Method through Bäcklund Transformation

by M. Mossa Al-Sawalha, Saima Noor, Mohammad Alqudah, Musaad S. Aldhabani and Roman Ullah

Fractal Fract. 2024, 8(9), 497; https://doi.org/10.3390/fractalfract8090497 - 23 Aug 2024

Viewed by 410

Abstract

The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement [...] Read more.

The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement of the characterization of dynamic waves while providing better modeling ability compared to integer types of derivatives. The solutions of the above-mentioned time–space fractional Date–Jimbo–Kashiwara–Miwa equation have tremendous importance in numerous scientific scenarios. The regular dynamical wave solutions of the aforementioned equation encompass three fundamental functions: trigonometric, hyperbolic, and rational functions will be among the topics covered. These solutions are graphically classified into three categories: compacton kink solitary wave solutions, kink soliton wave solutions and anti-kink soliton wave solutions. In addition, to explore the impact of the fractional parameter (

α

) on those solutions,

2 D

plots are utilized, while

3 D

plots are applied to present the solutions involving the integer-order derivatives. Full article

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

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25 pages, 8771 KiB

Open AccessArticle

Mathematical Modeling of Alzheimer’s Drug Donepezil Hydrochloride Transport to the Brain after Oral Administration

by Corina S. Drapaca

Fractal Fract. 2024, 8(9), 496; https://doi.org/10.3390/fractalfract8090496 - 23 Aug 2024

Viewed by 321

Abstract

Alzheimer’s disease (AD) is a progressive degenerative disorder that causes behavioral changes, cognitive decline, and memory loss. Currently, AD is incurable, and the few available medicines may, at best, improve symptoms or slow down AD progression. One main challenge in drug delivery to [...] Read more.

Alzheimer’s disease (AD) is a progressive degenerative disorder that causes behavioral changes, cognitive decline, and memory loss. Currently, AD is incurable, and the few available medicines may, at best, improve symptoms or slow down AD progression. One main challenge in drug delivery to the brain is the presence of the blood–brain barrier (BBB), a semi-permeable layer around cerebral capillaries controlling the influx of blood-borne particles into the brain. In this paper, a mathematical model of drug transport to the brain is proposed that incorporates two mechanisms of BBB crossing: transcytosis and diffusion. To account for the structural damage and accumulation of harmful waste in the brain caused by AD, the diffusion is assumed to be anomalous and is modeled using spatial Riemann–Liouville fractional-order derivatives. The model’s parameters are taken from published experimental observations of the delivery to mice brains of the orally administered AD drug donepezil hydrochloride. Numerical simulations suggest that drug delivery modalities should depend on the BBB fitness and anomalous diffusion and be tailored to AD severity. These results may inspire novel brain-targeted drug carriers for improved AD therapies. Full article

(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional, 2nd Edition)

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Fractal Fract., Volume 8, Issue 9 (September 2024) – 25 articles

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