Fractal and Fractional

23 pages, 17782 KiB

Open AccessArticle

Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator

by Yasir A. Madani, Zeeshan Ali, Mohammed Rabih, Amer Alsulami, Nidal H. E. Eljaneid, Khaled Aldwoah and Blgys Muflh

Fractal Fract. 2025, 9(1), 55; https://doi.org/10.3390/fractalfract9010055 - 20 Jan 2025

Viewed by 639

Abstract

This paper presents a new SIRS model for recurrent childhood diseases under the Caputo fractional difference operator. The existence theory is established using Brouwer’s fixed-point theorem and the Banach contraction principle, providing a comprehensive mathematical foundation for the model. Ulam stability is demonstrated [...] Read more.

This paper presents a new SIRS model for recurrent childhood diseases under the Caputo fractional difference operator. The existence theory is established using Brouwer’s fixed-point theorem and the Banach contraction principle, providing a comprehensive mathematical foundation for the model. Ulam stability is demonstrated using nonlinear functional analysis. Sensitivity analysis is conducted based on the variation of each parameter, and the basic reproduction number

(R_{0})

is introduced to assess local stability at two equilibrium points. The stability analysis indicates that the disease-free equilibrium point is stable when

R_{0} < 1

, while the endemic equilibrium point is stable when

R_{0} > 1

and otherwise unstable. Numerical simulations demonstrate the model’s effectiveness in capturing realistic scenarios, particularly the recurrent patterns observed in some childhood diseases. Full article

(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)

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13 pages, 1418 KiB

Open AccessArticle

Phased Fractional Low-Order Moment-Based Doppler Shift Estimation in the Presence of Interference Signals and Impulsive Noise

by Bo Ni, Mengjia Wang, Jiacheng Zhang, Ying Zhang and Tao Liu

Fractal Fract. 2025, 9(1), 54; https://doi.org/10.3390/fractalfract9010054 - 20 Jan 2025

Viewed by 623

Abstract

Doppler shift estimation continues to be a critical challenge of utmost significance in both theoretical research and practical engineering applications. Many innovators have crafted solutions specific to this issue, with notable contributions across various signals and scenarios. Given that cyclostationary signals are prevalent [...] Read more.

Doppler shift estimation continues to be a critical challenge of utmost significance in both theoretical research and practical engineering applications. Many innovators have crafted solutions specific to this issue, with notable contributions across various signals and scenarios. Given that cyclostationary signals are prevalent in both artificial and natural phenomena, we propose a novel framework based on the phased fractional lower-order moment (PFLOM) for estimating Doppler shift in mixed cyclostationary signals. During the estimation process, a more realistic impulse noise model is examined in contrast to the ideal Gaussian noise typically assumed in conventional methods. This approach is meticulously derived through a series of detailed steps in line with cyclostationary signal processing and PFLOM principles. Furthermore, an extensive simulation has been conducted to validate the efficacy and robustness of our proposed method. It is anticipated that the concept and method presented here could be applied more broadly due to its solid theoretical underpinnings. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing, 2nd Edition)

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25 pages, 477 KiB

Open AccessArticle

Topology of Locally and Non-Locally Generalized Derivatives

by Dimiter Prodanov

Fractal Fract. 2025, 9(1), 53; https://doi.org/10.3390/fractalfract9010053 - 20 Jan 2025

Viewed by 574

Abstract

This article investigates the continuity of derivatives of real-valued functions from a topological perspective. This is achieved by the characterization of their sets of discontinuity. The same principle is applied to Gateaux derivatives and gradients in Euclidean spaces. This article also introduces a [...] Read more.

This article investigates the continuity of derivatives of real-valued functions from a topological perspective. This is achieved by the characterization of their sets of discontinuity. The same principle is applied to Gateaux derivatives and gradients in Euclidean spaces. This article also introduces a generalization of the derivatives from the perspective of the modulus of continuity and characterizes their sets of discontinuities. There is a need for such generalizations when dealing with physical phenomena, such as fractures, shock waves, turbulence, Brownian motion, etc. Full article

(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)

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25 pages, 722 KiB

Open AccessArticle

Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–Liouville Fractional Integral Inequalities

by Abdul Mateen, Wali Haider, Asia Shehzadi, Hüseyin Budak and Bandar Bin-Mohsin

Fractal Fract. 2025, 9(1), 52; https://doi.org/10.3390/fractalfract9010052 - 18 Jan 2025

Viewed by 787

Abstract

This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable [...] Read more.

This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings. Full article

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

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Numerical analysis of Theorem 3: (<bold>a</bold>) 3D-plot, (<bold>b</bold>) 2D-plot for <inline-formula> <mml:math id="mm199"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>, computed and plotted with Mathematica. Full article ">Figure 2
Numerical analysis of Theorem 3: (<bold>a</bold>) 3D-plot, (<bold>b</bold>) 2D-plot for <inline-formula> <mml:math id="mm200"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Θ</mml:mi> <mml:mfenced open="(" close=")"> <mml:mi>ϖ</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>ϖ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>, computed and plotted with Wolfram Mathematica. Full article ">Figure 3
Numerical analysis of Theorem 3: (<bold>a</bold>) 3D-plot, (<bold>b</bold>) 2D-plot for for <inline-formula> <mml:math id="mm201"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Θ</mml:mi> <mml:mfenced open="(" close=")"> <mml:mi>ϖ</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>ϖ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>, computed and plotted with Wolfram Mathematica. Full article ">Figure 4
Numerical analysis of Theorem 6: (<bold>a</bold>) 3D-plot, (<bold>b</bold>) 2D-plot for <inline-formula> <mml:math id="mm202"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Θ</mml:mi> <mml:mfenced open="(" close=")"> <mml:mi>ϖ</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>r</mml:mi> <mml:mi>c</mml:mi> <mml:mi>T</mml:mi> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϖ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>, when <inline-formula> <mml:math id="mm203"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>, computed and plotted with Wolfram Mathematica. Full article ">

21 pages, 103718 KiB

Open AccessArticle

The Fractal Dimension, Structure Characteristics, and Damage Effects of Multi-Scale Cracks on Sandstone Under Triaxial Compression

by Pengjin Yang, Shengjun Miao, Kesheng Li, Xiangfan Shang, Pengliang Li and Meifeng Cai

Fractal Fract. 2025, 9(1), 51; https://doi.org/10.3390/fractalfract9010051 - 17 Jan 2025

Viewed by 689

Abstract

To study the influence of the spatial distribution and structure of multi-scale cracks on the mechanical behavior of rocks, triaxial compression tests and cyclic triaxial complete loading and unloading tests were conducted on sandstone, with real-time wave velocity monitoring and CT scan testing. [...] Read more.

To study the influence of the spatial distribution and structure of multi-scale cracks on the mechanical behavior of rocks, triaxial compression tests and cyclic triaxial complete loading and unloading tests were conducted on sandstone, with real-time wave velocity monitoring and CT scan testing. The quantitative classification criteria for multi-scale cracks on sandstone were established, and the constraint effect of confining pressure was analyzed. The crack with a length less than 0.1 mm is considered a small-scale crack, 0.1–1 mm is a medium-scale crack, and larger than 1 mm is a large-scale crack. As the confining pressure increases, the spatial fractal dimension of large-scale cracks decreases, while that of medium-scale cracks increases, and that of small-scale cracks remains stable. The respective nonlinear models of the aspect ratio were established with the length and density of multi-scale cracks. The results indicate significant differences in the effects of cracks of different scales on rock damage. The distribution density of medium-scale cracks in the failed specimen is higher, which is the main reason to produce damage. The small-scale cracks mainly originate from relatively uniform initial cracks in rocks, mainly distributed in medium-density and low-density areas. The results of this research provide important insights into how to quantitatively evaluate the damage of rocks. Full article

(This article belongs to the Section Engineering)

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26 pages, 1846 KiB

Open AccessArticle

Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context

by Yousuf Alkhezi and Ahmad Shafee

Fractal Fract. 2025, 9(1), 50; https://doi.org/10.3390/fractalfract9010050 - 17 Jan 2025

Viewed by 743

Abstract

The proposed study seeks to investigate various analytical and numerical techniques for solving fractional differential equations, with a particular focus on their applications in mathematical modeling and scientific research within the field of algebra. This study intends to investigate methods such as the [...] Read more.

The proposed study seeks to investigate various analytical and numerical techniques for solving fractional differential equations, with a particular focus on their applications in mathematical modeling and scientific research within the field of algebra. This study intends to investigate methods such as the Aboodh transform iteration method and the Aboodh residual power series method, specifically for addressing the Jaulent–Miodek system of partial differential equations. By analyzing the behavior of fractional-order differential equations and their solutions, this research seeks to contribute to a deeper understanding of complex mathematical phenomena. Furthermore, this study examines the role of the Caputo operator in fractional calculus, offering insights into its significance in modeling real-world systems within the algebraic context. Through this research, novel approaches for solving fractional differential equations are developed, offering essential tools for researchers in diverse fields of science and engineering, including algebraic applications. Full article

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

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27 pages, 16020 KiB

Open AccessArticle

Pore Structure and Its Fractal Dimension: A Case Study of the Marine Shales of the Niutitang Formation in Northwest Hunan, South China

by Wei Jiang, Yang Zhang, Tianran Ma, Song Chen, Yang Hu, Qiang Wei and Dingxiang Zhuang

Fractal Fract. 2025, 9(1), 49; https://doi.org/10.3390/fractalfract9010049 - 17 Jan 2025

Viewed by 541

Abstract

To analyze the pore structure and fractal characteristics of marine shale in the lower Cambrian Niutitang Formation in northwestern Hunan Province, China, the pore characteristics of shale were characterized using total organic carbon (TOC) content, field emission scanning electron microscopy (FESEM), X-ray diffraction [...] Read more.

To analyze the pore structure and fractal characteristics of marine shale in the lower Cambrian Niutitang Formation in northwestern Hunan Province, China, the pore characteristics of shale were characterized using total organic carbon (TOC) content, field emission scanning electron microscopy (FESEM), X-ray diffraction (XRD), low temperature nitrogen adsorption (LT-N₂GA) and methane adsorption experiments. The pore surface and pore space fractal dimensions of samples were calculated, respectively. The influencing factors of fractal dimensions and their impact on the adsorption of shale reservoirs were discussed. The results indicate the Niutitang Formation shale mainly develops four types of pores: organic pores, intragranular pores, intergranular pores and microcracks. The pores have a large specific surface area (SSA), primarily consisting of mesopores. The fractal dimensions are calculated using the FHH model and the XS model. The fractal dimensions (D₂ and D_f) are greater than D₁, indicating that the pore surface with larger pore size is rougher, and the pore structure of shale is complex. The pore volume (PV), SSA, and TOC show positive correlations with the fractal dimensions but negative correlations with APS. There is no obvious correlation between fractal dimensions and quartz content, while clay minerals show a negative correlation with D₂ and D_f. This is mainly because clay mineral particles are small in size and have weak resistance to compaction. The pyrite content is positively correlated with the fractal dimensions because pyrite promotes the development of organic, intergranular, and mold pores. According to Pearson correlation analysis, the main influencing factors of the pore surface fractal dimension are PV, SSA, and APS. The main influencing factors of the pore space fractal dimension are APS and the content of clay minerals. Further analysis of the influence of the fractal dimension on the adsorption capacity of shale reveals that the fractal dimensions are positively correlated with Langmuir volume, indicating that fractal dimensions can be used as a quantitative target for evaluating shale gas reservoirs. 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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26 pages, 7227 KiB

Open AccessArticle

Uncertainty-Based Scale Identification and Process–Topography Interaction Analysis via Bootstrap: Application to Grit Blasting

by François Berkmans, Julie Lemesle, Robin Guibert, Michal Wieczorowski, Christopher Brown and Maxence Bigerelle

Fractal Fract. 2025, 9(1), 48; https://doi.org/10.3390/fractalfract9010048 - 17 Jan 2025

Viewed by 618

Abstract

Finding the relevant scale to observe the influence of a process is one of the most important purposes of multiscale surface characterization. This study investigates various methods to determine a pertinent scale for evaluating the relationship between the relative area and grit blasting [...] Read more.

Finding the relevant scale to observe the influence of a process is one of the most important purposes of multiscale surface characterization. This study investigates various methods to determine a pertinent scale for evaluating the relationship between the relative area and grit blasting pressure. Several media types were tested alongside two different methods for calculating the relative area and three bootstrapping approaches for scale determination through regression. Comparison with the existing literature highlights innovations in roughness parameter characterization, particularly the advantages of relative area over traditional parameters like Sa. This study also discusses the relevance of different media types in influencing surface topography. Additionally, insights from a similar study on the multiscale Sdq parameter and blasting pressure correlation are integrated, emphasizing a scale relevance akin to our Sdr method’s 120 µm cut-off length. Overall, our findings suggest a pertinent scale of 10,000 µm² for the Patchwork method and a 120 µm cut-off length for the Sdr method, derived from bootstrapping on residual regression across all media. At the relevant scale, every value of R² inferior to 0.83 is not significant with the threshold of 5% for the two methods of calculation of the relative area. This study enhances the understanding of how media types and blasting pressures impact surface topography, offering insights for refining material processing and surface treatment strategies. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Materials Science)

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19 pages, 11512 KiB

Open AccessArticle

Finite-Time Synchronization of Fractional-Order Complex-Valued Multi-Layer Network via Adaptive Quantized Control Under Deceptive Attacks

by Lulu Xu, Juan Yu, Cheng Hu, Kailong Xiong and Tingting Shi

Fractal Fract. 2025, 9(1), 47; https://doi.org/10.3390/fractalfract9010047 - 17 Jan 2025

Viewed by 504

Abstract

This article investigates the problem of finite-time synchronization of fractional-order complex-valued random multi-layer networks without decomposing them into two real-valued systems. Firstly, by promoting real-valued signum functions, sign functions on the complex-valued domain are introduced. Simultaneously, quantization functions in the complex-valued domain are [...] Read more.

This article investigates the problem of finite-time synchronization of fractional-order complex-valued random multi-layer networks without decomposing them into two real-valued systems. Firstly, by promoting real-valued signum functions, sign functions on the complex-valued domain are introduced. Simultaneously, quantization functions in the complex-valued domain are also introduced, and several related formulas for sign functions and quantization functions in complex-valued domain are established. Under the framework of the given sign function and quantization function, an adaptive quantized control scheme with or without deception attacks is designed. According to the finite-time theorem, Lyapunov function, and graph theory methods, some sufficient criteria for realizing finite-time synchronization in complex-valued fractional-order multi-layer networks have been obtained. Furthermore, the setting time of finite-time synchronization is effectively evaluated. Eventually, the reliability of our results and the practicality of control strategies are verified through numerical examples. Full article

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18 pages, 337 KiB

Open AccessArticle

Existence Results of Nonlocal Fractional Integro-Neutral Differential Inclusions with Infinite Delay

by Madeaha Alghanmi and Shahad Alqurayqiri

Fractal Fract. 2025, 9(1), 46; https://doi.org/10.3390/fractalfract9010046 - 16 Jan 2025

Viewed by 565

Abstract

This article addresses a new class of delayed fractional multivalued problems complemented with nonlocal boundary conditions. In view of infinite delay theory, we convert the inclusion problem into a fixed-point multivalued problem, defined in an appropriate phase space. Then, sufficient criteria for the [...] Read more.

This article addresses a new class of delayed fractional multivalued problems complemented with nonlocal boundary conditions. In view of infinite delay theory, we convert the inclusion problem into a fixed-point multivalued problem, defined in an appropriate phase space. Then, sufficient criteria for the existence of solutions are established for the convex case of the given problem using the nonlinear Leray–Schauder alternative type, while Covitz and Nadler’s theorem is applied for nonconvex multivalued functions. Finally, the results are illustrated through examples. Full article

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

17 pages, 1608 KiB

Open AccessArticle

Dynamics of Fractional-Order Three-Species Food Chain Model with Vigilance Effect

by Vinoth Seralan, Rajarathinam Vadivel, Nallappan Gunasekaran and Taha Radwan

Fractal Fract. 2025, 9(1), 45; https://doi.org/10.3390/fractalfract9010045 - 16 Jan 2025

Viewed by 570

Abstract

This study examines a Caputo-type fractional-order food chain model, considering the Holling type II functional response with the vigilance effect. The model explores the interaction dynamics of the food chain model, which consists of prey, middle predators, and top predators. Additionally, habitat complexity [...] Read more.

This study examines a Caputo-type fractional-order food chain model, considering the Holling type II functional response with the vigilance effect. The model explores the interaction dynamics of the food chain model, which consists of prey, middle predators, and top predators. Additionally, habitat complexity is integrated into the model, which is assumed to reduce predation rates by lowering the encounter rates between predators and prey. All possible feasible equilibrium points are determined and the stability of our proposed model is explored near the equilibrium points. To support the analytical findings, numerical simulation results are given in terms of time series, phase portraits, and bifurcation diagrams. It is discovered that the proposed model can become more stable under a fractional-order derivative. Moreover, the interplay between the vigilance effect and habitat complexity is shown to influence the existence of stable and periodic dynamics. Full article

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18 pages, 4101 KiB

Open AccessArticle

Design and Optimization Application of Cut Blasting Parameters for One-Time Completion of Blind Shaft

by Yifeng Zhang, Yongsheng Jia, Nan Jiang, Quanming Xie, Lin Yuan, Yongbo Wu and Zehui Xu

Fractal Fract. 2025, 9(1), 44; https://doi.org/10.3390/fractalfract9010044 - 16 Jan 2025

Viewed by 534

Abstract

The one-time completion blasting technology for blind shafts is widely used in underground mining, for safety reasons. Efficient blind shaft excavation relies on reasonable cutting blasting technology. To optimize blasting parameters, the impact of explosion stress waves and gases on rock fragmentation is [...] Read more.

The one-time completion blasting technology for blind shafts is widely used in underground mining, for safety reasons. Efficient blind shaft excavation relies on reasonable cutting blasting technology. To optimize blasting parameters, the impact of explosion stress waves and gases on rock fragmentation is quantitatively analyzed using explosion stress wave theory. A calculation model for the radius R₁ of the crushed zone and the radius R₂ of the fractured zone in rock under the combined action of borehole cutting stress waves and blasting gases is derived and established. Combined with practical engineering examples and the determination method of compensation coefficient C_f, three types of linear cutting patterns, namely six-hole bucket cutting, seven-hole bucket cutting, and nine-hole bucket cutting, are designed. The post-blasting cavity volume and crack length of these three different cutting methods are calculated and analyzed using numerical simulation. Quantitative description of the distribution pattern of blasting-induced cracks in the simulation results of three cutting methods using the box-counting fractal dimension method are presented. Based on this analysis, the nine-hole bucket cutting is selected as the optimal scheme and validated through field application of cutting blasting. The results indicate that the nine-hole bucket cutting blasting scheme for one-time completion of blind shafts, with a designed hole depth of 8 m and a blasthole utilization rate of 93.7%, is an efficient and reasonable technical solution. Full article

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9 pages, 495 KiB

Open AccessArticle

Reconstructing the Heat Transfer Coefficient in the Inverse Fractional Stefan Problem

by Agata Chmielowska, Rafał Brociek and Damian Słota

Fractal Fract. 2025, 9(1), 43; https://doi.org/10.3390/fractalfract9010043 - 16 Jan 2025

Viewed by 516

Abstract

This paper presents an algorithm for solving the inverse fractional Stefan problem. The considered inverse problem consists of determining the heat transfer coefficient at one of the boundaries of the considered region. The additional information necessary for solving the inverse problem is the [...] Read more.

This paper presents an algorithm for solving the inverse fractional Stefan problem. The considered inverse problem consists of determining the heat transfer coefficient at one of the boundaries of the considered region. The additional information necessary for solving the inverse problem is the set of temperature values in selected points of the region. The fractional derivative with respect to time used in the considered Stefan problem is of the Caputo type. The direct problem was solved by using the alternating phase truncation method adapted to the model with the fractional derivative. To solve the inverse problem, the ant colony algorithm was used. This paper contains an example illustrating the accuracy and stability of the presented algorithm. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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Figure 1
Scheme of the two-phase problem. Full article ">Figure 2
The exact and disturbed input data presented for the measurements taken every <math display="inline"><semantics> <mrow> <mn>1</mn> <mspace width="0.166667em"/> </mrow> </semantics></math>s. Full article ">Figure 3
The reconstructed values of the heat transfer coefficient h for the exact measurement data (a) and data disturbed by the 2% error (b) for different numbers of measurements. Full article ">Figure 4
The temperature course at the measurement point for the measurements disturbed by the 1% error (a) and 2% error (b) (measurements taken every <math display="inline"><semantics> <mrow> <mn>1</mn> <mspace width="0.166667em"/> </mrow> </semantics></math>s; solid line—the exact course; dotted line—the reconstructed course). Full article ">Figure 5
The temperature distribution at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>100</mn> <mspace width="0.166667em"/> </mrow> </semantics></math>s for the exact measurements (a) and the measurements disturbed by the 1% error (b) (measurements taken every <math display="inline"><semantics> <mrow> <mn>1</mn> <mspace width="0.166667em"/> </mrow> </semantics></math>s; solid line—the exact course; dotted line—the reconstructed course). Full article ">

14 pages, 278 KiB

Open AccessArticle

Existence and Uniqueness Results for a Class of Fractional Integro-Stochastic Differential Equations

by Ayed. R. A. Alanzi, Shokrya S. Alshqaq, Raouf Fakhfakh and Abdellatif Ben Makhlouf

Fractal Fract. 2025, 9(1), 42; https://doi.org/10.3390/fractalfract9010042 - 15 Jan 2025

Viewed by 519

Abstract

The objective of this paper is to demonstrate the existence and uniqueness (EU) of solutions to a class of Fractional Integro-Stochastic Differential Equations (FISDEs) by utilizing the fixed-point technique (FPT) and stochastic techniques. Additionally, the paper proves the continuous dependence (CD) of solutions [...] Read more.

The objective of this paper is to demonstrate the existence and uniqueness (EU) of solutions to a class of Fractional Integro-Stochastic Differential Equations (FISDEs) by utilizing the fixed-point technique (FPT) and stochastic techniques. Additionally, the paper proves the continuous dependence (CD) of solutions on the initial data. We examine the Hyers–Ulam stability (HUS) of FISDEs by applying Gronwall inequalities. Two theoretical examples are presented to demonstrate our findings. Full article

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

29 pages, 5470 KiB

Open AccessArticle

Discrete-Time Design of Fractional Delay-Based Repetitive Controller with Sliding Mode Approach for Uncertain Linear Systems with Multiple Periodic Signals

by Edi Kurniawan, Azka M. Burrohman, Purwowibowo Purwowibowo, Sensus Wijonarko, Tatik Maftukhah, Jalu A. Prakosa, Dadang Rustandi, Enggar B. Pratiwi and Amaliyah Az-Zukhruf

Fractal Fract. 2025, 9(1), 41; https://doi.org/10.3390/fractalfract9010041 - 15 Jan 2025

Viewed by 558

Abstract

In this paper, a discrete-time design of a fractional internal model-based repetitive controller with a sliding mode approach is presented for uncertain linear systems subject to repetitive trajectory and periodic disturbance. The proposed algorithm, named a fractional delay-based repetitive sliding mode controller (FD-RSMC), [...] Read more.

In this paper, a discrete-time design of a fractional internal model-based repetitive controller with a sliding mode approach is presented for uncertain linear systems subject to repetitive trajectory and periodic disturbance. The proposed algorithm, named a fractional delay-based repetitive sliding mode controller (FD-RSMC), aims to enhance tracking accuracy, transient response, and robustness against parametric variations beyond what is offered by conventional repetitive controllers. First, a fractional delay-based repetitive controller (FD-RC) that allows the periodic delay steps to be noninteger is presented to improve the trajectory tracking accuracy and good disturbance compensation of multiple periodic signals. Second, a sliding mode control (SMC) with a discrete-time reaching law is systematically incorporated into FD-RC to improve transient response, especially during the learning period of FD-RC, and also to provide system robustness against model uncertainties. Finally, the stability proof of the closed-loop system with the proposed controller is assessed based on a delayed-sliding mode-reaching condition. Finally, comparative simulation studies are presented to demonstrate the superior performance of the proposed controller. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)

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24 pages, 21684 KiB

Open AccessArticle

An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets

by Khairul Habib Alam, Yumnam Rohen, Anita Tomar, Naeem Saleem, Maggie Aphane and Asima Razzaque

Fractal Fract. 2025, 9(1), 40; https://doi.org/10.3390/fractalfract9010040 - 15 Jan 2025

Viewed by 734

Abstract

This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity [...] Read more.

This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function

sin (u^{m}) - α u + β

, where

u, α,

β \in C

and

m \geq 2

. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractals)

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21 pages, 639 KiB

Open AccessArticle

Finite-Time Cluster Synchronization of Fractional-Order Complex-Valued Neural Networks Based on Memristor with Optimized Control Parameters

by Qi Chang, Rui Wang and Yongqing Yang

Fractal Fract. 2025, 9(1), 39; https://doi.org/10.3390/fractalfract9010039 - 14 Jan 2025

Viewed by 536

Abstract

The finite-time cluster synchronization (FTCS) of fractional-order complex-valued (FOCV) neural network has attracted wide attention. It is inconvenient and difficult to decompose complex-valued neural networks into real parts and imaginary parts. This paper addresses the FTCS of coupled memristive neural networks (CMNNs), which [...] Read more.

The finite-time cluster synchronization (FTCS) of fractional-order complex-valued (FOCV) neural network has attracted wide attention. It is inconvenient and difficult to decompose complex-valued neural networks into real parts and imaginary parts. This paper addresses the FTCS of coupled memristive neural networks (CMNNs), which are FOCV systems with a time delay. A controller is designed with a complex-valued sign function to achieve FTCS using a non-decomposition approach, which eliminates the need to separate the complex-valued system into its real and imaginary components. By applying fractional-order stability theory, some conditions are derived for FTCS based on the proposed controller. The settling time, related to the system’s initial values, can be computed using the Mittag–Leffler function. We further investigate the optimization of control parameters by formulating an optimization model, which is solved using particle swarm optimization (PSO) to determine the optimal control parameters. Finally, a numerical example and a comparative experiment are both provided to verify the theoretical results and optimization method. Full article

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17 pages, 2621 KiB

Open AccessArticle

Comparative Studies of Nonlinear Models and Their Applications to Magmatic Evolution and Crustal Growth of the Huai’an Terrane in the North China Craton

by Qiuming Cheng and Min Gao

Fractal Fract. 2025, 9(1), 38; https://doi.org/10.3390/fractalfract9010038 - 14 Jan 2025

Viewed by 546

Abstract

Power-law, inverse exponential and logarithmic models are widely used as empirical tools to describe anomalies in spatial and temporal geodynamic processes. However, the lack of clear interpretation of the relationships and distinctions among these models often makes their selection challenging, leaving them as [...] Read more.

Power-law, inverse exponential and logarithmic models are widely used as empirical tools to describe anomalies in spatial and temporal geodynamic processes. However, the lack of clear interpretation of the relationships and distinctions among these models often makes their selection challenging, leaving them as empirical tools to be validated by data. This paper introduces these nonlinear functions derived from a unified differential equation, with parameters that reflect their relative nonlinearities and singularities, enabling their comparative application. By applying these functions to analyze magmatic events of the Huai’an Terrane, this study reveals two major crustal growth and reworking events between 2.6 and 1.7 Ga, each exhibiting distinctive nonlinear characteristics. The power-law function highlights strong nonlinearity and singularity during phases of intense magmatic activity, while logarithmic and exponential functions effectively characterize transitions between different tectonic processes. Geochemical data, including U-Pb zircon dating and Lu-Hf isotopic analyses, further validate the models by delineating distinct phases of crustal growth and reworking within the Trans-North China Orogen. The findings help connect the anomalies of frequency of magmatic events with the tectonic processes, providing important insights into the evolution processes of the North China Craton. Full article

(This article belongs to the Section Engineering)

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17 pages, 345 KiB

Open AccessArticle

Numerical Algorithm for Coupled Fixed Points in Normed Spaces with Applications to Fractional Differential Equations and Economics

by Lifang Guo, Salha Alshaikey, Abeer Alshejari, Muhammad Din and Umar Ishtiaq

Fractal Fract. 2025, 9(1), 37; https://doi.org/10.3390/fractalfract9010037 - 14 Jan 2025

Viewed by 579

Abstract

This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This [...] Read more.

This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This class of operators expands on existing cyclic contractions, including interpolative Kannan mappings, interpolative Reich–Rus–Ćirić contractions, and other known contractions in the literature. We demonstrate the existence and uniqueness of fixed points for these operators and provide an example to illustrate our findings. Moreover, we discuss the applications of our results in solving nonlinear integral equations. Furthermore, we introduce the idea of a coupled interpolative enriched cyclic Reich–Rus–Ćirić operator and establish the existence of a strongly coupled fixed-point theorem for this contraction. Finally, we provide an application to fractional differential equations to show the validity of the main result. Full article

(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)

38 pages, 16379 KiB

Open AccessArticle

Hyperbolic Sine Function Control-Based Finite-Time Bipartite Synchronization of Fractional-Order Spatiotemporal Networks and Its Application in Image Encryption

by Lvming Liu, Haijun Jiang, Cheng Hu, Haizheng Yu, Siyu Chen, Yue Ren, Shenglong Chen and Tingting Shi

Fractal Fract. 2025, 9(1), 36; https://doi.org/10.3390/fractalfract9010036 - 13 Jan 2025

Viewed by 562

Abstract

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be [...] Read more.

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law’s outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application, 2nd Edition)

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24 pages, 3839 KiB

Open AccessArticle

Design of a Novel Fractional Whale Optimization-Enhanced Support Vector Regression (FWOA-SVR) Model for Accurate Solar Energy Forecasting

by Abdul Wadood, Hani Albalawi, Aadel Mohammed Alatwi, Hafeez Anwar and Tariq Ali

Fractal Fract. 2025, 9(1), 35; https://doi.org/10.3390/fractalfract9010035 - 11 Jan 2025

Viewed by 666

Abstract

This study presents a novel Fractional Whale Optimization Algorithm-Enhanced Support Vector Regression (FWOA-SVR) framework for solar energy forecasting, addressing the limitations of traditional SVR in modeling complex relationships within data. The proposed framework incorporates fractional calculus in the Whale Optimization Algorithm (WOA) to [...] Read more.

This study presents a novel Fractional Whale Optimization Algorithm-Enhanced Support Vector Regression (FWOA-SVR) framework for solar energy forecasting, addressing the limitations of traditional SVR in modeling complex relationships within data. The proposed framework incorporates fractional calculus in the Whale Optimization Algorithm (WOA) to improve the balance between exploration and exploitation during hyperparameter tuning. The FWOA-SVR model is comprehensively evaluated against traditional SVR, Long Short-Term Memory (LSTM), and Backpropagation Neural Network (BPNN) models using training, validation, and testing datasets. Experimental results show that FWOA-SVR achieves superior performance with the lowest MSE values (0.036311, 0.03942, and 0.03825), RMSE values (0.19213, 0.19856, and 0.19577), and the highest R² values (0.96392, 0.96104, and 0.96192) for training, validation, and testing, respectively. These results highlight the significant improvements of FWOA-SVR in prediction accuracy and efficiency, surpassing benchmark models in capturing complex patterns within the data. The findings highlight the effectiveness of integrating fractional optimization techniques into machine learning frameworks for advancing solar energy forecasting solutions. Full article

(This article belongs to the Special Issue Fractional-Order Learning Systems: Theory, Algorithms, and Emerging Applications)

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20 pages, 8752 KiB

Open AccessArticle

Fractional Electrodamage in A549 Human Lung Cancer Cells

by Hilario Martines-Arano, Jose Alberto Arano-Martinez, Manuel Alejandro Mosso-Pani, Alejandra Valdivia-Flores, Martin Trejo-Valdez, Blanca Estela García-Pérez and Carlos Torres-Torres

Fractal Fract. 2025, 9(1), 34; https://doi.org/10.3390/fractalfract9010034 - 10 Jan 2025

Viewed by 659

Abstract

Fractional electrodamage in A549 human lung cancer cells was analyzed by introducing a non-integer order parameter to model the influence of electrical stimulation on cellular behavior. Numerical simulations were conducted to evaluate the conversion of electrical energy to heat within A549 cancer cells, [...] Read more.

Fractional electrodamage in A549 human lung cancer cells was analyzed by introducing a non-integer order parameter to model the influence of electrical stimulation on cellular behavior. Numerical simulations were conducted to evaluate the conversion of electrical energy to heat within A549 cancer cells, emphasizing the electrocapacitive effects and electrical conductivity in modulating dielectric properties. Using the Riemann–Liouville fractional calculus framework, experimental results were accurately fitted, demonstrating the non-integer nature of electrodamage processes. The study identified a strong dependency of electrical behavior on frequency, revealing a critical role of fractional dynamics in the dielectric breakdown and susceptibility of A549 cells to voltage changes. These findings advance our understanding of cellular responses to electrical fields and provide insights into applications in cancer diagnostics, monitoring, and potential therapeutic treatments. Full article

(This article belongs to the Special Issue Fractal and Fractional Analysis in Biomedical Sciences and Engineering)

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21 pages, 16644 KiB

Open AccessArticle

A Time–Frequency Composite Recurrence Plots-Based Series Arc Fault Detection Method for Photovoltaic Systems with Different Operating Conditions

by Zhendong Yin, Hongxia Ouyang, Junchi Lu, Li Wang and Shanshui Yang

Fractal Fract. 2025, 9(1), 33; https://doi.org/10.3390/fractalfract9010033 - 8 Jan 2025

Viewed by 573

Abstract

Series arc faults (SAFs) pose a significant threat to the safety of photovoltaic (PV) systems. However, the complex operating conditions of PV systems make accurate SAF detection challenging. To tackle this issue, this article proposes a SAF detection method based on time–frequency composite [...] Read more.

Series arc faults (SAFs) pose a significant threat to the safety of photovoltaic (PV) systems. However, the complex operating conditions of PV systems make accurate SAF detection challenging. To tackle this issue, this article proposes a SAF detection method based on time–frequency composite recurrence plots (TFCRPs). Initially, variational mode decomposition (VMD) is employed to decompose the current into distinct modes. Subsequently, the proposed TFCRP transforms these modes into two-dimensional matrices, enabling the measurement of composite similarity between different phase states. Lastly, extra tree (ET) is utilized to fuse the fractional recurrence entropy (FRE) and the singular values extracted from the matrices, thereby achieving SAF detection. Experimental results indicate that the proposed method achieves a detection accuracy of 98.75% and can accurately detect SAFs under various operating conditions. Comparisons with different methods further highlight the advancement of the proposed method. Furthermore, the detection time of the proposed method (209 ms) meets the requirements of standard UL1699B. Full article

(This article belongs to the Special Issue Implementations and Applications of Algorithms Based on Fractional Calculus to Engineering Problems)

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18 pages, 4905 KiB

Open AccessArticle

A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams

by Mariela C. Bravo-Sánchez, Luis M. Palacios-Pineda, José L. Gómez-Color, Oscar Martínez-Romero, Imperio A. Perales-Martínez, Daniel Olvera-Trejo, Jorge A. Estrada-Díaz and Alex Elías-Zúñiga

Fractal Fract. 2025, 9(1), 32; https://doi.org/10.3390/fractalfract9010032 - 8 Jan 2025

Viewed by 576

Abstract

This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of [...] Read more.

This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of the dynamic response behavior of two low-density foams to sustain impact loads, fractional differential equations of motion are used to qualitatively and quantitatively describe the dynamic response behavior, assuming restoring forces for each foam characterized, respectively, by a polynomial of heptic degree and by a trigonometric tangential function. A two-scale transform is employed to solve the mathematical model and predict the material’s behavior under impact loads, accounting for the fractal structure of the material’s molecular configuration. To assess the accuracy of the mathematical model, we performed impact tests considering eight dropping heights and two plate weights. We found good predictions from the mathematical models compared to experimental data when the fractal derivatives were between 1.86 and 1.9, depending on the cushioning material used. The accuracy of the theoretical predictions achieved using fractal calculus elucidates how to predict multiscale phenomena associated with foam heterogeneity across space, density, and average pore size, which influence the foam chain’s molecular motion during impact loading conditions. Full article

(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications)

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24 pages, 2990 KiB

Open AccessArticle

Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers

by Sarfaraz Ahmed, Ujala Rehman, Jianbo Fei, Muhammad Irslan Khalid and Xiangsheng Chen

Fractal Fract. 2025, 9(1), 31; https://doi.org/10.3390/fractalfract9010031 - 8 Jan 2025

Viewed by 710

Abstract

A nonlinear

(3 + 1)

-dimensional nonlinear Geng equation that can be utilized to explain the dynamics of shallow-water waves in fluids is given special attention. Various wave solutions are produced with the aid of the Hirota bilinear and Cole–Hopf transformation [...] Read more.

A nonlinear

(3 + 1)

-dimensional nonlinear Geng equation that can be utilized to explain the dynamics of shallow-water waves in fluids is given special attention. Various wave solutions are produced with the aid of the Hirota bilinear and Cole–Hopf transformation techniques. By selecting the appropriate polynomial function and implementing the distinct transformations in bilinear form, bright lump waves, dark lump waves, and rogue waves (RWs) are generated. A positive quadratic transformation and cosine function are combined in Hirota bilinear form to evaluate the RW solutions. Typically, RWs have crests that are noticeably higher than those of surrounding waves. These waves are also known as killer, freak, or monster waves. The lump periodic solutions (LPSs) are obtained using a combination of the cosine and positive quadratic functions. The lump-one stripe solutions are computed by using a mix of positive quadratic and exponential transformations to the governing equation. The lump two-stripe solutions are obtained by using a mix of positive quadratic and exponential transformations to the governing equation. The interactional solutions of lump, kink, and periodic wave solutions are obtained. Additionally, mixed solutions with butterfly waves, X-waves and lump waves are computed. The Ma breather (MB), Kuznetsov–Ma breather (KMB), and generalized breathers GBs are generated. Furthermore, solitary wave solution is obtained and a relation for energy of the wave via ansatz function technique. Full article

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22 pages, 16370 KiB

Open AccessArticle

The Unbalanced Control Research of Fractional-Order Cascaded H-Bridge Multilevel STATCOM

by Junhua Xu, Guopeng He, Songqin Tang, Zheng Gong, Chunwei Wang and Yue Lan

Fractal Fract. 2025, 9(1), 30; https://doi.org/10.3390/fractalfract9010030 - 7 Jan 2025

Viewed by 617

Abstract

Recent research on fractional-order cascaded H-bridge multilevel static compensator (FCHM-STATCOM) indicates that it has better performance than the traditional cascaded H-bridge multilevel static compensator (CHM-STATCOM). The existing FCHM-STATCOM control system lacks some special control links for dealing with unbalanced operative situations, which is [...] Read more.

Recent research on fractional-order cascaded H-bridge multilevel static compensator (FCHM-STATCOM) indicates that it has better performance than the traditional cascaded H-bridge multilevel static compensator (CHM-STATCOM). The existing FCHM-STATCOM control system lacks some special control links for dealing with unbalanced operative situations, which is demanded by power systems. This paper improves the existing FCHM-STATCOM control system to satisfy the demand for power systems, creating the potential for it to be applied in the electrical industry. The improvement of the FCHM-STATCOM control system is based on traditional CHM-STATCOM control systems, introducing special control loops for unbalanced operative situations. The improved FCHM-STATCOM control system constructed in this paper consists of an outer control loop that can apply special strategies for different control objectives, two inner control loops, respectively, for positive- and negative-sequence currents, and an inter-phase balancing control loop for balancing its arm operation. At the end of this paper, the results of digital simulations verify the FCHM-STATCOM complete control system’s capacity to regulate its negative-sequence currents and balance its arm operation to deal with unbalanced operative situations. Moreover, based on different strategies, the control system shows effectiveness in eliminating power oscillations and current balancing. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing, 2nd Edition)

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19 pages, 10969 KiB

Open AccessArticle

Fish-Tail Structured Fractal Monopole Printed Antenna with Dual Broadband Characteristics for Sub–6GHz 5G and X–Band Radar Applications

by Guntamukkala Yaminisasi, Pokkunuri Pardhasaradhi, Nagandla Prasad, Boddapati Taraka Phani Madhav, Abeer D. Algarni, Sudipta Das and Mohammed El Ghzaoui

Fractal Fract. 2025, 9(1), 29; https://doi.org/10.3390/fractalfract9010029 - 7 Jan 2025

Viewed by 652

Abstract

This article presents a printed antenna, designed with a fractal-shaped patch with fish-tail structured outer edges, a tapered feedline, and a rectangular notch-based defected partial ground structure (DPGS). The presented design has been printed on a FR-4 substrate, which has a dielectric constant [...] Read more.

This article presents a printed antenna, designed with a fractal-shaped patch with fish-tail structured outer edges, a tapered feedline, and a rectangular notch-based defected partial ground structure (DPGS). The presented design has been printed on a FR-4 substrate, which has a dielectric constant of 4.4 and a loss tangent of 0.035. The overall dimension of the proposed antenna is 24 × 40 × 1.6 mm³. The proposed fractal antenna achieved dual broad-band functionality by maintaining the compact size of the radiator. The designed fractal radiator can operate at three distinct resonant frequencies (3.22, 7.64, and 9.41 GHz), covering two distinct frequency bands, extending from 2.5 to 4.2 GHz and 7 to 9.8 GHz. A thorough parametric analysis has been carried out using CST Studio suite 2019 licensed version to achieve better performance in terms of S₁₁ (dB), radiation efficiency, and gain over the operating frequency range. The operating bands fall within the S, C, and X bands to support sub-6GHz 5G and Radar applications at the microwave frequency range. Full article

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21 pages, 1282 KiB

Open AccessArticle

Computational Study of a Fractional-Order HIV Epidemic Model with Latent Phase and Treatment

by Sana Abdulkream Alharbi and Nada A. Almuallem

Fractal Fract. 2025, 9(1), 28; https://doi.org/10.3390/fractalfract9010028 - 7 Jan 2025

Viewed by 602

Abstract

In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is [...] Read more.

In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is evaluated. We analyze the model’s global asymptotic stability using the Lyapunov function and LaSalle’s invariance principle. To identify the primary factors affecting the dynamics of HIV/AIDS, a sensitivity analysis of the model parameters is conducted. We also examine a fractional-order HIV model using the Caputo fractional differential operator. Through qualitative analysis and applications, we determine the existence and uniqueness of the model’s solutions. We derive some results from the fixed-point theorem and Ulam–Hyers stability. Ultimately, the obtained numerical simulation results are in agreement with the analytical outcomes obtained from the model analysis. Our findings illustrate the efficacy of the fractional model in depicting the dynamics of the HIV/AIDS epidemic and offering critical insights for the formulation of effective control strategies. The results show that early intervention and treatment in the latent phase of infection can decrease the spread of the disease and its progression to AIDS, as well as increase the success of treatment strategies. Full article

(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)

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40 pages, 3314 KiB

Open AccessReview

Multifractal Applications in Hydro-Climatology: A Comprehensive Review of Modern Methods

by Shamseena Vahab and Adarsh Sankaran

Fractal Fract. 2025, 9(1), 27; https://doi.org/10.3390/fractalfract9010027 - 6 Jan 2025

Viewed by 963

Abstract

Complexity evaluation of hydro-climatic datasets is a challenging but essential pre-requisite for accurate modeling and subsequent planning. Changes in climate and anthropogenic interventions amplify the complexity of hydro-climatic time-series. Understanding persistence and fractal features may help us to develop new and robust modeling [...] Read more.

Complexity evaluation of hydro-climatic datasets is a challenging but essential pre-requisite for accurate modeling and subsequent planning. Changes in climate and anthropogenic interventions amplify the complexity of hydro-climatic time-series. Understanding persistence and fractal features may help us to develop new and robust modeling frameworks which can work well under non-stationary and non-linear environments. Classical fractal hydrology, rooted in statistical physics, has been developed since the 1980s and the modern alternatives based on de-trending, complex network, and time–frequency principles have been developed since 2002. More specifically, this review presents the procedures of Multifractal Detrended Fluctuation Analysis (MFDFA) and Arbitrary Order Hilbert Spectral Analysis (AOHSA), along with their applications in the field of hydro-climatology. Moreover, this study proposes a complex network-based fractal analysis (CNFA) framework for the multifractal analysis of daily streamflows as an alternative. The case study proves the efficacy of CNMFA and shows that it has the flexibility to be applied in visibility and inverted visibility schemes, which is effective in complex datasets comprising both high- and low-amplitude fluctuations. The comprehensive review showed that more than 75% of the literature focuses on characteristic analysis of the time-series using MFDFA rather than modeling. Among the variables, about 70% of studies focused on analyzing fine-resolution streamflow and rainfall datasets. This study recommends the use of CNMF in hydro-climatology and advocates the necessity of knowledge integration from multiple fields to enhance the multifractal modeling applications. This study further asserts that transforming the characterization into operational hydrology is highly warranted. Full article

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Diagram showing the connections of Peng (1994) [<a href="#B18-fractalfract-09-00027" class="html-bibr">18</a>] and Kantelhardt et al. (2002) [<a href="#B19-fractalfract-09-00027" class="html-bibr">19</a>] which laid the foundation of detrended fluctuation for fractal analysis. The radius of the circles is in proportion to the citation count based on Google scholar data. The connected papers’ diagram is generated using <a href="https://www.connectedpapers.com/" target="_blank">https://www.connectedpapers.com/</a>, accessed on 23 December 2024. Full article ">Figure 2
The overall methodology for the multifractal analysis of streamflow data using CNMF framework. In the symbolic plots of degree distribution, Renyi graph and singularity graph, the blue colored dots represents the results by VG analysis and red colored dots represents the results by UDVG scheme. Full article ">Figure 3
The Degree Distribution Curves for the overall streamflow data analysis for the selected stations of (a) Perumannu, (b) Pattazhy and (c) Ramamangalam. Full article ">Figure 4
The Renyi graphs and singularity spectra of the complete length daily streamflow data of the selected stations (a) Perummanu (b) Pattazhy and (c) Ramamangalam by VG and UDVG methods. The upper panels show the Renyi graphs and lower panels show the singularity spectra. Full article ">Figure 5
The UDVG Renyi Graphs and singularity graphs for the stream gauge stations of (a) Perumannu (b) Pattazhy and (c) Ramamangalam for seasonal flows. Full article ">Figure 6
Human Water Climate nexus. Full article ">Figure 7
Citations received by based papers of modern multifractal algorithms. Full article ">Figure 8
Keywords for cross-cutting research in hydro-complexity. Full article ">

17 pages, 4370 KiB

Open AccessArticle

Discrete Element Study of Particle Size Distribution Shape Governing Critical State Behavior of Granular Material

by Mingdong Jiang, Daniel Barreto, Zhi Ding and Kaifang Yang

Fractal Fract. 2025, 9(1), 26; https://doi.org/10.3390/fractalfract9010026 - 6 Jan 2025

Viewed by 709

Abstract

Granular soil is a porous medium composed of particles with different sizes and self-similar structures, exhibiting fractal characteristics. It is well established that variations in these fractal properties, such as particle size distribution (PSD), significantly influence the mechanical behavior of the soil. In [...] Read more.

Granular soil is a porous medium composed of particles with different sizes and self-similar structures, exhibiting fractal characteristics. It is well established that variations in these fractal properties, such as particle size distribution (PSD), significantly influence the mechanical behavior of the soil. In this paper, a three-dimensional (3D) Discrete Element Method (DEM) is applied to study the mechanical and critical-state behavior of the idealized granular assemblages, in which various PSD shape parameters are considered, including the coefficient of uniformity (C_u), the coefficient of curvature (C_c), and the coefficient of size span (C_s). In addition, the same PSDs but with different mean particle sizes (D₅₀) are also employed in the numerical simulations to examine the particle size effect on the mechanical behavior of the granular media. Numerical triaxial tests are carried out by imposing axial compression under constant mean effective pressure conditions. A unique critical-state stress ratio in p′-q space is observed, indicating that the critical friction angle is independent of the shape of the PSD. However, in the e-p′ plane, the critical state line (CSL) shifts downward and rotates counterclockwise, as the grading becomes more widely distributed, i.e., the increasing coefficient of span (C_s). Additionally, a decrease in the coefficient of curvature (C_c) would also move the CSL downward but with negligible rotation. However, it is found that the variations in the mean particle size (D₅₀) and coefficient of uniformity (C_u) do not affect the position of the CSL in the e-p′ plane. The numerical findings may shed some light on the development of constitutive models of sand that undergo variations in the grading due to crushing and erosion, and address fractal problems related to micro-mechanics in soils. Full article

(This article belongs to the Special Issue Fractal and Fractional Models in Soil Mechanics)

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Fractal Fract., Volume 9, Issue 1 (January 2025) – 55 articles

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