Fractal and Fractional

19 pages, 558 KB

Open AccessArticle

New Jacobi Galerkin Operational Matrices of Derivatives: A Highly Accurate Method for Solving Two-Point Fractional-Order Nonlinear Boundary Value Problems with Robin Boundary Conditions

by Hany Mostafa Ahmed

Fractal Fract. 2025, 9(11), 686; https://doi.org/10.3390/fractalfract9110686 (registering DOI) - 24 Oct 2025

Abstract

A novel numerical scheme is developed in this work to approximate solutions (APPSs) for nonlinear fractional differential equations (FDEs) governed by Robin boundary conditions (RBCs). The methodology is founded on a spectral collocation method (SCM) that uses a set of basis functions derived [...] Read more.

A novel numerical scheme is developed in this work to approximate solutions (APPSs) for nonlinear fractional differential equations (FDEs) governed by Robin boundary conditions (RBCs). The methodology is founded on a spectral collocation method (SCM) that uses a set of basis functions derived from generalized shifted Jacobi (GSJ) polynomials. These basis functions are uniquely formulated to satisfy the homogeneous form of RBCs (HRBCs). Key to this approach is the establishment of operational matrices (OMs) for ordinary derivatives (Ods) and fractional derivatives (Fds) of the constructed polynomials. The application of this framework effectively reduces the given FDE and its RBC to a system of nonlinear algebraic equations that are solvable by standard numerical routines. We provide theoretical assurances of the algorithm’s efficacy by establishing its convergence and conducting an error analysis. Finally, the efficacy of the proposed algorithm is demonstrated through three problems, with our APPSs compared against exact solutions (ExaSs) and existing results by other methods. The results confirm the high accuracy and efficiency of the scheme. Full article

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

12 pages, 8321 KB

Open AccessArticle

Design of High-Gain Linear Polarized Fabry–Perot Antenna Based on Minkowski Fractal Structure

by Wei Hu, Liangfu Peng, Tao Tang, Maged A. Aldhaeebi and Thamer S. Almoneef

Fractal Fract. 2025, 9(11), 685; https://doi.org/10.3390/fractalfract9110685 - 24 Oct 2025

Abstract

High-gain linear polarized antennas are widely used in wireless communications. However, the insertion loss of the feed network increases, limiting the potential for enhancing antenna gain. In this paper, a high-gain linear polarized Fabry–Perot (FP) antenna based on a fractal structure, which consisted [...] Read more.

High-gain linear polarized antennas are widely used in wireless communications. However, the insertion loss of the feed network increases, limiting the potential for enhancing antenna gain. In this paper, a high-gain linear polarized Fabry–Perot (FP) antenna based on a fractal structure, which consisted of a metasurface and 2 × 2 array antenna structure, was designed. The spacing between the metasurface structure and array antenna was a free space half-wavelength, forming an FP antenna with a high gain. The self-similarity of the fractal structure allowed miniaturization of the structure. The proposed antenna and metasurface structural units comprised a first-order Minkowski fractal structure. The antenna unit was further miniaturized by including a square gap structure in its unit structure, while its gain was improved by using an air dielectric layer as the dielectric substrate of the antenna unit. The antenna unit formed a 2 × 2 array antenna through a 1–4 feeding network. The reliability of the array antenna performance was verified by processing and measuring the antenna structure. Experimental results showed that the −10 dB working bandwidth of the antenna is 5.71–5.89 GHz, while at 5.8 GHz its gain is 16.5 dBi. The radiation efficiency is over 90%. The experimental results were consistent with the simulation results. The proposed antenna exhibits high gain and is suitable for short-distance wireless communication systems and other fields. Full article

(This article belongs to the Section Engineering)

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23 pages, 6605 KB

Open AccessArticle

Wintertime Cross-Correlational Structures Between Sea Surface Temperature Anomaly and Atmospheric-and-Oceanic Fields in the East/Japan Sea Under Arctic Oscillation

by Gyuchang Lim and Jong-Jin Park

Fractal Fract. 2025, 9(11), 684; https://doi.org/10.3390/fractalfract9110684 (registering DOI) - 23 Oct 2025

Abstract

The winter Arctic Oscillation (AO) modulates the East Asian climate and the East/Japan Sea (EJS) thermodynamics, yet the local, scale-dependent air–sea couplings remain unclear. Using 30 years of daily fields (1993–2022), we map at each grid point, the cross-persistence and scale-dependent cross-correlations between [...] Read more.

The winter Arctic Oscillation (AO) modulates the East Asian climate and the East/Japan Sea (EJS) thermodynamics, yet the local, scale-dependent air–sea couplings remain unclear. Using 30 years of daily fields (1993–2022), we map at each grid point, the cross-persistence and scale-dependent cross-correlations between sea surface temperature anomalies (SSTA) and (i) atmospheric anomalies, (ii) turbulent heat-flux anomalies (sensible and latent), and (iii) oceanic anomalies. Detrended Fluctuation/Cross-Correlation Analyses (DFA/DCCA, 5–50 days) yield the Hurst exponent (

H, h_{X Y}

) and the DCCA coefficient (

ρ_{d c c a}

). Significance is assessed with iterative-AAFT surrogates and Benjamini–Hochberg false discovery rate (FDR). Three robust features emerge: (1) during AO+, the East Korean Bay–Subpolar Front corridor shows large SSTA variance and high long-term memory (

H \geq

1.5); (2) turbulent heat-flux anomalies are anti-phased with SSTA and show little cross-persistence; (3) among oceanic fields, SSHA and meridional geostrophic velocity provide the most AO-robust positive coupling. Within a fractal frame, DFA slopes (

1 < H < 2

) quantify local self-similarity; interpreting winter anomalies as fBm implies a fractal-dimension proxy

D = 3 - H

, so AO+ hot spots exhibit

D \leq 1.5

. These fractal maps, together with

ρ_{d c c a}

, offer a compact way to pre-locate marine-heatwave-prone regions. The grid-point, FDR-controlled DFA/DCCA approach is transferable to other marginal seas. Full article

(This article belongs to the Special Issue Time-Fractal and Fractional Models in Physics and Engineering)

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15 pages, 277 KB

Open AccessArticle

Finite-Time Stability for a Class of Fractional Itô–Doob Stochastic Time Delayed Systems

by Wissam Ghoul, Hussien Albala, Hamid Boulares, Faycal Bouchelaghem and Abdelkader Moumen

Fractal Fract. 2025, 9(11), 683; https://doi.org/10.3390/fractalfract9110683 - 23 Oct 2025

Abstract

This paper addresses the finite-time stability of a class of fractional Itô–Doob stochastic systems with time delays. Novel stability criteria are established using a combination of Gronwall-type, Hölder’s, and Burkholder–Davis–Gundy (BDG) inequalities, thereby generalizing classical integer-order stability theory to the fractional domain. Furthermore, [...] Read more.

This paper addresses the finite-time stability of a class of fractional Itô–Doob stochastic systems with time delays. Novel stability criteria are established using a combination of Gronwall-type, Hölder’s, and Burkholder–Davis–Gundy (BDG) inequalities, thereby generalizing classical integer-order stability theory to the fractional domain. Furthermore, the analysis uniquely integrates stochastic perturbations and time delays, providing a comprehensive framework for systems exhibiting both memory and randomness. The effectiveness of the proposed approach is demonstrated through a numerical example of a three-dimensional stochastic delayed system with fractional dynamics. Full article

(This article belongs to the Special Issue Stochastic Models, Fractional Calculus and Non-Local Operators: Theoretical Results and Applications)

38 pages, 13235 KB

Open AccessArticle

Hardware-in-the-Loop Experimental Validation of a Fault-Tolerant Control System for Quadcopter UAV Motor Faults

by Muhammad Abdullah, Adil Zulfiqar, Muhammad Zeeshan Babar, Jamal Hussain Arman, Ghulam Hafeez, Ahmed S. Alsafran and Muhyaddin Rawa

Fractal Fract. 2025, 9(11), 682; https://doi.org/10.3390/fractalfract9110682 - 23 Oct 2025

Abstract

In this paper, a hybrid fault-tolerant control (FTC) system for quadcopter unmanned aerial vehicles (UAVs) is proposed to counteract the deterioration of the performance of the quadcopter due to motor faults. A robust and adaptive approach to controlling fault conditions is simulated by [...] Read more.

In this paper, a hybrid fault-tolerant control (FTC) system for quadcopter unmanned aerial vehicles (UAVs) is proposed to counteract the deterioration of the performance of the quadcopter due to motor faults. A robust and adaptive approach to controlling fault conditions is simulated by combining an integral back-stepping controller for translational motion and a nonlinear observer-based sliding-mode controller for rotational motion, and then implemented on an FPGA. Finally, motor faults are treated as disturbances and are successfully compensated by the controller to ensure safe and high-performance flight. Simulations were taken at 0%, 10%, 30%, and 50% motor faults to test how effective the proposed FTC system is. After simulations, the controller’s real-time performance and reliability were validated through hardware-in-the-loop (HIL) experiments. The results validated that the proposed hybrid controller can guarantee stable flight and precision tracking of the desired trajectory when any single motor fails up to the order of 50%. It shows that the controller is of high fault tolerance and robustness, which will be a potential solution for improving the reliability of UAVs in fault-prone conditions. Full article

(This article belongs to the Special Issue Advanced Fractional Sliding Mode Control for Dynamic and Robotic Systems)

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19 pages, 2412 KB

Open AccessArticle

Stability Analysis and Bifurcation Control of a Fractional Order Predator Prey System

by Zhue Wei, Fufeng Wu, Guangming Xue, Funing Lin and Heng Liu

Fractal Fract. 2025, 9(11), 681; https://doi.org/10.3390/fractalfract9110681 - 23 Oct 2025

Abstract

Fractional-order predator–prey models provide superior ecological fidelity by capturing intrinsic memory effects. However, the fractional derivative order introduces an additional dimension to parameter space, which is unexplored in existing dynamical analyses. This paper delves into the bifurcation behaviors of a two-dimensional fractional-order predator–prey [...] Read more.

Fractional-order predator–prey models provide superior ecological fidelity by capturing intrinsic memory effects. However, the fractional derivative order introduces an additional dimension to parameter space, which is unexplored in existing dynamical analyses. This paper delves into the bifurcation behaviors of a two-dimensional fractional-order predator–prey model, taking into account the maturation period of larvae. By utilizing time delay as the bifurcation parameter, the characteristic equation is analyzed to derive conditions for stability and bifurcation. Meanwhile, the introduction to fractional-order theory endows the predator–prey system with improved historical dependence. The relationship between order and bifurcation behavior in the fractional-order predator–prey model is discussed in simulation experiments, which provides a reference for the establishment of the fractional-order system. Additionally, this paper introduces a controller to control the bifurcation behavior through feedback-gain parameters. Simulation results not only exhibit the dynamic characteristics of fractional-order models, but also demonstrate the success of controllers over bifurcation behaviors. Full article

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17 pages, 340 KB

Open AccessArticle

Semi-Rings, Semi-Vector Spaces, and Fractal Interpolation

by Peter Massopust

Fractal Fract. 2025, 9(11), 680; https://doi.org/10.3390/fractalfract9110680 - 23 Oct 2025

Abstract

In this paper, we introduce fractal interpolation on complete semi-vector spaces. This approach is motivated by the requirements of the preservation of positivity or monotonicity of functions for some models in approximation and interpolation theory. The setting in complete semi-vector spaces does not [...] Read more.

In this paper, we introduce fractal interpolation on complete semi-vector spaces. This approach is motivated by the requirements of the preservation of positivity or monotonicity of functions for some models in approximation and interpolation theory. The setting in complete semi-vector spaces does not requite additional assumptions but is intrinsically built into the framework. For the purposes of this paper, fractal interpolation in the complete semi-vector spaces

C^{+}

and

L_{p}^{+}

is considered. Full article

(This article belongs to the Special Issue Applications of Fractal Interpolation in Mathematical Functions)

23 pages, 3312 KB

Open AccessArticle

Automatic Picking Method for the First Arrival Time of Microseismic Signals Based on Fractal Theory and Feature Fusion

by Huicong Xu, Kai Li, Pengfei Shan, Xuefei Wu, Shuai Zhang, Zeyang Wang, Chenguang Liu, Zhongming Yan, Liang Wu and Huachuan Wang

Fractal Fract. 2025, 9(11), 679; https://doi.org/10.3390/fractalfract9110679 - 23 Oct 2025

Abstract

Microseismic signals induced by mining activities often have low signal-to-noise ratios, and traditional picking methods are easily affected by noise, making accurate identification of P-wave arrivals difficult. To address this problem, this study proposes an adaptive denoising algorithm based on wavelet-threshold-enhanced Complete Ensemble [...] Read more.

Microseismic signals induced by mining activities often have low signal-to-noise ratios, and traditional picking methods are easily affected by noise, making accurate identification of P-wave arrivals difficult. To address this problem, this study proposes an adaptive denoising algorithm based on wavelet-threshold-enhanced Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and develops an automatic P-wave arrival picking method incorporating fractal box dimension features, along with a corresponding accuracy evaluation framework. The raw microseismic signals are decomposed using the improved CEEMDAN method, with high-frequency intrinsic mode functions (IMFs) processed by wavelet-threshold denoising and low- and mid-frequency IMFs retained for reconstruction, effectively suppressing background noise and enhancing signal clarity. Fractal box dimension is applied to characterize waveform complexity over short and long-time windows, and by introducing fractal derivatives and short-long window differences, abrupt changes in local-to-global complexity at P-wave arrivals are revealed. Energy mutation features are extracted using the short-term/long-term average (STA/LTA) energy ratio, and noise segments are standardized via Z-score processing. A multi-feature weighted fusion scoring function is constructed to achieve robust identification of P-wave arrivals. Evaluation metrics, including picking error, mean absolute error, and success rate, are used to comprehensively assess the method’s performance in terms of temporal deviation, statistical consistency, and robustness. Case studies using microseismic data from a mining site show that the proposed method can accurately identify P-wave arrivals under different signal-to-noise conditions, with automatic picking results highly consistent with manual labels, mean errors within the sampling interval (2–4 ms), and a picking success rate exceeding 95%. The method provides a reliable tool for seismic source localization and dynamic hazard prediction in mining microseismic monitoring. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering, Second Edition)

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21 pages, 416 KB

Open AccessArticle

On Generalized Wirtinger Inequalities for (k,ψ)-Caputo Fractional Derivatives and Applications

by Muhammad Samraiz, Humaira Javaid and Ishtiaq Ali

Fractal Fract. 2025, 9(11), 678; https://doi.org/10.3390/fractalfract9110678 - 22 Oct 2025

Abstract

The primary aim of this study is to establish new Wirtinger-type inequalities involving fractional derivatives, which are essential tools in analysis and applied mathematics. We derive generalized Wirtinger-type inequalities incorporating the

(k, ψ)

-Caputo fractional derivatives using Taylor’s expansion. The [...] Read more.

The primary aim of this study is to establish new Wirtinger-type inequalities involving fractional derivatives, which are essential tools in analysis and applied mathematics. We derive generalized Wirtinger-type inequalities incorporating the

(k, ψ)

-Caputo fractional derivatives using Taylor’s expansion. The inequalities are derived in

L_{p}

spaces

(p > 1)

through Hölder’s inequality. A detailed analytical discussion is provided to further examine the derived inequalities. The theoretical findings are validated through numerical examples and graphical representations. Furthermore, the novelty and applicability of the proposed technique are demonstrated through the applications of the resulting inequalities to derive new results related to the arithmetic–geometric mean inequality. Full article

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

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Fractal Fract., Volume 9, Issue 11 (November 2025) – 9 articles

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