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FracGrad: A Discretized Riemann–Liouville Fractional Integral Approach to Gradient Accumulation for Deep Learning
Fractional Bagley-Torvik Problem Analysis with Hilfer Fractional Derivatives and Pettis Function Space
Empirical Comparison of Neural Network Architectures for Prediction of Software Development Effort and Duration

Journal Description

Fractal and Fractional

Fractal and Fractional is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering, published monthly online by MDPI.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, and other databases.
Journal Rank: JCR - Q1 (Mathematics, Interdisciplinary Applications) / CiteScore - Q1 (Analysis)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 19.3 days after submission; acceptance to publication is undertaken in 2.8 days (median values for papers published in this journal in the second half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Impact Factor: 3.3 (2024); 5-Year Impact Factor: 3.2 (2024)

Imprint Information Journal Flyer Open Access ISSN: 2504-3110

Latest Articles

41 pages, 6499 KB

Open AccessArticle

Cascaded Optimized Fractional Controller for Green Hydrogen-Based Microgrids with Mitigating False Data Injection Attacks

by Nadia A. Nagem, Mokhtar Aly, Emad A. Mohamed, Aisha F. Fareed, Dokhyl M. Alqahtani and Wessam A. Hafez

Fractal Fract. 2026, 10(1), 55; https://doi.org/10.3390/fractalfract10010055 - 13 Jan 2026

Green hydrogen production and the use of fuel cells (FCs) in microgrid (MG) systems have become viable and feasible solutions due to their continuous cost reduction and advancements in technology. Furthermore, green hydrogen electrolyzers and FC can mitigate fluctuations in renewable energy generation [...] Read more.

Green hydrogen production and the use of fuel cells (FCs) in microgrid (MG) systems have become viable and feasible solutions due to their continuous cost reduction and advancements in technology. Furthermore, green hydrogen electrolyzers and FC can mitigate fluctuations in renewable energy generation and various demand-related disturbances. Proper incorporation of electrolyzers and FCs can enhance load frequency control (LFC) in MG systems. However, they are subjected to multiple false data injection attacks (FDIAs), which can deteriorate MG stability and availability. Moreover, most existing LFC control schemes—such as conventional PID-based methods, single-degree-of-freedom fractional-order controllers, and various optimization-based structures—lack robustness against coordinated and multi-point FDIAs, leading to significant degradation in frequency regulation performance. This paper presents a new, modified, multi-degree-of-freedom, cascaded fractional-order controller for green hydrogen-based MG systems with high fluctuating renewable and demand sources. The proposed LFC is a cascaded control structure that combines a 1+TID controller with a filtered fractional-order PID controller (FOPIDF), namely the cascaded 1+TID/FOPIDF LFC control. Furthermore, another tilt-integrator derivative electric vehicle (EV) battery frequency regulation controller is proposed to benefit from EVs installed in MG systems. The proposed cascaded 1+TID/FOPIDF LFC control and EV TID LFC methods are designed using the powerful capability of the exponential distribution optimizer (EDO), which determines the optimal set of design parameters, leading to guaranteed optimal performance. The effectiveness of the newly proposed cascaded 1+TID/FOPIDF LFC control and design approach employing multi-generational-based two-area MG systems is studied by taking into account a variety of projected scenarios of FDIAs and renewable/load fluctuation scenarios. In addition, performance comparisons with some featured controllers are provided in the paper. For example, in the case of fluctuation in RESs, the measured indices are as follows: ISE (1.079, 0.5306, 0.3515, 0.0104); IAE (15.011, 10.691, 9.527, 1.363); ITSE (100.613, 64.412, 53.649, 1.323); and ITAE (2120, 1765, 1683, 241.32) for TID, FOPID, FOTID, and proposed, respectively, which confirm superior frequency deviation mitigation using the proposed optimized cascaded 1+TID/FOPIDF and EV TID LFC control method. Full article

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28 pages, 647 KB

Open AccessArticle

A Generalized Fractional Legendre-Type Differential Equation Involving the Atangana–Baleanu–Caputo Derivative

by Muath Awadalla and Dalal Alhwikem

Fractal Fract. 2026, 10(1), 54; https://doi.org/10.3390/fractalfract10010054 - 13 Jan 2026

This paper introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem [...] Read more.

This paper introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem is reformulated as an equivalent Volterra integral equation, and existence and uniqueness of classical solutions are established via the Banach fixed-point theorem, supported by a proved Lipschitz estimate for the ABC derivative. A constructive solution representation is obtained through a Volterra–Neumann series, explicitly revealing the role of Mittag–Leffler functions. We prove that the fractional solutions converge uniformly to the classical Legendre polynomials as the fractional order approaches unity, with a quantitative convergence rate of order

O (1 - α)

under mild regularity assumptions on the Volterra kernel. A fully reproducible quadrature-based numerical scheme is developed, with explicit kernel formulas and implementation algorithms provided in appendices. Numerical experiments for the quadratic Legendre mode confirm the theoretical convergence and illustrate the smooth interpolation between fractional and classical regimes. An application to time-fractional diffusion in spherical coordinates demonstrates that the operator arises naturally in physical models, providing a mathematically consistent tool for extending classical angular analysis to fractional settings with memory. Full article

14 pages, 383 KB

Open AccessArticle

From Mathematics to Art: A Petri Net Representation of the Fibonacci Sequence and Its Fractal Geometry

by David Mailland and Iwona Grobelna

Fractal Fract. 2026, 10(1), 53; https://doi.org/10.3390/fractalfract10010053 - 13 Jan 2026

Mathematics, as Bertrand Russell noted, possesses both truth and beauty. In this work, we revisit the classical Fibonacci recurrence thanks to a minimal Petri net. Starting from a minimal layered construction that mirrors the second-order additive rule [...] Read more.

Mathematics, as Bertrand Russell noted, possesses both truth and beauty. In this work, we revisit the classical Fibonacci recurrence thanks to a minimal Petri net. Starting from a minimal layered construction that mirrors the second-order additive rule

F_{n} = F_{n - 1} + F_{n - 2}

, we show that the marking dynamics of the associated net generate a combinatorial triangle whose parity structure reveals a self-similar, Sierpiński-like pattern. To the best of our knowledge, this oblique fractal geometry has never been formally documented. We provide a formal definition of the underlying Petri net, analyse its computational properties, and explore the emergence of higher-order harmonics when token markings are considered modulo primes. The study highlights how a classical recurrence gives rise to previously unnoticed geometric regularities at the intersection of mathematics and art. Beyond its mathematical interest, the construction illustrates how minimal Petri net dynamics can be used as formal specification patterns for distributed, event-driven systems. Full article

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44 pages, 635 KB

Open AccessArticle

A Fractional q-Rung Orthopair Fuzzy Tensor Framework for Dynamic Group Decision-Making: Application to Smart City Renewable Energy Planning

by Muhammad Bilal, Chaoqian Li, A. K. Alzahrani and A. K. Aljahdali

Fractal Fract. 2026, 10(1), 52; https://doi.org/10.3390/fractalfract10010052 - 13 Jan 2026

In complex decision-making scenarios, such as smart city renewable energy project selection, decision-makers must contend with multi-dimensional uncertainty, conflicting expert opinions, and evolving temporal dynamics. This study introduces a novel Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT)-based group decision-making methodology that integrates the flexibility [...] Read more.

In complex decision-making scenarios, such as smart city renewable energy project selection, decision-makers must contend with multi-dimensional uncertainty, conflicting expert opinions, and evolving temporal dynamics. This study introduces a novel Fractional q-Rung Orthopair Fuzzy Tensor (Fq-ROFT)-based group decision-making methodology that integrates the flexibility of q-rung orthopair fuzzy sets with tensorial representation and fractional-order dynamics. The proposed framework allows for the modeling of positive and negative membership degrees in a multi-dimensional, time-dependent structure while capturing memory effects inherent in expert evaluations. A detailed case study involving six renewable energy alternatives and six criteria demonstrates the method’s ability to aggregate expert opinions, compute fractional dynamic scores, and provide robust, reliable rankings. Comparative analysis with existing approaches, including classical q-ROFSs, intuitionistic fuzzy sets, and weighted sum methods, highlights the superior discriminative power, consistency, and dynamic sensitivity of the Fq-ROFT approach. Sensitivity analysis confirms the robustness of the top-ranked alternatives under variations in expert weights and fractional orders and membership perturbations. The study concludes by discussing the advantages, limitations, and future research directions of the proposed methodology, establishing Fq-ROFT as a powerful tool for dynamic, high-dimensional, and uncertain group decision-making applications. Full article

18 pages, 3053 KB

Open AccessArticle

Dynamics and Chaos Analysis of the Fractional-Order Lü System Using a Hybrid Approach

by Mohamed Elbadri, Naseam Al-kuleab, Rania Saadeh, Mohamed Hafez and Mohamed A. Abdoon

Fractal Fract. 2026, 10(1), 51; https://doi.org/10.3390/fractalfract10010051 - 13 Jan 2026

In this study, an analysis of fractional-order Lü systems is performed through a framework approach consisting of analytical solution strategies in combination with numerical methods. On the analytical methodology front, the recently developed form of the new generalized differential transform method (NGDTM) is [...] Read more.

In this study, an analysis of fractional-order Lü systems is performed through a framework approach consisting of analytical solution strategies in combination with numerical methods. On the analytical methodology front, the recently developed form of the new generalized differential transform method (NGDTM) is adopted for its efficiency in providing an approximate solution with high capability in tracking the behavior of these systems. On the other hand, the Grünwald–Letnikov via Riemann–Liouville scheme (GLNS) is adopted within this study as one of its tools in confirming whether chaos exists within these systems. The performance and accuracy of the proposed method are also rigorously tested, and comparisons are made numerically with the Adams–Bashforth–Moulton method, which is used here as a standard method for validation purposes. It is clear from the results that the combination of analytical and numerical methods can greatly enhance both the speed of computation and the accuracy of results. Additionally, the proposed method or approach is found to be quite robust and accurate and can thus be employed for analyzing various fractional dynamical systems that display chaotic attractors. The proposed method can also be expanded upon in the future for analyzing complex models in science and engineering. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Chaotic and Complex Systems)

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23 pages, 831 KB

Open AccessArticle

Stability for Caputo–Hadamard Fractional Uncertain Differential Equation

by Shida Peng, Zhi Li, Jun Zhang, Yuncong Zhu and Liping Xu

Fractal Fract. 2026, 10(1), 50; https://doi.org/10.3390/fractalfract10010050 - 12 Jan 2026

This paper focuses on the Caputo-Hadamard fractional uncertain differential equations (CH-FUDEs) governed by Liu processes, which combine the Caputo–Hadamard fractional derivative with uncertain differential equations to describe dynamic systems involving memory characteristics and uncertain information. Within the framework of uncertain theory, this Liu [...] Read more.

This paper focuses on the Caputo-Hadamard fractional uncertain differential equations (CH-FUDEs) governed by Liu processes, which combine the Caputo–Hadamard fractional derivative with uncertain differential equations to describe dynamic systems involving memory characteristics and uncertain information. Within the framework of uncertain theory, this Liu process serves as the counterpart to Brownian motion. We establish some new Bihari type fractional inequalities that are easy to apply in practice and can be considered as a more general tool in some situations. As applications of those inequalities, we establish the well-posedness of a proposed class of equations under specified non-Lipschitz conditions. Building upon this result, we establish the notions of stability in distribution and stability in measure solutions to CH-FUDEs, deriving sufficient conditions to ensure these stability properties. Finally, the theoretical findings are verified through two numerical examples. Full article

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

25 pages, 518 KB

Open AccessArticle

Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications

by Saad Ihsan Butt, Mohammed Alammar and Youngsoo Seol

Fractal Fract. 2026, 10(1), 49; https://doi.org/10.3390/fractalfract10010049 - 12 Jan 2026

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex [...] Read more.

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. Full article

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

20 pages, 937 KB

Open AccessArticle

Quasi-Consensus of Fractional-Order Multi-Agent Systems with Mixed Delays and External Disturbance via Impulsive Pinning Control

by Tao Chen, Zhiwen Fu, Caimao Su, Ning Chen and Wanli Lin

Fractal Fract. 2026, 10(1), 48; https://doi.org/10.3390/fractalfract10010048 - 12 Jan 2026

In this brief, the quasi-consensus issue is examined for fractional-order multi-agent systems (FROMASs) subject to both mixed delays and external disturbances, employing an impulsive pinning control (IPC) strategy. Unlike mainstream pinning strategies with fixed nodes or static rules, a dynamic pinning mechanism based [...] Read more.

In this brief, the quasi-consensus issue is examined for fractional-order multi-agent systems (FROMASs) subject to both mixed delays and external disturbances, employing an impulsive pinning control (IPC) strategy. Unlike mainstream pinning strategies with fixed nodes or static rules, a dynamic pinning mechanism based on consensus error distances is proposed, which adaptively adjusts the set of pinned nodes at each impulsive instant. By integrating the Razumikhin method, Lyapunov stability theory, and the comparison system approach, sufficient conditions for achieving quasi-consensus of FROMASs are established. Moreover, the convergence bound of consensus errors is quantitatively estimated and shown to be explicitly determined by the intensity of external disturbances. This paper organically integrates the dynamic node pinning mechanism, FRO impulsive control, and Razumikhin stability analysis. It effectively handles mixed delays and external disturbances while significantly reducing control costs. Finally, numerical simulations show that the synchronization errors are strictly bounded within the threshold

M = 13.6372

, which effectively validates both the proposed control scheme and the theoretical analysis. Full article

(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)

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15 pages, 2130 KB

Open AccessArticle

Characterization and Analysis of Hybrid Fractal Antennas for Multiband Communication and Radar Applications

by Abdelbasset Azzouz, Rachid Bouhmidi, Mehr E. Munir, Moustafa M. Nasralla and Mohammed Chetioui

Fractal Fract. 2026, 10(1), 47; https://doi.org/10.3390/fractalfract10010047 - 12 Jan 2026

This work introduces the development and performance analysis of a hybrid fractal antenna combining a Koch snowflake outer geometry with a center slot patterned as a Sierpinski rectangular carpet. The antenna is fabricated on an FR4 board (

ε_{r} = 4.7

, [...] Read more.

This work introduces the development and performance analysis of a hybrid fractal antenna combining a Koch snowflake outer geometry with a center slot patterned as a Sierpinski rectangular carpet. The antenna is fabricated on an FR4 board (

ε_{r} = 4.7

,

\tan δ = 0.0197

) with dimensions

40 \times 60 \times 0.8

mm³. Electromagnetic simulations are performed using Ansys HFSS v15, revealing seven distinct resonances at 2.11, 3.06, 5.78, 6.94, 8.48, 9.23, and 9.56 GHz. The corresponding impedance bandwidths are 90, 37, 67, 100, 90, 130, and 220 MHz, with return losses of −14, −12, −16, −10, −30, −16, and −17 dB, and VSWR values ranging from 1.06 to 1.80. The gains at these resonances are 3.92, 8.24, 6.90, 11.66, 19.38, 16.76, and 12.06 dBi. Frequency allocation analysis indicates compatibility with UMTS/LTE (2.11 GHz), S-band 5G and radar (3.06 GHz), ISM/UNII-3 Wi-Fi and ITS (5.78 GHz), C-band satellite uplink (6.94 GHz), and X-band radar/satellite downlink (8.48–9.56 GHz). The proposed geometry demonstrates wide multi-band coverage, making it a strong candidate for integration into multi-standard communication and radar platforms requiring compact, broadband, and high-directivity performance. Full article

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27 pages, 1460 KB

Open AccessArticle

Optimal Fractional Order PID Controller Design for Hydraulic Turbines Using a Multi-Objective Imperialist Competitive Algorithm

by Mohamed Nejlaoui, Abdullah Alghafis and Nasser Ayidh Alqahtani

Fractal Fract. 2026, 10(1), 46; https://doi.org/10.3390/fractalfract10010046 - 11 Jan 2026

This paper introduces a novel approach for designing a Fractional Order Proportional-Integral-Derivative (FOPID) controller for the Hydraulic Turbine Regulating System (HTRS), aiming to overcome the challenge of tuning its five complex parameters (K_p, K_i, K_d, λ [...] Read more.

This paper introduces a novel approach for designing a Fractional Order Proportional-Integral-Derivative (FOPID) controller for the Hydraulic Turbine Regulating System (HTRS), aiming to overcome the challenge of tuning its five complex parameters (K_p, K_i, K_d, λ and μ). The design is formulated as a multi-objective optimization problem, minimized using the Multi-Objective Imperialist Competitive Algorithm (MOICA). The goal is to minimize two key transient performance metrics: the Integral of Squared Error (ISE) and the Integral of the Time Multiplied Squared Error (ITSE). MOICA efficiently generates a Pareto-front of non-dominated solutions, providing control system designers with diverse trade-off options. The resulting optimal FOPID controller demonstrated superior robustness when evaluated against simulated variations in key HTRS parameters (mg, eg and Tw). Comparative simulations against an optimally tuned integer-order PID and established literature methods (FOPID-GA, FOPID-MOPSO and FOPID-MOHHO) confirm the enhanced dynamic response and stable operation of the MOICA-based FOPID. The MOICA-tuned FOPID demonstrated superior performance for Setpoint Tracking, achieving up to a 26% faster settling speed (ITSE) and an 8% higher accuracy (ISE). Furthermore, for Disturbance Rejection, it showed enhanced robustness, leading to up to a 23% quicker recovery speed (ITSE) and an 18.9% greater error suppression (ISE). Full article

(This article belongs to the Section Engineering)

6 pages, 185 KB

Open AccessEditorial

Advances in Fractional Modeling and Computation

by Yuri Dimitrov, Slavi Georgiev, Venelin Todorov and Jordan Hristov

Fractal Fract. 2026, 10(1), 45; https://doi.org/10.3390/fractalfract10010045 - 11 Jan 2026

Fractional calculus, the study of derivatives and integrals of arbitrary order [1–3] [...] Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

25 pages, 546 KB

Open AccessArticle

Dynamic Analysis and Optimal Prevention Strategies for Monkeypox Spread Modeled via the Mittag–Leffler Kernel

by Mine Yurtoğlu, Dilara Yapışkan, Ebenezer Bonyah, Beyza Billur İskender Eroğlu, Derya Avcı and Delfim F. M. Torres

Fractal Fract. 2026, 10(1), 44; https://doi.org/10.3390/fractalfract10010044 - 10 Jan 2026

Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, [...] Read more.

Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, the eradication of smallpox over time and thus the lack of vaccination has led to the widespread and clinical importance of monkeypox. Although mathematical epidemiology research on the disease is complementary to clinical studies, it has attracted attention in the last few years. The present study aims to discuss the indispensable effects of three control strategies such as vaccination, treatment, and quarantine to prevent the monkeypox epidemic modeled via the Atangana–Baleanu operator. The main purpose is to determine optimal control measures planned to reduce the rates of exposed and infected individuals at the minimum costs. For the controlled model, the existence-uniqueness of the solutions, stability, and sensitivity analysis, and numerical optimal solutions are exhibited. The optimal system is numerically solved using the Adams-type predictor–corrector method. In the numerical simulations, the efficacy of the vaccination, treatment, and quarantine controls is evaluated in separate analyzes as single-, double-, and triple-control strategies. The results demonstrate that the most effective strategy for achieving the aimed outcome is the simultaneous application of vaccination, treatment, and quarantine controls. Full article

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

31 pages, 17740 KB

Open AccessArticle

HR-UMamba++: A High-Resolution Multi-Directional Mamba Framework for Coronary Artery Segmentation in X-Ray Coronary Angiography

by Xiuhan Zhang, Peng Lu, Zongsheng Zheng and Wenhui Li

Fractal Fract. 2026, 10(1), 43; https://doi.org/10.3390/fractalfract10010043 - 9 Jan 2026

Coronary artery disease (CAD) remains a leading cause of mortality worldwide, and accurate coronary artery segmentation in X-ray coronary angiography (XCA) is challenged by low contrast, structural ambiguity, and anisotropic vessel trajectories, which hinder quantitative coronary angiography. We propose HR-UMamba++, a U-Mamba-based framework [...] Read more.

Coronary artery disease (CAD) remains a leading cause of mortality worldwide, and accurate coronary artery segmentation in X-ray coronary angiography (XCA) is challenged by low contrast, structural ambiguity, and anisotropic vessel trajectories, which hinder quantitative coronary angiography. We propose HR-UMamba++, a U-Mamba-based framework centered on a rotation-aligned multi-directional state-space scan for modeling long-range vessel continuity across multiple orientations. To preserve thin distal branches, the framework is equipped with (i) a persistent high-resolution bypass that injects undownsampled structural details and (ii) a UNet++-style dense decoder topology for cross-scale topological fusion. On an in-house dataset of 739 XCA images from 374 patients, HR-UMamba++ is evaluated using eight segmentation metrics, fractal-geometry descriptors, and multi-view expert scoring. Compared with U-Net, Attention U-Net, HRNet, U-Mamba, DeepLabv3+, and YOLO11-seg, HR-UMamba++ achieves the best performance (Dice 0.8706, IoU 0.7794, HD95 16.99), yielding a relative Dice improvement of 6.0% over U-Mamba and reducing the deviation in fractal dimension by up to 57% relative to U-Net. Expert evaluation across eight angiographic views yields a mean score of 4.24 ± 0.49/5 with high inter-rater agreement. These results indicate that HR-UMamba++ produces anatomically faithful coronary trees and clinically useful segmentations that can serve as robust structural priors for downstream quantitative coronary analysis. Full article

(This article belongs to the Special Issue Fractional and Fractal Methods in Biomedical Imaging and Time Series Learning)

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34 pages, 3388 KB

Open AccessArticle

A Fractional-Order Spatiotemporal Unified Energy Framework for Non-Repetitive LiDAR Point Cloud Registration

by Qi Yang, Dongwei Li, Minghao Li and Lu Liu

Fractal Fract. 2026, 10(1), 42; https://doi.org/10.3390/fractalfract10010042 - 9 Jan 2026

Non-repetitive scanning LiDARs provide high coverage yet exhibit irregular sampling patterns, which destabilize local features and correspondences. To address this, we propose a novel spatiotemporal unified energy framework that integrates fractional calculus into rigid pose estimation. Spatially, we introduce a Riesz fractional regularization [...] Read more.

Non-repetitive scanning LiDARs provide high coverage yet exhibit irregular sampling patterns, which destabilize local features and correspondences. To address this, we propose a novel spatiotemporal unified energy framework that integrates fractional calculus into rigid pose estimation. Spatially, we introduce a Riesz fractional regularization term to impose non-local smoothness constraints on the residual field, mitigating structural inconsistencies. Temporally, we design a Grünwald–Letnikov fractional dynamics solver that leverages long-memory effects of historical gradients to reduce the risk of being trapped in local minima. Comparative experiments on the Stanford 3D, MVTec ITODD, and HomebrewedDB (HB) datasets demonstrate that our method significantly outperforms state-of-the-art geometric and learning-based approaches. Specifically, it maintains a success rate exceeding 90% even under severe sampling perturbations where traditional methods fail. Ablation studies further validate that the introduction of non-local spatial constraints and historical gradient memory significantly reshapes the energy landscape, ensuring robust convergence. This work provides a rigorous theoretical foundation for applying fractional operators to point cloud processing. Full article

(This article belongs to the Special Issue Fractional Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image Processing Algorithms, Second Edition)

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16 pages, 7986 KB

Open AccessArticle

Transfer Learning Fractional-Order Recurrent Neural Network for MPPT Under Weak PV Generation Conditions

by Umair Hussan, Mudasser Hassan, Umar Farooq, Huaizhi Wang and Muhammad Ahsan Ayub

Fractal Fract. 2026, 10(1), 41; https://doi.org/10.3390/fractalfract10010041 - 8 Jan 2026

Photovoltaic generation systems (PVGSs) face significant efficiency challenges under partial shading conditions and rapidly changing irradiance due to the limitations of conventional maximum power point tracking (MPPT) methods. To address these challenges, this paper proposes a Transfer Learning-based Fractional-Order Recurrent Neural Network (TL-FRNN) [...] Read more.

Photovoltaic generation systems (PVGSs) face significant efficiency challenges under partial shading conditions and rapidly changing irradiance due to the limitations of conventional maximum power point tracking (MPPT) methods. To address these challenges, this paper proposes a Transfer Learning-based Fractional-Order Recurrent Neural Network (TL-FRNN) for robust global maximum power point (GMPP) tracking across diverse operating conditions. The incorporation of fractional-order dynamics introduces long-term memory and non-local behavior, enabling smoother state evolution and improved discrimination between local and global maxima, particularly under weak and partially shaded conditions. The proposed approach leverages Caputo fractional derivatives with Grünwald–Letnikov approximation to capture the history-dependent behavior of PVGSs while implementing a parameter-partitioning strategy that separates shared features from task-specific parameters. The architecture employs a multi-head design with GMPP regression and partial shading classification capabilities, trained through a two-stage process of pretraining on general PV data followed by efficient fine-tuning on target systems with limited site-specific data. The TL-FRNN achieved 99.2% tracking efficiency with 98.7% GMPP detection accuracy, reducing convergence time by 53% compared to state-of-the-art alternatives while requiring 72% less retraining time through transfer learning. This approach represents a significant advancement in adaptive, intelligent MPPT control for real-world photovoltaic energy-harvesting systems. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)

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21 pages, 8534 KB

Open AccessArticle

On the Convergence Rate of the Caputo Fractional Difference Logistic Map of Nilpotent Matrices

by Rasa Smidtaite, Algirdas Kazlauskas, Mark Edelman and Minvydas Ragulskis

Fractal Fract. 2026, 10(1), 40; https://doi.org/10.3390/fractalfract10010040 - 8 Jan 2026

The convergence rate of the Caputo fractional difference logistic map of nilpotent matrices is investigated in this paper. The divergence rate of the auxiliary parameters governing the dynamics of nilpotents is exponential and is multiple to the Lyapunov exponent of the scalar non-fractional [...] Read more.

The convergence rate of the Caputo fractional difference logistic map of nilpotent matrices is investigated in this paper. The divergence rate of the auxiliary parameters governing the dynamics of nilpotents is exponential and is multiple to the Lyapunov exponent of the scalar non-fractional map. However, the convergence of the Caputo fractional difference logistic map of nilpotent matrices to a stable fixed point is governed by the interplay between the convergence rate of the scalar fractional map (the power law rate) and the exponential convergence induced by the nilpotent matrices. It is demonstrated that convergence is determined by the competition between the power law and exponential mechanisms, a feature not captured by scalar fractional maps, with higher-order auxiliary parameters diverging exponentially at increasingly higher rates. This paper provides insight into the complex dynamics of fractional maps of nilpotent matrices. Full article

(This article belongs to the Special Issue Nonlinear Fractional Maps: Dynamics and Control)

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16 pages, 528 KB

Open AccessArticle

A Novel Approach to the Dynamics of a Fractional-Order Neural Networks with Delay Through Two-Point Self-Mapped Contraction

by Sumati Kumari Panda, Nalleboyina Vijaya, Amer Hassan Albargi and Jamshaid Ahmad

Fractal Fract. 2026, 10(1), 39; https://doi.org/10.3390/fractalfract10010039 - 8 Jan 2026

The paper explores the uniform stability and equilibrium characteristics of a class of neural networks of fractional order with time delay. The two-point self-mapped contraction theorem is used in order to establish sufficient conditions for the uniform stability of the system. Further, the [...] Read more.

The paper explores the uniform stability and equilibrium characteristics of a class of neural networks of fractional order with time delay. The two-point self-mapped contraction theorem is used in order to establish sufficient conditions for the uniform stability of the system. Further, the existence, uniqueness, and uniform stability of the equilibrium point are established. A set of numerical examples is presented to proclaim the validity and usefulness of the proposed results. Full article

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

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19 pages, 9151 KB

Open AccessArticle

On a Friction Oscillator of Integer and Fractional Order; Stick–Slip Attractors

by Marius-F. Danca

Fractal Fract. 2026, 10(1), 38; https://doi.org/10.3390/fractalfract10010038 - 7 Jan 2026

This paper investigates a friction oscillator model in both its Integer-Order and Fractional-Order formulations. The lack of classical solutions for the governing differential equations with discontinuous right-hand sides is addressed by adopting a Differential Inclusion framework. Using Filippov regularization, the discontinuity is replaced [...] Read more.

This paper investigates a friction oscillator model in both its Integer-Order and Fractional-Order formulations. The lack of classical solutions for the governing differential equations with discontinuous right-hand sides is addressed by adopting a Differential Inclusion framework. Using Filippov regularization, the discontinuity is replaced by a set-valued map satisfying appropriate regularity conditions. Selection theory is then applied to construct a Lipschitz-continuous, single-valued function that approximates the set-valued map. This procedure reformulates the discontinuous initial value problem as a continuous, single-valued one, thereby providing a rigorous justification for the proposed approximation method. Numerical simulations are performed to study stick–slip attractors in both the Integer-Order and Fractional-Order cases. The results demonstrate that, in contrast to the Integer-Order system, periodic attractors cannot occur in the Fractional-Order regime. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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27 pages, 375 KB

Open AccessArticle

An Efficient and Accurate Numerical Approach for Fractional Bagley–Torvik Equations: Hermite Polynomials Combined with Least Squares

by Heba S. Osheba, Mohamed A. Ramadan and Taha Radwan

Fractal Fract. 2026, 10(1), 37; https://doi.org/10.3390/fractalfract10010037 - 7 Jan 2026

This paper proposes an efficient and accurate numerical framework for solving fractional Bagley–Torvik equations, which model viscoelastic and memory-dependent dynamic systems. The method combines the Hermite polynomial approximation with a least-squares optimization scheme to achieve high-accuracy solutions. By leveraging the analytical properties of [...] Read more.

This paper proposes an efficient and accurate numerical framework for solving fractional Bagley–Torvik equations, which model viscoelastic and memory-dependent dynamic systems. The method combines the Hermite polynomial approximation with a least-squares optimization scheme to achieve high-accuracy solutions. By leveraging the analytical properties of Hermite polynomials, Caputo fractional derivatives were computed efficiently, avoiding the complexities of direct fractional differentiation. The resulting weighted least-squares formulation transforms the problem into a stable algebraic system. Numerical results confirm the method’s superior accuracy, rapid convergence, and robustness compared with existing techniques, demonstrating its potential for broader applications in fractional-order boundary value problems. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

14 pages, 6937 KB

Open AccessArticle

Existence of Heteroclinic Orbits in Fractional-Order and Integer-Order Coupled Lorenz Systems

by Guiyao Ke, Jun Pan, Feiyu Hu and Haijun Wang

Fractal Fract. 2026, 10(1), 36; https://doi.org/10.3390/fractalfract10010036 - 7 Jan 2026

Applying two Lyapunov functions and the concepts of

α

-/

ω

-limit sets, this paper reexamines fractional-order and integer-order coupled Lorenz systems and simultaneously proves the existence of twelve heteroclinic orbits, i.e., four ones to

S_{0}

and [...] Read more.

Applying two Lyapunov functions and the concepts of

α

-/

ω

-limit sets, this paper reexamines fractional-order and integer-order coupled Lorenz systems and simultaneously proves the existence of twelve heteroclinic orbits, i.e., four ones to

S_{0}

and

S_{5, 6, 7, 8}

, four pairs of ones to

S_{1}

and

S_{5, 7}

,

S_{3}

and

S_{5, 6}

,

S_{2}

and

S_{6, 8}

,

S_{4}

and

S_{7, 8}

when

r - 1 > 0

,

b \geq 2 σ > 0

and

a c < 0

. These orbits have not been reported in existing studies on coupled Lorenz-type systems and are verified via numerical simulations. The findings not only uncover new dynamics of the Lorenz system family and expand the application scope of Lyapunov-based methods but also provide insights into heteroclinic orbits of other fractional-order and integer-order Lorenz-like counterparts. Full article

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