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Fractal Fract
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Implementations and Applications of Algorithms Based on Fractional Calculus to...
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Implementations and Applications of Algorithms Based on Fractional Calculus to Engineering Problems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: 31 July 2026 | Viewed by 4950

Special Issue Editors

Dr. José de Jesús Rangel Magdaleno

E-Mail Website
Guest Editor
Digital Systems Group, National Institute for Astrophysics, Optics and Electronics, Puebla 72840, Mexico
Interests: FPGA; instrumentation; mechatronics; digital systems; fault detection; fractional order
Special Issues, Collections and Topics in MDPI journals
Dr. Oscar Martínez-Fuentes

E-Mail Website
Guest Editor
Department of Electronics, National Institute for Astrophysics, Optics and Electronics, Puebla 72840, Mexico
Interests: fractional-order systems; Lyapunov stability theory; nonlinear observers and synchronization; chaos and non-conventional operators
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent years, the scientific community has focused its attention on using fractional calculus and differential equations as tools to develop various algorithms for signal processing and their corresponding applications, such as the development and implementation of complex algorithms associated with chaotic systems in embedded systems, which allows for the use of diverse applications and practical solutions in engineering problems.

Various research works have proven that applying fractional calculus facilitates the development of novel and more robust solutions compared to solutions with integer calculus. This enables efficient algorithms to be implemented and applied to engineering problems such as automatic control, fault detection, encryption, monitoring, intelligent systems, and image processing.  

Therefore, developing efficient algorithms based on fractional calculus is a novel engineering challenge due to several considerations such as real-time processing, quantification error, memory resources, finite word length, parallel processing, power consumption, maximum operation frequency, and bitrate, to mention a few.

This Special Issue provides a forum for presenting new and improved algorithms or techniques based on fractional calculus for applications in current engineering problems. We invite authors to contribute original research articles exploring the latest developments of fractional-order systems. Topics may include, but are not limited to, the following:

  • Embedded systems;
  • Robot control;
  • Biomedical applications;
  • Autonomous vehicles;
  • Neural networks;
  • Chaotic systems;
  • Image processing;
  • Nonlinear observers;
  • Power systems;
  • Signal processing.

Dr. José de Jesús Rangel Magdaleno
Dr. Oscar Martínez-Fuentes
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional calculus
  • fractional-order systems
  • algorithm implementation
  • algorithm development
  • signal processing
  • image processing
  • fault detection

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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32 pages, 12648 KB  
Article
Fractional-Order-Enhanced Dual-View Representation and VibrMamba–VMamba Collaborative Modeling for Gearbox Fault Diagnosis
by Fengyun Xie, Kang Niu, Zeyan Song, Shulei Wang, Huihang Chen and Ying Cao
Fractal Fract. 2026, 10(5), 342; https://doi.org/10.3390/fractalfract10050342 - 19 May 2026
Viewed by 102
Abstract
Gearbox fault diagnosis under controlled bench-test conditions with known speed variations and noise interference remains challenging because nonstationarity, background noise, and operating-condition fluctuations can easily submerge weak localized fault features. To address this issue, this study proposes a fault diagnosis method based on [...] Read more.
Gearbox fault diagnosis under controlled bench-test conditions with known speed variations and noise interference remains challenging because nonstationarity, background noise, and operating-condition fluctuations can easily submerge weak localized fault features. To address this issue, this study proposes a fault diagnosis method based on a fractional-order-enhanced dual-view representation and VibrMamba–VMamba collaborative modeling. First, this study introduces a Grünwald–Letnikov fractional-order differential enhancement module with a fractional order of α=0.6 to strengthen fault-sensitive impulsive components and improve the representation of nonstationary vibration signals. The framework then uses the enhanced signal to construct dual-view inputs: a fractional-order-enhanced one-dimensional vibration sequence and a fractional-order-enhanced synchrosqueezing transform (SST) time–frequency image. Subsequently, the framework constructs a VibrMamba temporal branch and a VMamba visual branch to extract dynamic temporal features and global structural features, respectively. Instead of using simple feature concatenation, this study designs a sample-adaptive collaborative fusion mechanism with gated weighting and cross-branch residual enhancement to integrate complementary temporal–visual representations. Bench-level experiments show that the proposed method achieves 98.90% diagnostic accuracy under clean test conditions and maintains 91.52% accuracy at −5 dB signal-to-noise ratio (SNR). These results should be interpreted as bench-level validation under controlled laboratory conditions rather than as direct evidence of field-level generalization. This framework provides a methodological solution that integrates fractional-order signal enhancement, dual-view representation, and Mamba-style collaborative state-space modeling for gearbox fault classification under controlled laboratory conditions with known speed variations and noise disturbances. 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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22 pages, 8053 KB  
Article
Rolling Bearing Fault Diagnosis Based on Fractional Constant Q Non-Stationary Gabor Transform and VMamba-Conv
by Fengyun Xie, Chengjie Song, Yang Wang, Minghua Song, Shengtong Zhou and Yuanwei Xie
Fractal Fract. 2025, 9(8), 515; https://doi.org/10.3390/fractalfract9080515 - 6 Aug 2025
Cited by 1 | Viewed by 1361
Abstract
Rolling bearings are prone to failure, meaning that research on intelligent fault diagnosis is crucial in relation to this key transmission component in rotating machinery. The application of deep learning (DL) has significantly advanced the development of intelligent fault diagnosis. This paper proposes [...] Read more.
Rolling bearings are prone to failure, meaning that research on intelligent fault diagnosis is crucial in relation to this key transmission component in rotating machinery. The application of deep learning (DL) has significantly advanced the development of intelligent fault diagnosis. This paper proposes a novel method for rolling bearing fault diagnosis based on the fractional constant Q non-stationary Gabor transform (FCO-NSGT) and VMamba-Conv. Firstly, a rolling bearing fault experimental platform is established and the vibration signals of rolling bearings under various working conditions are collected using an acceleration sensor. Secondly, a kurtosis-to-entropy ratio (KER) method and the rotational kernel function of the fractional Fourier transform (FRFT) are proposed and applied to the original CO-NSGT to overcome the limitations of the original CO-NSGT, such as the unsatisfactory time–frequency representation due to manual parameter setting and the energy dispersion problem of frequency-modulated signals that vary with time. A lightweight fault diagnosis model, VMamba-Conv, is proposed, which is a restructured version of VMamba. It integrates an efficient selective scanning mechanism, a state space model, and a convolutional network based on SimAX into a dual-branch architecture and uses inverted residual blocks to achieve a lightweight design while maintaining strong feature extraction capabilities. Finally, the time–frequency graph is inputted into VMamba-Conv to diagnose rolling bearing faults. This approach reduces the number of parameters, as well as the computational complexity, while ensuring high accuracy and excellent noise resistance. The results show that the proposed method has excellent fault diagnosis capabilities, with an average accuracy of 99.81%. By comparing the Adjusted Rand Index, Normalized Mutual Information, F1 Score, and accuracy, it is concluded that the proposed method outperforms other comparison methods, demonstrating its effectiveness and superiority. 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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21 pages, 16644 KB  
Article
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
Cited by 3 | Viewed by 2001
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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