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Axioms
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Fractional-Order Equations and Optimization Models in Engineering
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Fractional-Order Equations and Optimization Models in Engineering

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (28 February 2023) | Viewed by 16350

Special Issue Editors

Dr. Jianke Zhang

E-Mail Website
Guest Editor
Department of Applied Mathematics, Xi'an University of Posts and Telecommunications, Xi’an 710061, China
Interests: optimization; evolutionary algorithms; fractional differential equations
Special Issues, Collections and Topics in MDPI journals
Dr. Cheng Chen

E-Mail Website
Guest Editor
Department of Mathematics, School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China
Interests: partial differential equations; group analysis; lie groups; conservation laws
Dr. Shimin Guo

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Interests: fractional differential equations; spectral method; computational plasma physics; partial differential equations

Special Issue Information

Dear Colleagues,

Fractional-order equations and optimization models have become two effective tools for modeling in engineering and scientific research in recent years. On one hand, optimization algorithms are often used in many numerical solutions of fractional-order equations; on the other hand, the constraints of many optimization problems in practice are fractional ordinary differential or partial differential equations. Therefore, fractional-order equations and optimization theories tend to merge with each other. In recent years, the fractional-order calculus and optimization theories have made great progress in various fields of science and mathematics, and they have received more and more attention. Different types of fractional-order equations and optimization models are studied from the perspective of analysis and numerical value, and some specific applications in science and technology have been put forward.

The purpose of this Special Issue is to bring together research on “Fractional-Order Equations and Optimization Models” using different models and methodologies. We firmly believe that this Special Issue will provide an opportunity to encourage intersection and strengthen this hot research area.

A partial list of topics:

  • Fractional-order differential/integral equations
  • Fuzzy fractional-order differential/integral equations
  • Fractional-order equations: theory, numerical methods, applications to science and engineering;
  • Fractional calculus and its applications;
  • Optimization problems with fractional partial differential equation constraints;
  • Characterization of solutions for optimization problems;
  • Optimality and duality of optimization problems;
  • Multi-objective optimization problem and its application;
  • Fuzzy optimization: theory, algorithms, applications;

We hope that this initiative will be attractive to researchers specialized in the abovementioned fields. Contributions may be submitted on a continuous basis before the deadline. After a peer-review process, submissions will be selected for publication based on their quality and relevance.

Dr. Jianke Zhang
Dr. Cheng Chen
Dr. Shimin Guo
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 100 words) can be sent to the Editorial Office for announcement on this website.

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. Axioms 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 2400 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-order equations
  • fuzzy fractional-order equations
  • fractional calculus
  • PDE constrained optimization problem
  • fuzzy optimization

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Published Papers (9 papers)

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Research

26 pages, 9619 KiB  
Article
Application of Whale Optimization Algorithm Based FOPI Controllers for STATCOM and UPQC to Mitigate Harmonics and Voltage Instability in Modern Distribution Power Grids
by Mohamed Metwally Mahmoud, Basiony Shehata Atia, Yahia M. Esmail, Sid Ahmed El Mehdi Ardjoun, Noha Anwer, Ahmed I. Omar, Faisal Alsaif, Sager Alsulamy and Shazly A. Mohamed
Axioms 2023, 12(5), 420; https://doi.org/10.3390/axioms12050420 - 26 Apr 2023
Cited by 41 | Viewed by 2171
Abstract
In recent modern power systems, the number of renewable energy systems (RESs) and nonlinear loads have become more prevalent. When these systems are connected to the electricity grid, they may face new difficulties and issues such as harmonics and non-standard voltage. The proposed [...] Read more.
In recent modern power systems, the number of renewable energy systems (RESs) and nonlinear loads have become more prevalent. When these systems are connected to the electricity grid, they may face new difficulties and issues such as harmonics and non-standard voltage. The proposed study suggests the application of a whale optimization algorithm (WOA) based on a fractional-order proportional-integral controller (FOPIC) for unified power quality conditioner (UPQC) and STATCOM tools. These operate best with the help of their improved control system, to increase the system’s reliability and fast dynamic response, and to decrease the total harmonic distortion (THD) for enhancing the power quality (PQ). In this article, three different configurations are studied and assessed, namely: (C1) WOA-based FOPIC for UPQC, (C2) WOA-based FOPIC for STATCOM, and (C3) system without FACTS, i.e., base case, to mitigate the mentioned drawbacks. C3 is also considered as a base case to highlight the main benefits of C1 and C2 in improving the PQ by reducing the %THD of the voltage and current system and improving the systems’ voltage waveforms. With C2, voltage fluctuation is decreased by 98%, but it nearly disappears in C1 during normal conditions. Additionally, during the fault period, voltage distortion is reduced by 95% and 100% with C2 and C1, respectively. Furthermore, when comparing C1 to C2 and C3 under regular conditions, the percentage reduction in THD is remarkable. In addition, C1 eliminates the need for voltage sag, and harmonic and current harmonic detectors, and it helps to streamline the control approach and boost control precision. The modeling and simulation of the prepared system are performed by MATLAB/Simulink. Finally, it can be concluded that the acquired results are very interesting and helpful in the recovery to the steady state of wind systems and nonlinear loads, thereby increasing their grid connection capabilities. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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22 pages, 7280 KiB  
Article
L1-Norm Robust Regularized Extreme Learning Machine with Asymmetric C-Loss for Regression
by Qing Wu, Fan Wang, Yu An and Ke Li
Axioms 2023, 12(2), 204; https://doi.org/10.3390/axioms12020204 - 15 Feb 2023
Cited by 1 | Viewed by 1761
Abstract
Extreme learning machines (ELMs) have recently attracted significant attention due to their fast training speeds and good prediction effect. However, ELMs ignore the inherent distribution of the original samples, and they are prone to overfitting, which fails at achieving good generalization performance. In [...] Read more.
Extreme learning machines (ELMs) have recently attracted significant attention due to their fast training speeds and good prediction effect. However, ELMs ignore the inherent distribution of the original samples, and they are prone to overfitting, which fails at achieving good generalization performance. In this paper, based on expectile penalty and correntropy, an asymmetric C-loss function (called AC-loss) is proposed, which is non-convex, bounded, and relatively insensitive to noise. Further, a novel extreme learning machine called L1 norm robust regularized extreme learning machine with asymmetric C-loss (L1-ACELM) is presented to handle the overfitting problem. The proposed algorithm benefits from L1 norm and replaces the square loss function with the AC-loss function. The L1-ACELM can generate a more compact network with fewer hidden nodes and reduce the impact of noise. To evaluate the effectiveness of the proposed algorithm on noisy datasets, different levels of noise are added in numerical experiments. The results for different types of artificial and benchmark datasets demonstrate that L1-ACELM achieves better generalization performance compared to other state-of-the-art algorithms, especially when noise exists in the datasets. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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24 pages, 5261 KiB  
Article
Improved Cascade Correlation Neural Network Model Based on Group Intelligence Optimization Algorithm
by Jun Deng, Qingxia Li and Wenhong Wei
Axioms 2023, 12(2), 164; https://doi.org/10.3390/axioms12020164 - 6 Feb 2023
Cited by 1 | Viewed by 1598
Abstract
The Cascade Correlation learning algorithm is a special supervised learning algorithm for artificial neural network architecture. The optimization algorithm in the traditional neural network has the disadvantages of a single optimization goal, slow convergence speed, and can easily fall into local area, which [...] Read more.
The Cascade Correlation learning algorithm is a special supervised learning algorithm for artificial neural network architecture. The optimization algorithm in the traditional neural network has the disadvantages of a single optimization goal, slow convergence speed, and can easily fall into local area, which cannot fully meet the key elements in the cascade correlation learning algorithm. In comparison, the group intelligence optimization algorithm can take into account these key elements in the optimization process at the same time, and obtain better optimization results. In this paper, we propose the single-objective optimization algorithm jDE-B and the multi-objective optimization algorithm MOEA-T, and improve the network expansion mode in the learning process of Cascade Correlation neural networks. We investigate the effect of applying the group intelligent optimization algorithm in the Cascade Correlation learning algorithm. Experimental results show that our improved algorithm is able to enhance the ability of the Cascade Correlation neural network to fit problems, reduce the number of hidden units and the depth of the network, and optimize the network structure. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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15 pages, 5180 KiB  
Article
Hybrid Projective Synchronization of Fractional-Order Extended Hindmarsh–Rose Neurons with Hidden Attractors
by Xuerong Shi and Zuolei Wang
Axioms 2023, 12(2), 157; https://doi.org/10.3390/axioms12020157 - 2 Feb 2023
Cited by 2 | Viewed by 1271
Abstract
In view of the diversity of stimulated current that neurons may experience, an extended Hindmarsh–Rose neuron model is proposed and the corresponding fractional-order neuron model, with no equilibrium point, is depicted. Additionally, various hidden attractors of the addressed neuron model are analyzed by [...] Read more.
In view of the diversity of stimulated current that neurons may experience, an extended Hindmarsh–Rose neuron model is proposed and the corresponding fractional-order neuron model, with no equilibrium point, is depicted. Additionally, various hidden attractors of the addressed neuron model are analyzed by changing system parameters and the order of fractional-order neuron system. Furthermore, hybrid projective synchronizations of the proposed neurons are investigated and schemes are obtained by designing suitable controllers according to fractional stability theory. Besides, the validity of the theoretical results is verified through numerical simulations. In short, the research results have potential application in revealing the dynamical behaviors of neuron system and controlling the behaviors of neuron into certain status. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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12 pages, 3926 KiB  
Article
Dynamical Analysis and Generalized Synchronization of a Novel Fractional-Order Hyperchaotic System with Hidden Attractor
by Li Xin, Xuerong Shi and Mingjie Xu
Axioms 2023, 12(1), 6; https://doi.org/10.3390/axioms12010006 - 22 Dec 2022
Cited by 4 | Viewed by 1518
Abstract
In this paper, hidden dynamical behaviors in a novel fractional-order hyperchaotic system without an equilibrium point are investigated. It is found that the chaotic system exhibits various hidden behaviors for different parameters, such as the hyperchaotic attractor, the chaotic attractor and the limit [...] Read more.
In this paper, hidden dynamical behaviors in a novel fractional-order hyperchaotic system without an equilibrium point are investigated. It is found that the chaotic system exhibits various hidden behaviors for different parameters, such as the hyperchaotic attractor, the chaotic attractor and the limit cycle. The behaviors are demonstrated via phase portraits and time evolution curves. Moreover, generalized synchronization of the systems is discussed, which can be realized by designing suitable controllers. Numerical simulations are carried out to verify the effectiveness of this synchronization scheme. By analyzing the synchronization performance, it is inferred that the lower the derivative order is, the less time is required to reach synchronization. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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19 pages, 1932 KiB  
Article
Multi-Objective Task Scheduling of Circuit Repair
by Shengyu Liu, Xiaogang Qi and Lifang Liu
Axioms 2022, 11(12), 714; https://doi.org/10.3390/axioms11120714 - 9 Dec 2022
Cited by 2 | Viewed by 1171
Abstract
With the development of technology and the increase of equipment usage intensity, the original support mode of circuit repair, with an ideal model and single objective, is no longer applicable. Therefore, we focus on improving the support mode of circuit repair in this [...] Read more.
With the development of technology and the increase of equipment usage intensity, the original support mode of circuit repair, with an ideal model and single objective, is no longer applicable. Therefore, we focus on improving the support mode of circuit repair in this article. First, in this article, we propose three rest strategies, and consider the scheduling optimization of flexible rest for repair teams, for the first time. We build a more scientific and comprehensive mathematical model for the task scheduling of circuit repair. Specifically, this model aims to maximize benefits and minimize risks during scheduling up to a certain moment, taking into account constraints, such as geographic information, resources, etc. Second, in this article, we design three hybrid algorithms, namely, NSGAII-2Opt-DE(N2D), SPEA2-2Opt-DE(S2D) and MOEA/D-2Opt-DE(M2D). Third, in this article, we design a comprehensive evaluation indicator, area. It mainly contributes to evaluation of the convergence speed of the multi-objective optimization algorithms. Finally, extensive computational experiments were conducted to verify the scientificity of the rest strategies, model, algorithms and evaluation indicator proposed in this article. The experimental results showed that our proposed N2D, S2D and M2D outperformed the existing algorithms, in terms of solution quality and convergence speed. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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12 pages, 5575 KiB  
Article
Hidden Dynamics and Hybrid Synchronization of Fractional-Order Memristive Systems
by Haipeng Jiang, Lizhou Zhuang, Cheng Chen and Zuolei Wang
Axioms 2022, 11(11), 645; https://doi.org/10.3390/axioms11110645 - 15 Nov 2022
Cited by 2 | Viewed by 1478
Abstract
A fractional-order memristive system without equilibrium is addressed. Hidden attractors in the proposed system are discussed and the coexistence of a hidden attractor is found. Via theoretical analysis, the hybrid synchronization of the proposed system with partial controllers is investigated using fractional stability [...] Read more.
A fractional-order memristive system without equilibrium is addressed. Hidden attractors in the proposed system are discussed and the coexistence of a hidden attractor is found. Via theoretical analysis, the hybrid synchronization of the proposed system with partial controllers is investigated using fractional stability theory. Numerical simulation verifies the validity of the hybrid synchronization scheme. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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17 pages, 396 KiB  
Article
A Fractional-Order SIR-C Cyber Rumor Propagation Prediction Model with a Clarification Mechanism
by Linna Li, Yuze Li and Jianke Zhang
Axioms 2022, 11(11), 603; https://doi.org/10.3390/axioms11110603 - 29 Oct 2022
Cited by 2 | Viewed by 1606
Abstract
As communication continues to develop, the high freedom and low cost of the communication network environment also make rumors spread more rapidly. If rumors are not clarified and controlled in time, it is very easy to trigger mass panic and undermine social stability. [...] Read more.
As communication continues to develop, the high freedom and low cost of the communication network environment also make rumors spread more rapidly. If rumors are not clarified and controlled in time, it is very easy to trigger mass panic and undermine social stability. Therefore, it is important to establish an efficient model for rumor propagation. In this paper, the impact of rumor clarifiers on the spread of rumors is considered and fractional order differentiation is introduced to solve the problem that traditional models do not take into account the “anomalous propagation” characteristics of information. A fractional-order Susceptible-Infected-Removal-Clarify (SIR-C) rumor propagation prediction model featuring the clarification mechanism is proposed. The existence and asymptotic stability conditions of the rumor-free equilibrium point (RFEP) E0; the boundary equilibrium points (BEPs) E1 and E2 are also given. Finally, the stability conditions and practical cases are verified by numerical simulations. The experimental results confirm the analysis of the theoretical study and the model fits well with the real-world case data with just minor deviations. As a result, the model can play a positive and effective role in rumor propagation prediction. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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19 pages, 2120 KiB  
Article
Novel Global Harmony Search Algorithm for General Linear Complementarity Problem
by Longquan Yong
Axioms 2022, 11(8), 370; https://doi.org/10.3390/axioms11080370 - 28 Jul 2022
Cited by 3 | Viewed by 1633
Abstract
Linear complementarity problem (LCP) is studied. After reforming general LCP as the system of nonlinear equations by NCP-function, LCP is equivalent to solving an unconstrained optimization model, which can be solved by a recently proposed algorithm named novel global harmony search (NGHS). NGHS [...] Read more.
Linear complementarity problem (LCP) is studied. After reforming general LCP as the system of nonlinear equations by NCP-function, LCP is equivalent to solving an unconstrained optimization model, which can be solved by a recently proposed algorithm named novel global harmony search (NGHS). NGHS algorithm can overcome the disadvantage of interior-point methods. Numerical results show that the NGHS algorithm has a higher rate of convergence than the other HS variants. For LCP with a unique solution, NGHS converges to its unique solution. For LCP with multiple solutions, NGHS can find as many solutions as possible. Meanwhile, for unsolvable LCP, all algorithms are terminated on the solution with the minimum error. Full article
(This article belongs to the Special Issue Fractional-Order Equations and Optimization Models in Engineering)
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