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Fractional Dynamical Systems and Its Applications in Science and Engineering
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Fractional Dynamical Systems and Its Applications in Science and Engineering

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 7224

Special Issue Editors

Dr. Pengyu Chen

E-Mail Website1 Website2
Guest Editor
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Interests: stochastic PDEs, nonlinear analysis; fractional dynamics systems; nonlocal evolution equations; random dynamical systems; random attractors; operator semigroups; nonlocal differential equations with delay, PDEs for symmetry, numerical analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Haiyong Qin

E-Mail Website
Guest Editor
School of Mathematics, Qilu Normal University, Jinan 250200, China
Interests: nonlinear analysis; fractional control systems; nonlinear functional analysis; fractional impulsive systems; control science and engineering
Dr. Haoyu Niu

E-Mail Website
Guest Editor
School of Engineering, University of California, Merced, CA 95340, USA
Interests: applied fractional calculus; control science and engineering; fractional calculus and fractional control

Special Issue Information

Dear Colleagues,

In applied mathematics and mathematical analysis, the fractional derivative is a derivative of any order, real or complex. During the last decade, fractional calculus has gained importance in both theoretical and applied aspects of many branches of science and engineering.

Recently, several fractional techniques have come into the spotlight due to the ability to model many complex natural phenomena that cannot be formulated in integer-order nonlinear evolution equations. The existence of this kind of calculus extends to many distinct fields, such as infectious disease epidemiology, neural networks, fluid mechanics, population ecology, solid-state physics, wave propagation in plasma physics, thermodynamics, condensed matter physics, nonlinear optics, civil engineering, quantum mechanics, plasma wave, soil mechanics, and so on. Fractional calculus is considered one of the most notable branches of science, where it provides a considerable number of explanations and details about the nonlocal property since it depends on both historical and current states of the problem in contrast to classical calculus, which depends on the current state only.

Based on the importance of this kind of calculus, many definitions have been being derived, such as fractional Riemann–Liouville derivatives, Caputo, the Caputo–Fabrizio definition, conformable fractional derivatives, and so on. Computational and numerical schemes can be applied to obtain various types of important results for these models.

It is worthwhile to publish a Special Issue on this topic to highlight recent advances achieved by mathematicians and engineers actively working in this field. The purpose of this Special Issue is to promote research in the field of fractional calculus and its applications to science and engineering. All original and high-quality manuscripts in the emerging field of fractional differential equations and their potential applications in all areas of science are welcome.

Potential topics include but are not limited to the following:

  • Survey on fractional differential equations;
  • Fractional differential equations with uncertainty;
  • Fractional dynamics systems;
  • Applications of fractional theory;
  • Existence, controllability, oscillatory ability, and stability of nonlinear systems;
  • Optimal controls for fractional functional differential equations;
  • Applications of fractional problems in science and engineering;
  • Fractional order nonlinear systems: modeling, analysis, and simulation;
  • Inequalities with applications to differential and integral equations;
  • Numerical methods and numerical analysis of fractional differential equations;
  • Other theoretical and experimental aspects of fractional differentiation and related approaches.

Dr. Pengyu Chen
Prof. Dr. Haiyong Qin
Dr. Haoyu Niu
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • survey on fractional differential equations
  • fractional differential equations with uncertainty
  • fractional dynamics systems
  • applications of fractional theory
  • existence, controllability, oscillatory ability, and stability of nonlinear systems
  • optimal controls for fractional functional differential equations
  • applications of fractional problems in science and engineering
  • fractional order nonlinear systems: modeling, analysis, and simulation
  • inequalities with applications to differential and integral equations
  • numerical methods and numerical analysis of fractional differential equations
  • other theoretical and experimental aspects of fractional differentiation and related approaches

Published Papers (5 papers)

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Research

19 pages, 334 KiB  
Article
Approximate Controllability for a Class of Semi-Linear Fractional Integro-Differential Impulsive Evolution Equations of Order 1 < α < 2 with Delay
by Daliang Zhao
Mathematics 2023, 11(19), 4069; https://doi.org/10.3390/math11194069 - 25 Sep 2023
Cited by 1 | Viewed by 711
Abstract
This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order 1<α<2 with delay in Banach spaces. Firstly, we study the existence of the PC-mild solution for our objective [...] Read more.
This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order 1<α<2 with delay in Banach spaces. Firstly, we study the existence of the PC-mild solution for our objective system via some characteristic solution operators related to the Mainardi’s Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems and Its Applications in Science and Engineering)
14 pages, 347 KiB  
Article
Fractional Langevin Coupled System with Stieltjes Integral Conditions
by Rafia Majeed, Binlin Zhang and Mehboob Alam
Mathematics 2023, 11(10), 2278; https://doi.org/10.3390/math11102278 - 13 May 2023
Cited by 2 | Viewed by 731
Abstract
This article outlines the necessary requirements for a coupled system of fractional order boundary value involving the Caputo fractional derivative, including its existence, uniqueness, and various forms of Ulam stability. We demonstrate the existence and uniqueness of the proposed coupled system by using [...] Read more.
This article outlines the necessary requirements for a coupled system of fractional order boundary value involving the Caputo fractional derivative, including its existence, uniqueness, and various forms of Ulam stability. We demonstrate the existence and uniqueness of the proposed coupled system by using the cone-type Leray–Schauder result and the Banach contraction principle. Based on the traditional method of nonlinear functional analysis, the stability is examined. An example is used to provide a clear illustration of our main results. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems and Its Applications in Science and Engineering)
15 pages, 3467 KiB  
Article
A Deep Learning Optimizer Based on Grünwald–Letnikov Fractional Order Definition
by Xiaojun Zhou, Chunna Zhao and Yaqun Huang
Mathematics 2023, 11(2), 316; https://doi.org/10.3390/math11020316 - 07 Jan 2023
Cited by 3 | Viewed by 1582
Abstract
In this paper, a deep learning optimization algorithm is proposed, which is based on the Grünwald–Letnikov (G-L) fractional order definition. An optimizer fractional calculus gradient descent based on the G-L fractional order definition (FCGD_G-L) is designed. Using the short-memory effect of the G-L [...] Read more.
In this paper, a deep learning optimization algorithm is proposed, which is based on the Grünwald–Letnikov (G-L) fractional order definition. An optimizer fractional calculus gradient descent based on the G-L fractional order definition (FCGD_G-L) is designed. Using the short-memory effect of the G-L fractional order definition, the derivation only needs 10 time steps. At the same time, via the transforming formula of the G-L fractional order definition, the Gamma function is eliminated. Thereby, it can achieve the unification of the fractional order and integer order in FCGD_G-L. To prevent the parameters falling into local optimum, a small disturbance is added in the unfolding process. According to the stochastic gradient descent (SGD) and Adam, two optimizers’ fractional calculus stochastic gradient descent based on the G-L definition (FCSGD_G-L), and the fractional calculus Adam based on the G-L definition (FCAdam_G-L), are obtained. These optimizers are validated on two time series prediction tasks. With the analysis of train loss, related experiments show that FCGD_G-L has the faster convergence speed and better convergence accuracy than the conventional integer order optimizer. Because of the fractional order property, the optimizer exhibits stronger robustness and generalization ability. Through the test sets, using the saved optimal model to evaluate, FCGD_G-L also shows a better evaluation effect than the conventional integer order optimizer. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems and Its Applications in Science and Engineering)
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40 pages, 10251 KiB  
Article
A Novel Control Hardware Architecture for Implementation of Fractional-Order Identification and Control Algorithms Applied to a Temperature Prototype
by Juan J. Gude and Pablo García Bringas
Mathematics 2023, 11(1), 143; https://doi.org/10.3390/math11010143 - 28 Dec 2022
Cited by 5 | Viewed by 1754
Abstract
In this paper, the conceptualization of a control hardware architecture aimed to the implementation of integer- and fractional-order identification and control algorithms is presented. The proposed hardware architecture combines the capability of implementing PC-based control applications with embedded applications on microprocessor- and FPGA-based [...] Read more.
In this paper, the conceptualization of a control hardware architecture aimed to the implementation of integer- and fractional-order identification and control algorithms is presented. The proposed hardware architecture combines the capability of implementing PC-based control applications with embedded applications on microprocessor- and FPGA-based real-time targets. In this work, the potential advantages of this hardware architecture over other available alternatives are discussed from different perspectives. The experimental prototype that has been designed and built to evaluate the control hardware architecture proposed in this work is also described in detail. The thermal-based process taking place in the prototype is characterized for being reconfigurable and exhibiting fractional behaviour, which results in a suitable equipment for the purpose of fractional-order identification and control. In order to demonstrate the applicability and effectiveness of the proposed control hardware architecture, integer- and fractional-order identification and control algorithms implemented in various control technologies have been applied to the temperature-based experimental prototype described before. Detailed discussion about results and identification and control issues are provided. The main contribution of this work is to provide an efficient and practical hardware architecture for implementing fractional-order identification and control algorithms in different control technologies, helping to bridge the gap between real-time hardware solutions and software-based simulations of fractional-order systems and controllers. Finally, some conclusions and concluding remarks are offered in the industrial context. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems and Its Applications in Science and Engineering)
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30 pages, 341 KiB  
Article
Existence and Uniqueness of Solutions to Four-Point Impulsive Fractional Differential Equations with p-Laplacian Operator
by Limin Chu, Weimin Hu, Youhui Su and Yongzhen Yun
Mathematics 2022, 10(11), 1852; https://doi.org/10.3390/math10111852 - 28 May 2022
Viewed by 1409
Abstract
In this paper, by using fixed-point theorems, the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations with p-Laplacian operators are studied. In addition, three examples are given to justify the conclusion. The interest of this [...] Read more.
In this paper, by using fixed-point theorems, the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations with p-Laplacian operators are studied. In addition, three examples are given to justify the conclusion. The interest of this paper is to study impulsive fractional differential equations with p-Laplacian operators. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems and Its Applications in Science and Engineering)
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