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Mathematics
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Fractional Modeling, Control, Analysis and Applications
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Fractional Modeling, Control, Analysis and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: 30 October 2024 | Viewed by 4678

Special Issue Editors

Prof. Dr. Ying Luo

E-Mail Website
Guest Editor
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
Interests: fractional order system and control; mechatronics; robotic system; digital twin
Prof. Dr. Fudong Ge

E-Mail Website
Guest Editor
Department of Computer Sciences, China University of Geosciences, Wuhan 430074, China
Interests: modeling and control for fractional-order pde systems
Dr. Jorge Muñoz

E-Mail Website
Guest Editor
RoboticsLab, University Carlos III of Madrid, Avenida Universidad 30, 28911 Madrid, Spain
Interests: robust control; adaptive control; fractional-order control; robotics; humanoid robots; soft robots
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

With the deep exploration of fractional calculus, fractional modeling, control, analysis and applications have drawn much attention and shown the increasingly important role of fractional calculus in various scientific fields. Fractional systems are believed to play an essential role in our day-to-day activities. A comprehensive study of fractional systems and related applications is a challenging problem, and which solution is impossible without the use of contemporary mathematical modeling, control and analysis techniques.

Therefore, this Special Issue focuses on the topics of fractional calculus, modeling and identification of fractional dynamical systems, fractional order control, and fractional order analysis for related applications.

This Special Issue will show the important theoretical significance and practical values of fractional calculus. Topics include but are not limited to:

  • Fractional system mathematical modeling and identification;
  • Fractional system modeling with artificial intelligence/machine learning;
  • Fractional-order PDE system modeling and control;
  • Fractional modeling and control on mechatronics;
  • Fractional control design, analysis and optimization;
  • Fractional applications on motion and servo systems; 
  • Fractional applications on electronics systems;
  • Fractional applications on robotic systems.

Interdisciplinary system modeling and controls in Engineering are particularly welcome in this Special Issue.  

Prof. Dr. Ying Luo
Prof. Dr. Fudong Ge
Dr. Jorge Muñoz
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

  • fractional systems
  • fractional calculus
  • fractional differential equations
  • fractional-order PDE systems
  • modeling and identification
  • control design and optimization
  • artificial intelligence/machine learning
  • mechatronics systems
  • motion and servo systems
  • robotic systems
  • PDE systems

Published Papers (5 papers)

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Research

20 pages, 2769 KiB  
Article
Application of Fractional Differential Equations for Modeling Bacteria Migration in Porous Medium
by Vladimir Chugunov and Sergei Fomin
Mathematics 2024, 12(5), 685; https://doi.org/10.3390/math12050685 - 26 Feb 2024
Viewed by 454
Abstract
One of the modern, recently developed mathematical approaches for modeling various complex chaotic processes (the bacteria migration is apparently one of them), is the application of fractional differential equations. Introduction of fractional derivatives is also a very effective approach for investigation of the [...] Read more.
One of the modern, recently developed mathematical approaches for modeling various complex chaotic processes (the bacteria migration is apparently one of them), is the application of fractional differential equations. Introduction of fractional derivatives is also a very effective approach for investigation of the reactive processes (growth of bacteria in our case). Our recent advances in application of fractional differential equations for modeling the anomalous transport of reactive and non-reactive contaminants (see our recent publications in the References) allow us to expect that the anomalous transport of growing bacteria can also be effectively described by the models with fractional derivatives. Based on these modern approaches, utilizing fractional differential equations, in this paper we developed a reliable mathematical model that could be properly calibrated and, consequently, provide an adequate description of the growing bacteria transport. This model accounts for the memory effects in the bacteria transport due to the random character of bacteria trapping and release by the porous matrix. Two types of bacteria in the saturated porous medium are considered: mobile and immobile bacteria. Bacteria in the mobile phase are migrating in the fluid and have the velocity of the bulk flow, whereas bacteria in the immobile phase are the bacteria that are captured by the porous matrix. These bacteria have zero velocity and can cause clogging of some pores (therefore, porosity is possibly not constant). Examining different conventional models and comparing computations based on these models, we show that this extremely complex character of bacteria transport cannot be described by the traditional approach based on classical partial differential equations. In this paper we suggest fractional differential equations as a simple but very effective tool that can be used for constructing the proper model capable of simulating all the above-mentioned effects associated with migration of alive bacteria. Using this approach, a reliable model of the growing bacteria transport in the porous medium is developed and validated by comparison with experimental laboratory results. We proved that this novel model can be properly linearized and calibrated, so that an excellent agreement with available experimental results can be achieved. This simple model can be used in many applications, for example, as a part of more general mathematical models for predicting the outcomes of the bioremediation of contaminated soils. Full article
(This article belongs to the Special Issue Fractional Modeling, Control, Analysis and Applications)
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15 pages, 2548 KiB  
Article
Employing a Fractional Basis Set to Solve Nonlinear Multidimensional Fractional Differential Equations
by Md. Habibur Rahman, Muhammad I. Bhatti and Nicholas Dimakis
Mathematics 2023, 11(22), 4604; https://doi.org/10.3390/math11224604 - 10 Nov 2023
Viewed by 569
Abstract
Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the [...] Read more.
Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the Bhatti-fractional polynomials basis set. The method involved approximating the desired solution and treated the resulting equation as a matrix equation. All fractional derivatives are considered in the Caputo sense. The resulting operational matrix was inverted, and the desired solution was obtained. The effectiveness of the method was demonstrated by solving two specific types of nonlinear multidimensional fractional differential equations. The results showed higher accuracy, with absolute errors ranging from 10−12 to 10−6 when compared with exact solutions. The proposed technique offered computational efficiency that could be implemented in various programming languages. The examples of two partial fractional differential equations were solved using Mathematica symbolic programming language, and the method showed potential for efficient resolution of fractional differential equations. Full article
(This article belongs to the Special Issue Fractional Modeling, Control, Analysis and Applications)
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18 pages, 1393 KiB  
Article
Kinetic Behavior and Optimal Control of a Fractional-Order Hepatitis B Model
by Tingting Xue, Xiaolin Fan and Yan Xu
Mathematics 2023, 11(17), 3642; https://doi.org/10.3390/math11173642 - 23 Aug 2023
Viewed by 795
Abstract
The fractional-order calculus model is suitable for describing real-world problems that contain non-local effects and have memory genetic effects. Based on the definition of the Caputo derivative, the article proposes a class of fractional hepatitis B epidemic model with a general incidence rate. [...] Read more.
The fractional-order calculus model is suitable for describing real-world problems that contain non-local effects and have memory genetic effects. Based on the definition of the Caputo derivative, the article proposes a class of fractional hepatitis B epidemic model with a general incidence rate. Firstly, the existence, uniqueness, positivity and boundedness of model solutions, basic reproduction number, equilibrium points, and local stability of equilibrium points are studied employing fractional differential equation theory, stability theory, and infectious disease dynamics theory. Secondly, the fractional necessary optimality conditions for fractional optimal control problems are derived by applying the Pontryagin maximum principle. Finally, the optimization simulation results of fractional optimal control problem are discussed. To control the spread of the hepatitis B virus, three control variables (isolation, treatment, and vaccination) are applied, and the optimal control theory is used to formulate the optimal control strategy. Specifically, by isolating infected and non-infected people, treating patients, and vaccinating susceptible people at the same time, the number of hepatitis B patients can be minimized, the number of recovered people can be increased, and the purpose of ultimately eliminating the transmission of hepatitis B virus can be achieved. Full article
(This article belongs to the Special Issue Fractional Modeling, Control, Analysis and Applications)
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30 pages, 5387 KiB  
Article
Modelling Fractional Advection–Diffusion Processes via the Adomian Decomposition
by Alberto Antonini and Valentina Anna Lia Salomoni
Mathematics 2023, 11(12), 2657; https://doi.org/10.3390/math11122657 - 11 Jun 2023
Viewed by 876
Abstract
When treating geomaterials, fractional derivatives are used to model anomalous dispersion or diffusion phenomena that occur when the mass transport media are anisotropic, which is generally the case. Taking into account anomalous diffusion processes, a revised Fick’s diffusion law is to be considered, [...] Read more.
When treating geomaterials, fractional derivatives are used to model anomalous dispersion or diffusion phenomena that occur when the mass transport media are anisotropic, which is generally the case. Taking into account anomalous diffusion processes, a revised Fick’s diffusion law is to be considered, where the fractional derivative order physically reflects the heterogeneity of the soil medium in which the diffusion phenomena take place. The solutions of fractional partial differential equations can be computed by using the so-called semi-analytical methods that do not require any discretization and linearization in order to obtain accurate results, e.g., the Adomian Decomposition Method (ADM). Such a method is innovatively applied for overcoming the critical issue of geometric nonlinearities in coupled saturated porous media and the potentialities of the approach are studied, as well as findings discussed. Full article
(This article belongs to the Special Issue Fractional Modeling, Control, Analysis and Applications)
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17 pages, 1515 KiB  
Article
Experimental Validation of Fractional PID Controllers Applied to a Two-Tank System
by Felipe de J. Sorcia-Vázquez, Jesse Y. Rumbo-Morales, Jorge A. Brizuela-Mendoza, Gerardo Ortiz-Torres, Estela Sarmiento-Bustos, Alan F. Pérez-Vidal, Erasmo M. Rentería-Vargas, Miguel De-la-Torre and René Osorio-Sánchez
Mathematics 2023, 11(12), 2651; https://doi.org/10.3390/math11122651 - 10 Jun 2023
Cited by 2 | Viewed by 1305
Abstract
An experimental validation of fractional-order PID (FOPID) controllers, which were applied to a two coupled tanks system, is presented in this article. Two FOPID controllers, a continuous FOPID (cFOPID) and a discrete FOPID (dFOPID), were implemented in real-time. The gains tuning process was [...] Read more.
An experimental validation of fractional-order PID (FOPID) controllers, which were applied to a two coupled tanks system, is presented in this article. Two FOPID controllers, a continuous FOPID (cFOPID) and a discrete FOPID (dFOPID), were implemented in real-time. The gains tuning process was accomplished by applying genetic algorithms while considering the cost function with respect to the tracking error and control effort. The gains optimization process was performed directly to the two-tanks non-linear model. The real-time implementation used a National Instruments PCIe-6321 card as a data acquisition system; for the interface, we used a Simulink Matlab and Simulink Desktop Real-Time Toolbox. The performance of the fractional controllers was compared with the performance of classical PID controllers. Full article
(This article belongs to the Special Issue Fractional Modeling, Control, Analysis and Applications)
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