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Fractal Fract
Special Issues
Analysis and Applications of Fractional Calculus and Mathematical Modelling
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Analysis and Applications of Fractional Calculus and Mathematical Modelling

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

Deadline for manuscript submissions: 31 March 2026 | Viewed by 2385

Special Issue Editor

Dr. Marcelo Kaminski Lenzi

E-Mail Website1 Website2
Guest Editor
Departamento de Engenharia Química, Universidade Federal do Paraná, Rua Coronel Francisco H. dos Santos, 100, Curitiba 81531-980, PR, Brazil
Interests: fractional calculus; process control; process instrumentation; diffusion phenomena; design of experiments
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

 Dear Colleagues,

Almost 330 years after Leibniz’s letter to L'Hopital, fractional calculus and derivatives of arbitrary order are still being used extensively and widely in both fundamental and applied research. Consequently, it is important to consider the following question:

How far can knowledge and scientific boundaries be pushed with the aid of fractional calculus?

This Special Issue, entitled “Analysis and Applications of Fractional Calculus and Mathematical Modelling”, intends to provide readers with state-of-the-art research publications showing ideas and challenges for future research and contribute to the beginning of collaboration and exchange among different research groups in fractional calculus worldwide in the future.

To achieve this goal, the Special Issue has two major branches. In the first one, manuscripts with fundamental research involving analytical mathematical methods, novel derivative definitions, and numerical methods for faster solutions, among others, are very welcome. In the second branch, applied research studies and scenarios will certainly aid in the development of a milestone Special Issue by investigating fractional calculus' applications to different research areas and fields, such as process systems engineering, transport phenomena, biological systems, electrical circuits, and materials science, among others.

From these two branches—fundamental and applied research—this Special Issue intends to become a valuable reference for both beginners and senior researchers in the academic world and also for practitioner professionals in industry.

Dr. Marcelo Kaminski Lenzi
Guest Editor

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. 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 differential and integral equations
  • fractional models
  • mathematical modelling
  • fractional dynamics
  • modeling simulation
  • process control
  • application of fractional calculus, focusing on modeling and optimization.

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

Published Papers (3 papers)

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17 pages, 742 KiB  
Article
Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives
by Lamia Loudahi, Amjad Ali, Jing Yuan, Jalil Ahmad, Lamiaa Galal Amin and Yunlan Wei
Fractal Fract. 2025, 9(5), 321; https://doi.org/10.3390/fractalfract9050321 - 19 May 2025
Viewed by 235
Abstract
Water pollution is a significant threat for human health, particularly in developed countries. This study advances the mathematical understanding of WP transmission dynamics by developing a fractional–fractal derivative framework with non-singular kernels and the Mittage–Leffler function, which successfully preserves the non-local behavior of [...] Read more.
Water pollution is a significant threat for human health, particularly in developed countries. This study advances the mathematical understanding of WP transmission dynamics by developing a fractional–fractal derivative framework with non-singular kernels and the Mittage–Leffler function, which successfully preserves the non-local behavior of pollutants. The fractional–fractal derivatives in sense of the Atangana–Baleanu–Caputo formulation inherently captures the non-local and memory-dependent behavior of pollutant diffusion, addressing limitations of classical differential operators. A novel parameter, γ, is introduced to represent the recovery rate of water systems through treatment processes, explicitly modeling the bridge between natural purification mechanisms and engineered remediation efforts. Furthermore, this study establishes stability analysis, and the existence and uniqueness of the solution are established through fixed-point theory to ensure the mathematical stability of the system. Moreover, a numerical scheme based on the Newton polynomial is formulated, by obtaining significant simulations of pollution dynamics under various conditions. Graphical results show the effect of important parameters on pollutant evolution, providing useful information about the behavior of the system. Full article
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus and Mathematical Modelling)
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17 pages, 2182 KiB  
Article
High-Performance Identification and Control of MIMO (Multiple Input—Multiple Output) Experimental Module with Fractional-Order Approach Application
by Alexandre Marques de Almeida, Alisson Luan Daga, Rafael Palma Setti Penteado Lanzarini, Ervin Kaminski Lenzi and Marcelo Kaminski Lenzi
Fractal Fract. 2025, 9(4), 226; https://doi.org/10.3390/fractalfract9040226 - 2 Apr 2025
Viewed by 296
Abstract
This paper focuses on the application of fractional calculus techniques in the identification and control of multivariable (multiple input—multiple output) systems (MIMO). By considering a previously reported experimental set-up similar to a greenhouse, this study proposes the open-loop identification of fractional order transfer [...] Read more.
This paper focuses on the application of fractional calculus techniques in the identification and control of multivariable (multiple input—multiple output) systems (MIMO). By considering a previously reported experimental set-up similar to a greenhouse, this study proposes the open-loop identification of fractional order transfer functions relating to the controlled and manipulated variables, which were validated by experimental data. Afterward, the theoretical analysis of Fractional-order Proportional and Integral (FOPI) closed-loop control for this MIMO system was carried out. An important aspect concerns the use of Particle Swarm Optimization (PSO) metaheuristic algorithm for optimization tasks, both in parameter estimation and controller tuning. Moreover, comparisons with integer order models and controllers (IOPID-IMC) were performed. The results demonstrate the superior performance and robustness of the FOPI-PSO fractional control, which achieves up to 79.6% reduction in ITAE and 72.1% reduction in ITSE criteria. Without the need for explicit decouplers, the decentralized FOPI-PSO control structure demonstrated effective handling of interactions between the temperature and humidity control loops, simplifying the control design while maintaining performance. The fractional-order controllers exhibited robustness to measurement noise, as evidenced by stable and precise control responses in the presence of experimental uncertainties. Additionally, the optimized tuning of FOPI controllers implicitly compensated for disturbances and setpoint changes without requiring additional feedforward mechanisms. This study contributes to a better understanding of fractional calculus applications in designing FO–MIMO systems and provides a practical framework for addressing the identified gaps in the field. Full article
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus and Mathematical Modelling)
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23 pages, 9619 KiB  
Article
Global Mittag-Leffler Attractive Sets, Boundedness, and Finite-Time Stabilization in Novel Chaotic 4D Supply Chain Models with Fractional Order Form
by Muhamad Deni Johansyah, Aceng Sambas, Muhammad Farman, Sundarapandian Vaidyanathan, Song Zheng, Bob Foster and Monika Hidayanti
Fractal Fract. 2024, 8(8), 462; https://doi.org/10.3390/fractalfract8080462 - 6 Aug 2024
Cited by 5 | Viewed by 1229
Abstract
This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger [...] Read more.
This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM). Full article
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus and Mathematical Modelling)
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