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Fractal Fract
Special Issues
Applications of Fractional Calculus in Modern Mathematical Modeling
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Applications of Fractional Calculus in Modern Mathematical Modeling

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

Deadline for manuscript submissions: 31 December 2025 | Viewed by 1662

Special Issue Editors

Prof. Dr. Feliz Manuel Minhós

E-Mail Website
Guest Editor
1. Department of Mathematics, School of Science and Technology, University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
2. Center for Research in Mathematics and Applications (CIMA), Institute for Advanced Studies and Research (IIFA), University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
Interests: differential and difference equations; dynamical systems; boundary value problems; topological and variational methods; fractional calculus; differential and integral equations
Special Issues, Collections and Topics in MDPI journals
Dr. Ali Raza

E-Mail Website
Guest Editor
1. Center for Research in Mathematics and Applications (CIMA), Institute for Advanced Studies and Research (IIFA), University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
2. Department of Physical Sciences, The University of Chenab, Gujrat, Pakistan
Interests: complex nonlinear stochastic dynamical systems; reduce-order models; machine learning; data assimilation; stochastic methods; numerical analysis of stochastic models; fractional calculus; differential and integral equations
Special Issues, Collections and Topics in MDPI journals
Dr. Muhammad Mohsin

E-Mail Website
Guest Editor
Department of Mathematics, Technische Universitat Chemnitz, 62, 09111 Chemnitz, Germany
Interests: computational biology; numerical analysis; non-linear dynamics; stochastic differential equations; stochastic methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus extends the notion of derivatives and integrals of arbitrary order and gives rise to a variety of complex models and analyzes. This is due to its ability to capture the effects of memory and many other effects that other forms of calculus simply cannot capture. Therefore, this Special Issue will aggregate the latest research that focuses on the most recent theoretical advances, computing methods, and wide applications of fractional calculus in the modern mathematical modeling of various fields such as physics, biology, engineering, and finance. It will be centered around the theory of fractional calculus, new analytical and numerical methods for solving fractional differential equations, fractional partial differential equations and their applications, the biological modeling of fractional calculus, control systems and engineering, the application of fractional differential calculus to financial and economic problems, the mathematics of fractional calculus methods, non-local operators in mathematical modeling, the epidemiology and treatment of infectious diseases, and the use of fractional operators and models of time-fractional derivatives.

Prof. Dr. Feliz Manuel Minhós
Dr. Ali Raza
Dr. Muhammad Mohsin
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. 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 equations (FDEs)
  • fractional delay differential equations (FDDEs)
  • stochastic fractional delay differential equations (SFDDEs)
  • memory effects
  • fractional partial differential equations (FRDE's)
  • stochastic fractional partial differential equations (SFPDEs)
  • existence and uniqueness of solutions for FDEs
  • stability analysis of FDEs
  • sufficient conditions for solvability of FDEs with multi-point type data
  • uniqueness of solution for FDEs with impulses given by functions with several variables
  • solvability of functional FDES in involving P-Laplacian
  • multiplicity results for singular FDES in biological models
  • numerical methods in fractional calculus
  • biological systems modeling through fractional calculus
  • efficient computational methods to solve fractional differential equations for biological systems
  • control theory
  • mathematical modeling in finance through fractional calculus
  • data-driven approaches through fractional calculus
  • machine learning with fractional operators
  • future research directions and open problems in the application of fractional calculus in mathematical modeling

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Published Papers (4 papers)

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Research

33 pages, 1387 KiB  
Article
Design of Non-Standard Finite Difference and Dynamical Consistent Approximation of Campylobacteriosis Epidemic Model with Memory Effects
by Ali Raza, Feliz Minhós, Umar Shafique, Emad Fadhal and Wafa F. Alfwzan
Fractal Fract. 2025, 9(6), 358; https://doi.org/10.3390/fractalfract9060358 - 29 May 2025
Viewed by 270
Abstract
Campylobacteriosis has been described as an ever-changing disease and health issue that is rather dangerous for different population groups all over the globe. The World Health Organization (WHO) reports that 33 million years of healthy living are lost annually, and nearly one in [...] Read more.
Campylobacteriosis has been described as an ever-changing disease and health issue that is rather dangerous for different population groups all over the globe. The World Health Organization (WHO) reports that 33 million years of healthy living are lost annually, and nearly one in ten persons have foodborne illnesses, including Campylobacteriosis. This explains why there is a need to develop new policies and strategies in the management of diseases at the intergovernmental level. Within this framework, an advanced stochastic fractional delayed model for Campylobacteriosis includes new stochastic, memory, and time delay factors. This model adopts a numerical computational technique called the Grunwald–Letnikov-based Nonstandard Finite Difference (GL-NSFD) scheme, which yields an exponential fitted solution that is non-negative and uniformly bounded, which are essential characteristics when working with compartmental models in epidemic research. Two equilibrium states are identified: the first is an infectious Campylobacteriosis-free state, and the second is a Campylobacteriosis-present state. When stability analysis with the help of the basic reproduction number R0 is performed, the stability of both equilibrium points depends on the R0 value. This is in concordance with the actual epidemiological data and the research conducted by the WHO in recent years, with a focus on the tendency to increase the rate of infections and the necessity to intervene in time. The model goes further to analyze how a delay in response affects the band of Campylobacteriosis spread, and also agrees that a delay in response is a significant factor. The first simulations of the current state of the system suggest that certain conditions can be achieved, and the eradication of the disease is possible if specific precautions are taken. The outcomes also indicate that enhancing the levels of compliance with the WHO-endorsed SOPs by a significant margin can lower infection rates significantly, which can serve as a roadmap to respond to this public health threat. Unlike most analytical papers, this research contributes actual findings and provides useful recommendations for disease management approaches and policies. Full article
(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)
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23 pages, 3829 KiB  
Article
Dissipativity Constraints in Zener-Type Time Dispersive Electromagnetic Materials of the Fractional Type
by Teodor M. Atanacković, Marko Janev, Milan Narandžić and Stevan Pilipović
Fractal Fract. 2025, 9(6), 342; https://doi.org/10.3390/fractalfract9060342 - 26 May 2025
Viewed by 162
Abstract
Thermodynamic constraints must be satisfied for the parameters of a constitutive relation, particularly for a model describing an electromagnetic (or any other) material with the intention of giving that model a physical meaning. We present sufficient conditions for the parameters of the constitutive [...] Read more.
Thermodynamic constraints must be satisfied for the parameters of a constitutive relation, particularly for a model describing an electromagnetic (or any other) material with the intention of giving that model a physical meaning. We present sufficient conditions for the parameters of the constitutive relation of an electromagnetic Zener-type fractional 2D and 3D anisotropic model so that a weak form of the thermodynamic (entropy) inequality is satisfied. Moreover, for such models, we analyze the corresponding thermodynamic constraints for field reconstruction and regularity in the 2D anisotropic case. This is carried out by the use of the matrix version of the Bochner theorem in the most general form, including generalized functions as elements of a matrix, which appear in that theorem. The given numerical results confirm the calculus presented in the paper. Full article
(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)
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16 pages, 402 KiB  
Article
A Simple Fractional Model with Unusual Dynamics in the Derivative Order
by Guillermo Fernández-Anaya, Francisco A. Godínez, Rogelio Valdés, Luis Alberto Quezada-Téllez and M. A. Polo-Labarrios
Fractal Fract. 2025, 9(4), 264; https://doi.org/10.3390/fractalfract9040264 - 21 Apr 2025
Viewed by 277
Abstract
Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: [...] Read more.
Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: (1) an integer-order differential equation that includes the state variable, (2) a differential equation that incorporates the state variable and features both integer- and fractional-order derivatives, and (3) fractional variable-order differential equations nested in the derivative orders. We prove a result that shows how the extended recursion of the last case is generalized. These examples illustrate the richness that simple dynamical systems can reveal through the order of their derivatives. Full article
(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)
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27 pages, 1455 KiB  
Article
Neutral Delayed Fractional Models in Financial Time Series: Insights into Borsa Istanbul Sectors Affected by the Kahramanmaraş Earthquake
by Ömer Akgüller, Mehmet Ali Balcı, Larissa Margareta Batrancea, Dilara Altan Koç and Anca Nichita
Fractal Fract. 2025, 9(3), 141; https://doi.org/10.3390/fractalfract9030141 - 24 Feb 2025
Viewed by 460
Abstract
This study examines the impact of the Kahramanmaraş Earthquake on four key sectors of Borsa Istanbul: Basic Metal, Insurance, Non-Metallic Mineral Products, and Wholesale and Retail Trade using neutral delayed fractional differential equations. Employing the Chebyshev collocation method, we numerically solved the neutral [...] Read more.
This study examines the impact of the Kahramanmaraş Earthquake on four key sectors of Borsa Istanbul: Basic Metal, Insurance, Non-Metallic Mineral Products, and Wholesale and Retail Trade using neutral delayed fractional differential equations. Employing the Chebyshev collocation method, we numerically solved the neutral delayed fractional differential equations with initial conditions scaled by each sector’s log difference standard deviation to accurately reflect market volatility. Fractional orders were derived from the Hurst exponent, and time delays were identified using average mutual information, autocorrelation function, and partial autocorrelation function methods. The results reveal significant changes post-earthquake, including reduced market persistence and increased volatility in the Basic Metal and Insurance sectors, contrasted by enhanced stability in the Non-Metallic Mineral Products sector. Neutral delayed fractional differential equations demonstrated superior performance over traditional models by effectively capturing memory and delay effects. This work underscores the efficacy of neutral delayed fractional differential equations in modeling financial resilience amid external shocks. Full article
(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)
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