Trends in Fractional Modelling in Science and Innovative Technologies

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 11107

Special Issue Editor

Special Issue Information

Dear Colleagues,

Fractional calculus has played an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. Modelling methods involving fractional operators are being continuously generalized and enhanced, especially during the last few decades.

Fractional calculus has an amazing history in the modelling of non-linear and anomalous problems in mathematics, physics, statistics, and engineering, involving a diversity of fractional-order integral and derivative operators, such as the ones named after Grunwald–Letnikov, Riemann–Liouville, Weyl, Caputo, Hadamard, Riesz, Erdelyi–Kober, etc. based on the power-law memory. Beyond this positive classical basis, in recent years new trends in fractional modelling involving operators with non-singular kernels were created to model dissipative phenomena that cannot be adequately modelled by fractional differential operators based on singular kernels.

The goal of this Special Issue is to report the latest progress in fractional calculus oriented towards scientific and engineering problems in the light of the classic (power-law) and modern trends (with non-singular kernels) thus allowing the scientific society to see what could be done and how in this amazing area of mathematical modelling. 

We kindly invite researchers working within the fields of theory, methods, and applications of these problems to submit their latest findings to this Special Issue.

The main topics of the collection include, but are not limited to:

  • Fractional modelling: broad aspects;
  • Solution techniques: analytical and numerical;
  • Fractional modelling: new trends, new fractional operators, mathematical properties of fractional operators;
  • Fractional-order ODEs, PDEs, and integro-differential equations involving new fractional operators;
  • Memory kernels to fractional operators: identification, construction, definitions of fractional operators on their basis and relevant properties;
  • Special functions of mathematical physics and applied mathematics associated with classical and new fractional operators;
  • Examples beyond the classical singular kernel applications: Non-power-law relaxations involving new operators;
  • Symmetry in fractional operators and models;
  • Thermodynamic compatibility of fractional models with singular and nonsingular kernels;
  • Diffusion models: broad aspects;
  • Local fractional calculus;
  • Discrete fractional calculus;
  • Heat, mass, and momentum transfer (fluid dynamics) with relaxations (power-law and bounded kernels);
  • Mechanics and rheology of solid materials and innovative fractional modelling;
  • Nano-applications of fractional modelling;
  • Biomechanical and biomedical applications of fractional calculus;
  • Chaos and complexity;
  • Control problems and model identifications with singular and non-singular fractional operators;
  • Electrochemical systems and alternative energy sources: models by fractional operators;
  • Fractional modelling of electrochemical and magnetic systems;
  • Dynamical and stochastic systems based upon fractional calculus with singular and non-singular fractional operators.

Prof. Dr. Jordan Hristov
Guest Editor

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Published Papers (7 papers)

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Editorial

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2 pages, 177 KiB  
Editorial
Special Issue “Trends in Fractional Modelling in Science and Innovative Technologies”
by Jordan Hristov
Symmetry 2023, 15(4), 884; https://doi.org/10.3390/sym15040884 - 8 Apr 2023
Cited by 1 | Viewed by 879
Abstract
Fractional calculus has played an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years [...] Full article
(This article belongs to the Special Issue Trends in Fractional Modelling in Science and Innovative Technologies)

Research

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21 pages, 3936 KiB  
Article
Application of a Machine Learning Algorithm for Evaluation of Stiff Fractional Modeling of Polytropic Gas Spheres and Electric Circuits
by Fawaz Khaled Alarfaj, Naveed Ahmad Khan, Muhammad Sulaiman and Abdullah M. Alomair
Symmetry 2022, 14(12), 2482; https://doi.org/10.3390/sym14122482 - 23 Nov 2022
Cited by 11 | Viewed by 1849
Abstract
Fractional polytropic gas sphere problems and electrical engineering models typically simulated with interconnected circuits have numerous applications in physical, astrophysical phenomena, and thermionic currents. Generally, most of these models are singular-nonlinear, symmetric, and include time delay, which has increased attention to them among [...] Read more.
Fractional polytropic gas sphere problems and electrical engineering models typically simulated with interconnected circuits have numerous applications in physical, astrophysical phenomena, and thermionic currents. Generally, most of these models are singular-nonlinear, symmetric, and include time delay, which has increased attention to them among researchers. In this work, we explored deep neural networks (DNNs) with an optimization algorithm to calculate the approximate solutions for nonlinear fractional differential equations (NFDEs). The target data-driven design of the DNN-LM algorithm was further implemented on the fractional models to study the rigorous impact and symmetry of different parameters on RL, RC circuits, and polytropic gas spheres. The targeted data generated from the analytical and numerical approaches in the literature for different cases were utilized by the deep neural networks to predict the numerical solutions by minimizing the differences in mean square error using the Levenberg–Marquardt algorithm. The numerical solutions obtained by the designed technique were contrasted with the multi-step reproducing kernel Hilbert space method (MS-RKM), Laplace transformation method (LTM), and Padé approximations. The results demonstrate the accuracy of the design technique as the DNN-LM algorithm overlaps with the actual results with minimum percentage absolute errors that lie between 108 and 1012. The extensive graphical and statistical analysis of the designed technique showed that the DNN-LM algorithm is dependable and facilitates the examination of higher-order nonlinear complex problems due to the flexibility of the DNN architecture and the effectiveness of the optimization procedure. Full article
(This article belongs to the Special Issue Trends in Fractional Modelling in Science and Innovative Technologies)
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21 pages, 2952 KiB  
Article
Fractional Stefan Problem Solving by the Alternating Phase Truncation Method
by Agata Chmielowska and Damian Słota
Symmetry 2022, 14(11), 2287; https://doi.org/10.3390/sym14112287 - 1 Nov 2022
Cited by 5 | Viewed by 1327
Abstract
The aim of this paper is the adaptation of the alternating phase truncation (APT) method for solving the two-phase time-fractional Stefan problem. The aim was to determine the approximate temperature distribution in the domain with the moving boundary between the solid and the [...] Read more.
The aim of this paper is the adaptation of the alternating phase truncation (APT) method for solving the two-phase time-fractional Stefan problem. The aim was to determine the approximate temperature distribution in the domain with the moving boundary between the solid and the liquid phase. The adaptation of the APT method is a kind of method that allows us to consider the enthalpy distribution instead of the temperature distribution in the domain. The method consists of reducing the whole considered domain to liquid phase by adding sufficient heat at each point of the solid and then, after solving the heat equation transformed to the enthalpy form in the obtained region, subtracting the heat that has been added. Next the whole domain is reduced to the solid phase by subtracting the sufficient heat from each point of the liquid. The heat equation is solved in the obtained region and, after that, the heat that had been subtracted is added at the proper points. The steps of the APT method were adapted to solve the equations with the fractional derivatives. The paper includes numerical examples illustrating the application of the described method. Full article
(This article belongs to the Special Issue Trends in Fractional Modelling in Science and Innovative Technologies)
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13 pages, 269 KiB  
Article
The Unique Solution for Sequential Fractional Differential Equations with Integral Multi-Point and Anti-Periodic Type Boundary Conditions
by Zhaocai Hao and Beibei Chen
Symmetry 2022, 14(4), 761; https://doi.org/10.3390/sym14040761 - 6 Apr 2022
Cited by 2 | Viewed by 1100
Abstract
In this paper, we obtain the existence of the unique solution of anti-periodic type (anti-symmetry) integral multi-point boundary conditions for sequential fractional differential equations. We apply the Banach contraction mapping principle to get the desired results. Our results specialize and extend some existing [...] Read more.
In this paper, we obtain the existence of the unique solution of anti-periodic type (anti-symmetry) integral multi-point boundary conditions for sequential fractional differential equations. We apply the Banach contraction mapping principle to get the desired results. Our results specialize and extend some existing results. Full article
(This article belongs to the Special Issue Trends in Fractional Modelling in Science and Innovative Technologies)
24 pages, 17062 KiB  
Article
A Fractional Approach to a Computational Eco-Epidemiological Model with Holling Type-II Functional Response
by B. Günay, Praveen Agarwal, Juan L. G. Guirao and Shaher Momani
Symmetry 2021, 13(7), 1159; https://doi.org/10.3390/sym13071159 - 28 Jun 2021
Cited by 8 | Viewed by 2037
Abstract
Eco-epidemiological can be considered as a significant combination of two research fields of computational biology and epidemiology. These problems mainly take ecological systems into account of the impact of epidemiological factors. In this paper, we examine the chaotic nature of a computational system [...] Read more.
Eco-epidemiological can be considered as a significant combination of two research fields of computational biology and epidemiology. These problems mainly take ecological systems into account of the impact of epidemiological factors. In this paper, we examine the chaotic nature of a computational system related to the spread of disease into a specific environment involving a novel differential operator called the Atangana–Baleanu fractional derivative. To approximate the solutions of this fractional system, an efficient numerical method is adopted. The numerical method is an implicit approximate method that can provide very suitable numerical approximations for fractional problems due to symmetry. Symmetry is one of the distinguishing features of this technique compared to other methods in the literature. Through considering different choices of parameters in the model, several meaningful numerical simulations are presented. It is clear that hiring a new derivative operator greatly increases the flexibility of the model in describing the different scenarios in the model. The results of this paper can be very useful help for decision-makers to describe the situation related to the problem, in a more efficient way, and control the epidemic. Full article
(This article belongs to the Special Issue Trends in Fractional Modelling in Science and Innovative Technologies)
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13 pages, 1543 KiB  
Article
Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
by Jocelyn Sabatier
Symmetry 2021, 13(6), 1099; https://doi.org/10.3390/sym13061099 - 21 Jun 2021
Cited by 16 | Viewed by 1861
Abstract
Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely [...] Read more.
Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed. Full article
(This article belongs to the Special Issue Trends in Fractional Modelling in Science and Innovative Technologies)
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Review

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47 pages, 584 KiB  
Review
Non-Local Kinetics: Revisiting and Updates Emphasizing Fractional Calculus Applications
by Jordan Hristov
Symmetry 2023, 15(3), 632; https://doi.org/10.3390/sym15030632 - 2 Mar 2023
Cited by 4 | Viewed by 1125
Abstract
Non-local kinetic problems spanning a wide area of problems where fractional calculus is applicable have been analyzed. Classical fractional kinetics based on the Continuum Time Random Walk diffusion model with the absence of stationary states, real-world problems from pharmacokinetics, and modern material processing [...] Read more.
Non-local kinetic problems spanning a wide area of problems where fractional calculus is applicable have been analyzed. Classical fractional kinetics based on the Continuum Time Random Walk diffusion model with the absence of stationary states, real-world problems from pharmacokinetics, and modern material processing have been reviewed. Fractional allometry has been considered a potential area of application. The main focus in the analysis has been paid to the memory functions in the convolution formulation, crossing from the classical power law to versions of the Mittag-Leffler function. The main idea is to revisit the non-local kinetic problems with an update updating on new issues relevant to new trends in fractional calculus. Full article
(This article belongs to the Special Issue Trends in Fractional Modelling in Science and Innovative Technologies)
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