Fractional Dynamics 2021

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

Deadline for manuscript submissions: closed (15 April 2022) | Viewed by 10224

Special Issue Editors

Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals
School of Mathematics and Statistics, Tianshui Normal University, 741000 Gansu, China
Interests: numerical and Computational methods in fractional differential equations; high-order numerical differential formulas for the fractional derivatives; high-order numerical algorithms for fractional differential equations
Department of Basic and Applied Sciences for Engineering, University of Rome 'La Sapienza', 00161 Rome, Italy
Interests: numerical analysis; approximation theory; spline; refinable function; numerical solution of fractional differential problem; numerical inverse problem
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the recent years, modeling, numerical simulation, and applications of Fractional Calculus have increasingly become very popular, particularly impressive concerning applications.  The basic ideas on fractional derivatives have achieved an incredibly  valuable status, while the variety of applications in mathematics, physics, engineering, economics, biology, and medicine have opened new challenging research fields. Clearly, these applications call for the development of suitable  mathematical tools to extract quantitative information from the models, newly reformulated in terms of fractional differential equations. This Special Issue will address timely subjects, such as a variety of dynamical systems governed by fractional differential equations, pertaining to:

  • the description of epidemiological models (typically based on fractional in time and/or in space ordinary and partial differential equations) where several populations interact. All this, exploiting big data and possibly machine learning techniques;
  • modeling viruses reproduction subject to genetic variations; interplay with a variety of vaccines;
  • earthquake modeling, all based on real data;
  • (fractional) control in engineering problems with industrial applications;
  • economical and financial science modeling, especially in pandemic times. Again, all this exploiting big data and possibly *machine learning* techniques;
  • fractional diffusion on (complex) networks; in particular, applications of neural networks to fractional (i.e., anomalous) diffusion equations.

Prof. Dr. Carlo Cattani
Prof. Dr. Hengfei Ding
Prof. Dr. Francesca Pitolli
Guest Editors

Manuscript Submission Information

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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
  • fractal
  • fractional dynamical systems
  • fractional partial differential equations

Published Papers (5 papers)

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Research

20 pages, 864 KiB  
Article
Understanding Dynamics and Bifurcation Control Mechanism for a Fractional-Order Delayed Duopoly Game Model in Insurance Market
Fractal Fract. 2022, 6(5), 270; https://doi.org/10.3390/fractalfract6050270 - 17 May 2022
Cited by 2 | Viewed by 1354
Abstract
Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is [...] Read more.
Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is a very meaningful work. In this present work, we set up a novel fractional-order delayed duopoly game model in insurance market and discuss the dynamics including existence and uniqueness, non-negativeness, and boundedness of solution for the established fractional-order delayed duopoly game model in insurance market. By selecting the delay as a bifurcation parameter, we build a new delay-independent condition ensuring the stability and creation of Hopf bifurcation of the built fractional-order delayed duopoly game model. Making use of a suitable definite function, we explore the globally asymptotic stability of the involved fractional-order delayed duopoly game model. By virtue of hybrid controller which includes state feedback and parameter perturbation, we can effectively control the stability and the time of creation of Hopf bifurcation for the involved fractional-order delayed duopoly game model. The research indicates that time delay plays an all-important role in stabilizing the system and controlling the time of onset of Hopf bifurcation of the involved fractional-order delayed duopoly game model. To check the rationality of derived primary conclusions, Matlab simulation plots are explicitly presented. The established results in this manuscript are wholly novel and own immense theoretical guiding significance in managing and operating insurance companies. Full article
(This article belongs to the Special Issue Fractional Dynamics 2021)
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23 pages, 522 KiB  
Article
Numerical Analysis of Local Discontinuous Galerkin Method for the Time-Fractional Fourth-Order Equation with Initial Singularity
Fractal Fract. 2022, 6(4), 206; https://doi.org/10.3390/fractalfract6040206 - 07 Apr 2022
Cited by 2 | Viewed by 1598
Abstract
In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time t=0. The Caputo time-fractional derivative with [...] Read more.
In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time t=0. The Caputo time-fractional derivative with derivative order α(0,1) is discretized by the well-known L1 formula on nonuniform meshes; for the spatial derivative, the local discontinuous Galerkin (LDG) finite element method is used. Based on the discrete fractional Gronwall’s inequality, we prove the stability of the proposed scheme and the optimal error estimate for the solution, i.e., (2α)-order accurate in time and (k+1)-order accurate in space, when piece-wise polynomials of degree at most k are used. Moreover, a second-order and nonuniform time-stepping scheme is developed for the fractional model. The scheme uses the L2-1σ formula for the time fractional derivative and the LDG method for the space approximation. The stability and temporal optimal second-order convergence of the scheme are also shown. Finally, some numerical experiments are presented to confirm the theoretical results. Full article
(This article belongs to the Special Issue Fractional Dynamics 2021)
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22 pages, 492 KiB  
Article
Power Law Kernel Analysis of MHD Maxwell Fluid with Ramped Boundary Conditions: Transport Phenomena Solutions Based on Special Functions
Fractal Fract. 2021, 5(4), 248; https://doi.org/10.3390/fractalfract5040248 - 01 Dec 2021
Cited by 19 | Viewed by 2105
Abstract
In this paper, a new approach to find exact solutions is carried out for a generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate, which is analyzed for ramped-wall temperature and velocity with constant concentration. The vertical plate is [...] Read more.
In this paper, a new approach to find exact solutions is carried out for a generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate, which is analyzed for ramped-wall temperature and velocity with constant concentration. The vertical plate is suspended in a porous medium and encounters the effects of radiation. An innovative definition of the time-fractional operator in power-law-kernel form is implemented to hypothesize the constitutive mass, energy, and momentum equations. The Laplace integral transformation technique is applied on a dimensionless form of governing partial differential equations by introducing some non-dimensional suitable parameters to establish the exact expressions in terms of special functions for ramped velocity, temperature, and constant-concentration fields. In order to validate the problem, the absence of the mass Grashof parameter led to the investigated solutions obtaining good agreement in existing literature. Additionally, several system parameters were used, such as as magnetic value M, Prandtl value Pr, Maxwell parameter λ, dimensionless time τ, Schmidt number “Sc”, fractional parameter α, andMass and Thermal Grashof numbers Gm and Gr, respectively, to examine their impacts on velocity, wall temperature, and constant concentration. Results are also discussed in detail and demonstrated graphically via Mathcad-15 software. A comprehensive comparative study between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. Full article
(This article belongs to the Special Issue Fractional Dynamics 2021)
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12 pages, 1461 KiB  
Article
Semi-Analytical Solutions for Fuzzy Caputo–Fabrizio Fractional-Order Two-Dimensional Heat Equation
Fractal Fract. 2021, 5(4), 139; https://doi.org/10.3390/fractalfract5040139 - 25 Sep 2021
Cited by 11 | Viewed by 1774
Abstract
In the analysis in this article, we developed a scheme for the computation of a semi-analytical solution to a fuzzy fractional-order heat equation of two dimensions having some external diffusion source term. For this, we applied the Laplace transform along with decomposition techniques [...] Read more.
In the analysis in this article, we developed a scheme for the computation of a semi-analytical solution to a fuzzy fractional-order heat equation of two dimensions having some external diffusion source term. For this, we applied the Laplace transform along with decomposition techniques and the Adomian polynomial under the Caputo–Fabrizio fractional differential operator. Furthermore, for obtaining a semi-analytical series-type solution, the decomposition of the unknown quantity and its addition established the said solution. The obtained series solution was calculated and approached the approximate solution of the proposed equation. For the validation of our scheme, three different examples have been provided, and the solutions were calculated in fuzzy form. All the three illustrations simulated two different fractional orders between 0 and 1 for the upper and lower portions of the fuzzy solution. The said fractional operator is nonsingular and global due to the presence of the exponential function. It globalizes the dynamical behavior of the said equation, which is guaranteed for all types of fuzzy solution lying between 0 and 1 at any fractional order. The fuzziness is also included in the unknown quantity due to the fuzzy number providing the solution in fuzzy form, having upper and lower branches. Full article
(This article belongs to the Special Issue Fractional Dynamics 2021)
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21 pages, 3444 KiB  
Article
An Experimental Approach towards Motion Modeling and Control of a Vehicle Transiting a Non-Newtonian Environment
Fractal Fract. 2021, 5(3), 104; https://doi.org/10.3390/fractalfract5030104 - 25 Aug 2021
Cited by 3 | Viewed by 1650
Abstract
The present work tackles the modeling of the motion dynamics of an object submerged in a non-Newtonian environment. The mathematical model is developed starting from already known Newtonian interactions between the submersible and the fluid. The obtained model is therefore altered through optimization [...] Read more.
The present work tackles the modeling of the motion dynamics of an object submerged in a non-Newtonian environment. The mathematical model is developed starting from already known Newtonian interactions between the submersible and the fluid. The obtained model is therefore altered through optimization techniques to describe non-Newtonian interactions on the motion of the vehicle by using real-life data regarding non-Newtonian influences on submerged thrusting. For the obtained non-Newtonian fractional order process model, a fractional order control approach is employed to sway the submerged object’s position inside the viscoelastic environment. The presented modeling and control methodologies are solidified by real-life experimental data used to validate the veracity of the presented concepts. The robustness of the control strategy is experimentally validated on both Newtonian and non-Newtonian environments. Full article
(This article belongs to the Special Issue Fractional Dynamics 2021)
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