The Craft of Fractional Modelling in Science and Engineering III

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

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 14981

Special Issue Editor

Special Issue Information

Dear Colleagues,

Fractional calculus has performed an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. Modeling methods involving fractional operators have continuously been generalized and enhanced, especially during the last few decades. Many operations in physics and engineering can be defined accurately using systems of differential equations containing different types of fractional derivatives.

The goal of this Special Issue is to report the latest progress and to present the craft of fractional modeling in science and engineering. We therefore invite researchers working within fields of theory, methods, and applications of these problems to submit their latest findings to this Special Issue.

The best articles from the collection will be selected by the guest-editor and the editorial board and will be published as a book.

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

  • Fractional modeling: broad aspects
  • Solution techniques: analytical and numerical
  • Memory kernels: identification, construction and definitions of new fractional operators
  • Diffusion models
  • Local fractional calculus
  • Discrete fractional calculus
  • Heat, mass and momentum transfer (fluid dynamics) with relaxations
  • Mechanics and rheology of solid materials
  • Nanoapplications of fractional modeling
  • Biomechanical and biomedical applications of fractional calculus
  • Chaos and complexity
  • Thermodynamic compatibility of fractional models
  • Control problems and model identifications with fractional operators
  • Electrochemical systems and alternative energy sources

Prof. Dr. Jordan Hristov
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.

Published Papers (7 papers)

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Editorial

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2 pages, 167 KiB  
Editorial
The Craft of Fractional Modelling in Science and Engineering: II and III
by Jordan Hristov
Fractal Fract. 2021, 5(4), 281; https://doi.org/10.3390/fractalfract5040281 - 20 Dec 2021
Cited by 1 | Viewed by 1624
Abstract
A comprehensive understanding of fractional systems plays a pivotal role in practical applications [...] Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)

Research

Jump to: Editorial

24 pages, 4041 KiB  
Article
Analysis of the Effects of the Viscous Thermal Losses in the Flute Musical Instruments
by Gaby Abou Haidar, Xavier Moreau and Roy Abi Zeid Daou
Fractal Fract. 2021, 5(1), 11; https://doi.org/10.3390/fractalfract5010011 - 19 Jan 2021
Cited by 2 | Viewed by 1850
Abstract
This article presents the third part of a larger project whose final objective is to study and analyse the effects of viscous thermal losses in a flute wind musical instrument. After implementing the test bench in the first phase and modelling and validating [...] Read more.
This article presents the third part of a larger project whose final objective is to study and analyse the effects of viscous thermal losses in a flute wind musical instrument. After implementing the test bench in the first phase and modelling and validating the dynamic behaviour of the simulator, based on the previously implemented test bench (without considering the losses in the system) in the second phase, this third phase deals with the study of the viscous thermal losses that will be generated within the resonator of the flute. These losses are mainly due to the friction of the air inside the resonator with its boundaries and the changes of the temperature within this medium. They are mainly affected by the flute geometry and the materials used in the fabrication of this instrument. After modelling these losses in the frequency domain, they will be represented using a system approach where the fractional order part is separated from the system’s transfer function. Thus, this representation allows us to study, in a precise way, the influence of the fractional order behaviour on the overall system. Effectively, the fractional behavior only appears much below the 20 Hz audible frequencies, but it explains the influence of this order on the frequency response over the range [20–20,000] Hz. Some simulations will be proposed to show the effects of the fractional order on the system response. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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10 pages, 391 KiB  
Article
Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation
by Rafał Brociek, Agata Chmielowska and Damian Słota
Fractal Fract. 2020, 4(3), 39; https://doi.org/10.3390/fractalfract4030039 - 06 Aug 2020
Cited by 5 | Viewed by 1864
Abstract
This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) [...] Read more.
This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm’s accuracy. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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18 pages, 439 KiB  
Article
Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation
by Emilia Bazhlekova and Ivan Bazhlekov
Fractal Fract. 2020, 4(3), 32; https://doi.org/10.3390/fractalfract4030032 - 08 Jul 2020
Cited by 10 | Viewed by 2113
Abstract
The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized [...] Read more.
The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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15 pages, 418 KiB  
Article
Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion
by Maike Antonio Faustino dos Santos
Fractal Fract. 2020, 4(2), 28; https://doi.org/10.3390/fractalfract4020028 - 22 Jun 2020
Cited by 13 | Viewed by 2303
Abstract
Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a [...] Read more.
Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a non-static stochastic resetting model in the context of comb structure that consists of a structure formed by backbone in x axis and branches in y axis. Then, we find the exact analytical solutions for marginal distribution concerning x and y axis. Moreover, we show the time evolution behavior to mean square displacements (MSD) in both directions. As a consequence, the model revels that until the system reaches the equilibrium, i.e., constant MSD, there is a Brownian diffusion in y direction, i.e., ( Δ y ) 2 t , and a crossover between sub and ballistic diffusion behaviors in x direction, i.e., ( Δ x ) 2 t 1 2 and ( Δ x ) 2 t 2 respectively. For static stochastic resetting, the ballistic regime vanishes. Also, we consider the idealized model according to the memory kernels to investigate the exponential and tempered power-law memory kernels effects on diffusive behaviors. In this way, we expose a rich class of anomalous diffusion process with crossovers among them. The proposal and the techniques applied in this work are useful to describe random walkers with non-static stochastic resetting on comb structure. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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11 pages, 2326 KiB  
Article
Simplified Mathematical Model for the Description of Anomalous Migration of Soluble Substances in Vertical Filtration Flow
by Vsevolod Bohaienko and Volodymyr Bulavatsky
Fractal Fract. 2020, 4(2), 20; https://doi.org/10.3390/fractalfract4020020 - 06 May 2020
Cited by 5 | Viewed by 1956
Abstract
Since the use of the fractional-differential mathematical model of anomalous geomigration process based on the MIM (mobile–immoble media) approach in engineering practice significantly complicates simulations, a corresponding simplified mathematical model is constructed. For this model, we state a two-dimensional initial-boundary value problem of [...] Read more.
Since the use of the fractional-differential mathematical model of anomalous geomigration process based on the MIM (mobile–immoble media) approach in engineering practice significantly complicates simulations, a corresponding simplified mathematical model is constructed. For this model, we state a two-dimensional initial-boundary value problem of convective diffusion of soluble substances under the conditions of vertical steady-state filtration of groundwater with free surface from a reservoir to a coastal drain. To simplify the domain of simulation, we use the technique of transition into the domain of complex flow potential through a conformal mapping. In the case of averaging filtration velocity over the domain of complex flow potential, an analytical solution of the considered problem is obtained. In the general case of a variable filtration velocity, an algorithm has been developed to obtain numerical solutions. The results of process simulation using the presented algorithm shows that the constructed mathematical model can be efficiently used to simplify and accelerate modeling process. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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12 pages, 484 KiB  
Article
Parametric Identification of Nonlinear Fractional Hammerstein Models
by Vineet Prasad, Kajal Kothari and Utkal Mehta
Fractal Fract. 2020, 4(1), 2; https://doi.org/10.3390/fractalfract4010002 - 30 Dec 2019
Cited by 8 | Viewed by 2540
Abstract
In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational [...] Read more.
In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational matrix of block pulse functions to reduce computational complexity. A special test signal is developed to isolate the identification of the nonlinear static function from that of the fractional-order linear dynamic system. The merit of the proposed technique is indicated by concurrent identification of the fractional order with linear system coefficients, algebraic representation of the immeasurable nonlinear static function output, and permitting use of non-iterative procedures for identification of the nonlinearity. The efficacy of the proposed method is exhibited through simulation at various signal-to-noise ratios. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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