Special Issue "The Craft of Fractional Modelling in Science and Engineering"

A special issue of Fractal and Fractional (ISSN 2504-3110).

Deadline for manuscript submissions: closed (31 December 2017)

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Guest Editor
Prof. Dr. Jordan Hristov

Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 1756 Sofia, Bulgaria
Website 1 | Website 2 | Website 3 | E-Mail
Interests: non-lineat transport phenomena; modelling; scaling; fractional calculus; heat and mass transfer; diffusion problems

Special Issue Information

Dear Colleagues,

Fractional calculus performs an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. The modelling methods involving fractional operators are continuously generalized and enhanced especially during the last few decades. Many operations in physics and engineering can be defined accurately by using systems of differential equations containing different type of fractional derivatives.

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

The best articles from the collection will be selected by the Guest Editor and the Editorial Board and published as a book.

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

  • Fractional modelling: a 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
  • Nano-applications of fractional modelling
  • 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
  • Electromagnetics
  • Fractional electric circuits

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 papers will be 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 quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) is waived for well-prepared manuscripts submitted to this issue. 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 modelling
  • applied models
  • solution techniques
  • memory kernels
  • fractional operator definitions
  • biomechanical and medical applications
  • control and identification
  • local fractional calculus
  • discrete fractional calculus
  • electrochemical systems

Published Papers (10 papers)

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Editorial

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Open AccessEditorial The Craft of Fractional Modeling in Science and Engineering 2017
Fractal Fract 2018, 2(2), 16; https://doi.org/10.3390/fractalfract2020016
Received: 12 April 2018 / Revised: 12 April 2018 / Accepted: 13 April 2018 / Published: 15 April 2018
Cited by 1 | PDF Full-text (195 KB) | HTML Full-text | XML Full-text

Research

Jump to: Editorial

Open AccessArticle Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis
Fractal Fract 2018, 2(1), 11; https://doi.org/10.3390/fractalfract2010011
Received: 28 December 2017 / Revised: 6 February 2018 / Accepted: 8 February 2018 / Published: 11 February 2018
Cited by 1 | PDF Full-text (1904 KB) | HTML Full-text | XML Full-text
Abstract
A fractional-calculus-based model is used to analyze the data obtained from the image analysis of mixtures of olive and soybean oil, which were quantified with the RGB color system. The model consists in a linear fractional differential equation, containing one fractional derivative of [...] Read more.
A fractional-calculus-based model is used to analyze the data obtained from the image analysis of mixtures of olive and soybean oil, which were quantified with the RGB color system. The model consists in a linear fractional differential equation, containing one fractional derivative of order α and an additional term multiplied by a parameter k. Using a hybrid parameter estimation scheme (genetic algorithm and a simplex-based algorithm), the model parameters were estimated as k = 3.42 ± 0.12 and α = 1.196 ± 0.027, while a correlation coefficient value of 0.997 was obtained. For the sake of comparison, parameter α was set equal to 1 and an integer order model was also studied, resulting in a one-parameter model with k = 3.11 ± 0.28. Joint confidence regions are calculated for the fractional order model, showing that the derivative order is statistically different from 1. Finally, an independent validation sample of color component B equal to 96 obtained from a sample with olive oil mass fraction equal to 0.25 is used for prediction purposes. The fractional model predicted the color B value equal to 93.1 ± 6.6. Full article
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Open AccessArticle Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model
Fractal Fract 2018, 2(1), 10; https://doi.org/10.3390/fractalfract2010010
Received: 20 November 2017 / Revised: 11 January 2018 / Accepted: 7 February 2018 / Published: 9 February 2018
Cited by 8 | PDF Full-text (407 KB) | HTML Full-text | XML Full-text
Abstract
This paper considers the Freedman model using the Liouville–Caputo fractional-order derivative and the fractional-order derivative with Mittag–Leffler kernel in the Liouville–Caputo sense. Alternative solutions via Laplace transform, Sumudu–Picard and Adams–Moulton rules were obtained. We prove the uniqueness and existence of the solutions for [...] Read more.
This paper considers the Freedman model using the Liouville–Caputo fractional-order derivative and the fractional-order derivative with Mittag–Leffler kernel in the Liouville–Caputo sense. Alternative solutions via Laplace transform, Sumudu–Picard and Adams–Moulton rules were obtained. We prove the uniqueness and existence of the solutions for the alternative model. Numerical simulations for the prediction and interaction between a unilingual and a bilingual population were obtained for different values of the fractional order. Full article
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Open AccessArticle Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium
Fractal Fract 2018, 2(1), 5; https://doi.org/10.3390/fractalfract2010005
Received: 19 December 2017 / Revised: 9 January 2018 / Accepted: 12 January 2018 / Published: 17 January 2018
Cited by 1 | PDF Full-text (3409 KB) | HTML Full-text | XML Full-text
Abstract
The control of thermal interfaces has gained importance in recent years because of the high cost of heating and cooling materials in many applications. Thus, the main focus in this work is to compare the second and third generations of the CRONE controller [...] Read more.
The control of thermal interfaces has gained importance in recent years because of the high cost of heating and cooling materials in many applications. Thus, the main focus in this work is to compare the second and third generations of the CRONE controller (French acronym of Commande Robuste d’Ordre Non Entier), which means a non-integer order robust controller, and to synthesize a robust controller that can fit several types of systems. For this study, the plant consists of a rectangular homogeneous bar of length L, where the heating element in applied on one boundary, and a temperature sensor is placed at distance x from that boundary (x is considered very small with respect to L). The type of material used is the third parameter, which may help in analyzing the robustness of the synthesized controller. The originality of this work resides in controlling a non-integer plant using a fractional order controller, as, so far, almost all of the systems where the CRONE controller has been implemented were of integer order. Three case studies were defined in order to show how and where each CRONE generation controller can be applied. These case studies were chosen in such a way as to influence the asymptotic behavior of the open-loop transfer function in the Black–Nichols diagram in order to point out the importance of respecting the conditions of the applications of the CRONE generations. Results show that the second generation performs well when the parametric uncertainties do not affect the phase of the plant, whereas the third generation is the most robust, even when both the phase and the gain variations are encountered. However, it also has some limitations, especially when the temperature to be controlled is far from the interface when the density of flux is applied. Full article
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Open AccessArticle Fractional Velocity as a Tool for the Study of Non-Linear Problems
Fractal Fract 2018, 2(1), 4; https://doi.org/10.3390/fractalfract2010004
Received: 27 December 2017 / Revised: 14 January 2018 / Accepted: 15 January 2018 / Published: 17 January 2018
Cited by 5 | PDF Full-text (434 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
Singular functions and, in general, Hölder functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocities as tools to characterize Hölder and singular functions, in particular. Fractional velocities are defined as limits [...] Read more.
Singular functions and, in general, Hölder functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocities as tools to characterize Hölder and singular functions, in particular. Fractional velocities are defined as limits of the difference quotients of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their non trivial values is disconnected. This can be used for example to model instantaneous interactions, for example Langevin dynamics. Examples are given by the De Rham and Neidinger’s singular functions, represented by limits of iterative function systems. Finally, the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated. Full article
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Open AccessArticle European Vanilla Option Pricing Model of Fractional Order without Singular Kernel
Fractal Fract 2018, 2(1), 3; https://doi.org/10.3390/fractalfract2010003
Received: 28 December 2017 / Revised: 14 January 2018 / Accepted: 14 January 2018 / Published: 16 January 2018
Cited by 12 | PDF Full-text (550 KB) | HTML Full-text | XML Full-text
Abstract
Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called [...] Read more.
Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo–Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black–Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time. Full article
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Open AccessArticle Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation
Fractal Fract 2017, 1(1), 17; https://doi.org/10.3390/fractalfract1010017
Received: 20 November 2017 / Revised: 8 December 2017 / Accepted: 9 December 2017 / Published: 12 December 2017
Cited by 2 | PDF Full-text (929 KB) | HTML Full-text | XML Full-text
Abstract
The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction [...] Read more.
The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction inverse problem. Additional information for the inverse problem was the temperature measurements obtained from porous aluminum. In this paper, the authors used a finite difference method to solve direct problems and the Real Ant Colony Optimization algorithm to find a minimum of certain functional (solve the inverse problem). Finally, the authors present the temperature values computed from the model and compare them with the measured data from real objects. Full article
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Open AccessArticle Series Solution of the Pantograph Equation and Its Properties
Fractal Fract 2017, 1(1), 16; https://doi.org/10.3390/fractalfract1010016
Received: 26 October 2017 / Revised: 28 November 2017 / Accepted: 30 November 2017 / Published: 8 December 2017
Cited by 1 | PDF Full-text (283 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we discuss the classical pantograph equation and its generalizations to include fractional order and the higher order case. The special functions are obtained from the series solution of these equations. We study different properties of these special functions and establish [...] Read more.
In this paper, we discuss the classical pantograph equation and its generalizations to include fractional order and the higher order case. The special functions are obtained from the series solution of these equations. We study different properties of these special functions and establish the relation with other functions. Further, we discuss some contiguous relations for these special functions. Full article
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Open AccessArticle Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model
Fractal Fract 2017, 1(1), 7; https://doi.org/10.3390/fractalfract1010007
Received: 21 September 2017 / Revised: 20 October 2017 / Accepted: 23 October 2017 / Published: 24 October 2017
Cited by 3 | PDF Full-text (392 KB) | HTML Full-text | XML Full-text
Abstract
Stokes’ first problem for a class of viscoelastic fluids with the generalized fractional Maxwell constitutive model is considered. The constitutive equation is obtained from the classical Maxwell stress–strain relation by substituting the first-order derivatives of stress and strain by derivatives of non-integer orders [...] Read more.
Stokes’ first problem for a class of viscoelastic fluids with the generalized fractional Maxwell constitutive model is considered. The constitutive equation is obtained from the classical Maxwell stress–strain relation by substituting the first-order derivatives of stress and strain by derivatives of non-integer orders in the interval ( 0 , 1 ] . Explicit integral representation of the solution is derived and some of its characteristics are discussed: non-negativity and monotonicity, asymptotic behavior, analyticity, finite/infinite propagation speed, and absence of wave front. To illustrate analytical findings, numerical results for different values of the parameters are presented. Full article
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Open AccessArticle Dynamics and Stability Results for Hilfer Fractional Type Thermistor Problem
Fractal Fract 2017, 1(1), 5; https://doi.org/10.3390/fractalfract1010005
Received: 22 August 2017 / Revised: 5 September 2017 / Accepted: 6 September 2017 / Published: 9 September 2017
Cited by 3 | PDF Full-text (284 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we study the dynamics and stability of thermistor problem for Hilfer fractional type. Classical fixed point theorems are utilized in deriving the results. Full article
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