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3 pages, 195 KiB

Open AccessEditorial

The Craft of Fractional Modeling in Science and Engineering 2017

by Jordan Hristov

Fractal Fract. 2018, 2(2), 16; https://doi.org/10.3390/fractalfract2020016 - 15 Apr 2018

Cited by 6 | Viewed by 2955

Abstract

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(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

Research

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12 pages, 1904 KiB

Open AccessArticle

Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis

by Ervin K. Lenzi, Andrea Ryba and Marcelo K. Lenzi

Fractal Fract. 2018, 2(1), 11; https://doi.org/10.3390/fractalfract2010011 - 11 Feb 2018

Cited by 2 | Viewed by 3306

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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14 pages, 407 KiB

Open AccessArticle

Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model

by José Francisco Gómez-Aguilar and Abdon Atangana

Fractal Fract. 2018, 2(1), 10; https://doi.org/10.3390/fractalfract2010010 - 09 Feb 2018

Cited by 29 | Viewed by 4274

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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19 pages, 3409 KiB

Open AccessArticle

Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium

by Xavier Moreau, Roy Abi Zeid Daou and Fady Christophy

Fractal Fract. 2018, 2(1), 5; https://doi.org/10.3390/fractalfract2010005 - 17 Jan 2018

Cited by 8 | Viewed by 3506

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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23 pages, 434 KiB

Open AccessArticle

Fractional Velocity as a Tool for the Study of Non-Linear Problems

by Dimiter Prodanov

Fractal Fract. 2018, 2(1), 4; https://doi.org/10.3390/fractalfract2010004 - 17 Jan 2018

Cited by 19 | Viewed by 3566

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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11 pages, 550 KiB

Open AccessArticle

European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

by Mehmet Yavuz and Necati Özdemir

Fractal Fract. 2018, 2(1), 3; https://doi.org/10.3390/fractalfract2010003 - 16 Jan 2018

Cited by 92 | Viewed by 8191

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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929 KiB

Open AccessArticle

Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation

by Rafał Brociek, Damian Słota, Mariusz Król, Grzegorz Matula and Waldemar Kwaśny

Fractal Fract. 2017, 1(1), 17; https://doi.org/10.3390/fractalfract1010017 - 12 Dec 2017

Cited by 18 | Viewed by 3402

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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283 KiB

Open AccessArticle

Series Solution of the Pantograph Equation and Its Properties

by Sachin Bhalekar and Jayvant Patade

Fractal Fract. 2017, 1(1), 16; https://doi.org/10.3390/fractalfract1010016 - 08 Dec 2017

Cited by 12 | Viewed by 3165

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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392 KiB

Open AccessArticle

Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model

by Emilia Bazhlekova and Ivan Bazhlekov

Fractal Fract. 2017, 1(1), 7; https://doi.org/10.3390/fractalfract1010007 - 24 Oct 2017

Cited by 11 | Viewed by 3351

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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284 KiB

Open AccessArticle

Dynamics and Stability Results for Hilfer Fractional Type Thermistor Problem

by D. Vivek, K. Kanagarajan and Seenith Sivasundaram

Fractal Fract. 2017, 1(1), 5; https://doi.org/10.3390/fractalfract1010005 - 09 Sep 2017

Cited by 17 | Viewed by 3276

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

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)

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