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Fractal Fract
Volume 4
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Fractal Fract., Volume 4, Issue 3 (September 2020) – 18 articles

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6 pages, 580 KiB  
Article
Integral Representation of Fractional Derivative of Delta Function
by Ming Li
Fractal Fract. 2020, 4(3), 47; https://doi.org/10.3390/fractalfract4030047 - 20 Sep 2020
Cited by 8 | Viewed by 2736
Abstract
Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the [...] Read more.
Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
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Figure 1

Figure 1
<p>Linear system.</p> Full article ">Figure 2
<p>A fractional system resulting from <math display="inline"><semantics> <mrow> <msup> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </semantics></math></p> Full article ">
15 pages, 6290 KiB  
Article
Fractal and Fractional Derivative Modelling of Material Phase Change
by Harry Esmonde
Fractal Fract. 2020, 4(3), 46; https://doi.org/10.3390/fractalfract4030046 - 14 Sep 2020
Cited by 8 | Viewed by 2373
Abstract
An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations [...] Read more.
An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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Figure 1

Figure 1
<p>Rheological models.</p> Full article ">Figure 2
<p>Bode plots of Maxwell and Kelvin Voigt systems.</p> Full article ">Figure 3
<p>Fractal network and equivalent system.</p> Full article ">Figure 4
<p>Power of <span class="html-italic">s</span> in fractal evolution.</p> Full article ">Figure 5
<p>Cure speed of adhesive versus bond gap [<a href="#B13-fractalfract-04-00046" class="html-bibr">13</a>].</p> Full article ">Figure 6
<p>Power of <span class="html-italic">s</span> in adhesive cure.</p> Full article ">Figure 7
<p>Schematic of squeeze film geometry showing a sample under test (not to scale).</p> Full article ">Figure 8
<p>Design of top plate (<b>left</b>) and entire test assembly (<b>right</b>).</p> Full article ">Figure 9
<p>Frequency response for Cream at 10 <span class="html-fig-inline" id="fractalfract-04-00046-i001"> <img alt="Fractalfract 04 00046 i001" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i001.png"/></span>, 20 <span class="html-fig-inline" id="fractalfract-04-00046-i002"> <img alt="Fractalfract 04 00046 i002" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i002.png"/></span> and 30 <span class="html-fig-inline" id="fractalfract-04-00046-i003"> <img alt="Fractalfract 04 00046 i003" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i003.png"/></span> min (experimental), and the modelled impedance <span class="html-fig-inline" id="fractalfract-04-00046-i004"> <img alt="Fractalfract 04 00046 i004" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i004.png"/></span>.</p> Full article ">Figure 10
<p>Frequency response for gelatine at 0 <span class="html-fig-inline" id="fractalfract-04-00046-i001"> <img alt="Fractalfract 04 00046 i001" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i001.png"/></span>, 4 <span class="html-fig-inline" id="fractalfract-04-00046-i002"> <img alt="Fractalfract 04 00046 i002" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i002.png"/></span> and 9 <span class="html-fig-inline" id="fractalfract-04-00046-i003"> <img alt="Fractalfract 04 00046 i003" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i003.png"/></span> min (experimental), and the modelled impedance <span class="html-fig-inline" id="fractalfract-04-00046-i004"> <img alt="Fractalfract 04 00046 i004" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i004.png"/></span>.</p> Full article ">Figure 11
<p>Frequency response near the gel point for gelatine at 0 <span class="html-fig-inline" id="fractalfract-04-00046-i001"> <img alt="Fractalfract 04 00046 i001" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i001.png"/></span>, 1 <span class="html-fig-inline" id="fractalfract-04-00046-i002"> <img alt="Fractalfract 04 00046 i002" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i002.png"/></span> and 2 <span class="html-fig-inline" id="fractalfract-04-00046-i003"> <img alt="Fractalfract 04 00046 i003" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i003.png"/></span> min (experimental), and the modelled impedance <span class="html-fig-inline" id="fractalfract-04-00046-i004"> <img alt="Fractalfract 04 00046 i004" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i004.png"/></span>.</p> Full article ">Figure 12
<p>Frequency response for ethyl vinyl acetate (EVA) at 97 °C <span class="html-fig-inline" id="fractalfract-04-00046-i001"> <img alt="Fractalfract 04 00046 i001" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i001.png"/></span>, 72 °C <span class="html-fig-inline" id="fractalfract-04-00046-i002"> <img alt="Fractalfract 04 00046 i002" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i002.png"/></span> and 36 °C <span class="html-fig-inline" id="fractalfract-04-00046-i003"> <img alt="Fractalfract 04 00046 i003" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i003.png"/></span> (experimental), and the modelled impedance <span class="html-fig-inline" id="fractalfract-04-00046-i004"> <img alt="Fractalfract 04 00046 i004" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i004.png"/></span>.</p> Full article ">Figure 13
<p>Experimental frequency response for EVA near the melting point, at 79 °C <span class="html-fig-inline" id="fractalfract-04-00046-i001"> <img alt="Fractalfract 04 00046 i001" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i001.png"/></span>, 72 °C <span class="html-fig-inline" id="fractalfract-04-00046-i002"> <img alt="Fractalfract 04 00046 i002" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i002.png"/></span>, 64 °C <span class="html-fig-inline" id="fractalfract-04-00046-i003"> <img alt="Fractalfract 04 00046 i003" src="/fractalfract/fractalfract-04-00046/article_deploy/html/images/fractalfract-04-00046-i003.png"/></span>.</p> Full article ">Figure 14
<p>Evolution of <math display="inline"><semantics> <mrow> <mi>β</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for gelatine.</p> Full article ">Figure 15
<p>Model fits to experimental data at 72 °C.</p> Full article ">Figure 16
<p>Evolution of <math display="inline"><semantics> <mrow> <mi>β</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for EVA.</p> Full article ">
15 pages, 296 KiB  
Article
Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus
by Arran Fernandez and Iftikhar Husain
Fractal Fract. 2020, 4(3), 45; https://doi.org/10.3390/fractalfract4030045 - 12 Sep 2020
Cited by 13 | Viewed by 2895
Abstract
Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, [...] Read more.
Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
18 pages, 554 KiB  
Article
Fractional SIS Epidemic Models
by Caterina Balzotti, Mirko D’Ovidio and Paola Loreti
Fractal Fract. 2020, 4(3), 44; https://doi.org/10.3390/fractalfract4030044 - 31 Aug 2020
Cited by 20 | Viewed by 3608
Abstract
In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with [...] Read more.
In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order α converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the α-SIS models. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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Figure 1

Figure 1
<p>Comparison between the solutions to the susceptible-infectious-susceptible (SIS) model and the explicit and numerical fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics></math>. The analysis shows correspondence between SIS model and the case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of our model. This result was expected, and it confirms the continuity wit respect to <math display="inline"><semantics> <mi>α</mi> </semantics></math> (see (P4)).</p> Full article ">Figure 2
<p>Comparison between the explicit and numerical fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Comparison between the explicit and numerical fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Comparison between the solutions to the SIS model and the fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics></math> (continuity w.r. to <math display="inline"><semantics> <mi>α</mi> </semantics></math>).</p> Full article ">Figure 5
<p>Comparison between the fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Comparison between the fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">
19 pages, 609 KiB  
Article
Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact
by Muhammad Farman, Ali Akgül, Dumitru Baleanu, Sumaiyah Imtiaz and Aqeel Ahmad
Fractal Fract. 2020, 4(3), 43; https://doi.org/10.3390/fractalfract4030043 - 21 Aug 2020
Cited by 32 | Viewed by 3206
Abstract
The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop [...] Read more.
The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system’s actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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Figure 1

Figure 1
<p><math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> interest rate with Caputo fractional derivative.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> investment demand with Caputo fractional derivative.</p> Full article ">Figure 3
<p><math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> price exponent with Caputo fractional derivative.</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> interest rate with ABC derivative.</p> Full article ">Figure 5
<p><math display="inline"><semantics> <mrow> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> investment demand with ABC derivative.</p> Full article ">Figure 6
<p><math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> price exponent with ABC derivative.</p> Full article ">Figure 7
<p><math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> interest rate with Caputo and ABC derivative.</p> Full article ">Figure 8
<p><math display="inline"><semantics> <mrow> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> investment demand with Caputo and ABC derivative.</p> Full article ">Figure 9
<p><math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> price exponent with Caputo and ABC derivative.</p> Full article ">Figure 10
<p><math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> price exponent with Caputo and ABC derivative.</p> Full article ">Figure 11
<p>Impact of critical minimum interest rate with <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Impact of critical minimum interest rate with <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>Impact of critical minimum interest rate with <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">
11 pages, 349 KiB  
Article
Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators
by Renat T. Sibatov and HongGuang Sun
Fractal Fract. 2020, 4(3), 42; https://doi.org/10.3390/fractalfract4030042 - 17 Aug 2020
Cited by 5 | Viewed by 2694
Abstract
The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–Planck equation, [...] Read more.
The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker–Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed. Full article
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Figure 1

Figure 1
<p>Schematic of the time-of-flight method in coplanar geometry.</p> Full article ">Figure 2
<p>Transient current for dispersive transport described by drift-diffusion equations with different fractional time-derivative operators. The curves are calculated by using Formula (<a href="#FD30-fractalfract-04-00042" class="html-disp-formula">30</a>). (<b>a</b>) Riemann-Liouville derivative, (<b>b</b>) tempered fractional operator, (<b>c</b>) Caputo-Fabrizio operator, (<b>d</b>) Atangana-Baleanu derivative. Here, <math display="inline"><semantics> <mi>α</mi> </semantics></math> is an order of fractional operators, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> truncation parameter, <span class="html-italic">v</span> drift velocity, <span class="html-italic">L</span> interelectrode distance.</p> Full article ">
8 pages, 2887 KiB  
Article
Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation
by Hulya Durur, Esin Ilhan and Hasan Bulut
Fractal Fract. 2020, 4(3), 41; https://doi.org/10.3390/fractalfract4030041 - 16 Aug 2020
Cited by 62 | Viewed by 3816
Abstract
This manuscript focuses on the application of the (m+1/G′)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are [...] Read more.
This manuscript focuses on the application of the (m+1/G′)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
►▼ Show Figures

Figure 1

Figure 1
<p>Three-dimensional and 2D graphs of Equation (15) for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>λ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>μ</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>τ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>ϖ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> values and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> for 2D.</p> Full article ">Figure 2
<p>Three-dimensional and 2D graphs for values <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>λ</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>μ</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>ρ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>τ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>ϖ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of Equation (17) and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> for 2D.</p> Full article ">Figure 3
<p>Three-dimensional and 2D graphs for values <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>λ</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>τ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>ϖ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of Equation (19) and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> for 2D.</p> Full article ">Figure 4
<p>Three-dimensional and 2D graphs for values <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>λ</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>μ</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>τ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>ϖ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of Equation (21) and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> for 2D.</p> Full article ">
5 pages, 218 KiB  
Viewpoint
Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?
by Jocelyn Sabatier
Fractal Fract. 2020, 4(3), 40; https://doi.org/10.3390/fractalfract4030040 - 11 Aug 2020
Cited by 23 | Viewed by 3374
Abstract
In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are [...] Read more.
In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
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 - 6 Aug 2020
Cited by 5 | Viewed by 2137
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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<p>Locations of the measurement points in cases of (<b>a</b>) four measurement points; (<b>b</b>) two measurement points.</p> Full article ">Figure 2
<p>The absolute errors of the solution <span class="html-italic">u</span> reconstruction for the measurement points <span class="html-italic">B</span> (<b>a</b>) and <span class="html-italic">C</span> (<b>b</b>) in the case of the measurements taken every 4 s.</p> Full article ">
10 pages, 294 KiB  
Article
A Stochastic Fractional Calculus with Applications to Variational Principles
by Houssine Zine and Delfim F. M. Torres
Fractal Fract. 2020, 4(3), 38; https://doi.org/10.3390/fractalfract4030038 - 1 Aug 2020
Cited by 10 | Viewed by 2993
Abstract
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, [...] Read more.
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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<p>Expectation of the extremal to the stochastic fractional problem of the calculus of variations of Example 2.</p> Full article ">
20 pages, 947 KiB  
Article
Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags
by Guido Maione
Fractal Fract. 2020, 4(3), 37; https://doi.org/10.3390/fractalfract4030037 - 27 Jul 2020
Cited by 6 | Viewed by 2521
Abstract
This paper concerns cascaded, shifted, fractional-order, lead compensators made by the serial connection of two stages introducing their respective phase leads in shifted adjacent frequency ranges. Adding up leads in these intervals gives a flat phase in a wide frequency range. Moreover, the [...] Read more.
This paper concerns cascaded, shifted, fractional-order, lead compensators made by the serial connection of two stages introducing their respective phase leads in shifted adjacent frequency ranges. Adding up leads in these intervals gives a flat phase in a wide frequency range. Moreover, the simple elements of the cascade can be easily realized by rational transfer functions. On this basis, a method is proposed in order to design a robust controller for a class of benchmark plants that are difficult to compensate due to monotonically increasing lags. The simulation experiments show the efficiency, performance and robustness of the approach. Full article
(This article belongs to the Special Issue Fractional Calculus in Control and Modelling)
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<p>Bode magnitude diagrams of the FLECs <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (solid lines) and of the second-order realizations <math display="inline"><semantics> <mrow> <msub> <mi>G</mi> <mrow> <mi>C</mi> <mi>H</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (dashed lines) for <math display="inline"><semantics> <mrow> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.7</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Bode phase diagrams of the FLECs <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (solid lines) and of the second-order realizations <math display="inline"><semantics> <mrow> <msub> <mi>G</mi> <mrow> <mi>C</mi> <mi>H</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (dashed lines) for <math display="inline"><semantics> <mrow> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.7</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Bode magnitude diagrams of <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (dashed lines), <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (dash-dotted lines) and <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (solid lines) for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ν</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>2</mn> </msub> <mspace width="4.pt"/> <mi>and </mi> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>12</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> <mspace width="4.pt"/> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mn>0.5</mn> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>2</mn> </msub> <mspace width="4.pt"/> <mi>and</mi> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>12</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> <mspace width="4.pt"/> </mrow> </semantics></math> 0.7 (curves <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <msub> <mrow/> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">c</mi> <msub> <mrow/> <mn>2</mn> </msub> <mspace width="4.pt"/> <mi>and</mi> <mspace width="4.pt"/> <mi mathvariant="normal">c</mi> <msub> <mrow/> <mn>12</mn> </msub> </mrow> </semantics></math>).</p> Full article ">Figure 4
<p>Bode phase diagrams of <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (dashed lines), <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (dash-dotted lines) and <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (solid lines) for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ν</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>2</mn> </msub> <mspace width="4.pt"/> <mi>and</mi> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>12</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> <mspace width="4.pt"/> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mn>0.5</mn> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>2</mn> </msub> <mspace width="4.pt"/> <mi>and</mi> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>12</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> <mspace width="4.pt"/> </mrow> </semantics></math> 0.7 (curves <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <msub> <mrow/> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">c</mi> <msub> <mrow/> <mn>2</mn> </msub> <mspace width="4.pt"/> <mi>and</mi> <mspace width="4.pt"/> <mi mathvariant="normal">c</mi> <msub> <mrow/> <mn>12</mn> </msub> </mrow> </semantics></math>).</p> Full article ">Figure 5
<p>Bode magnitude diagrams of <math display="inline"><semantics> <msubsup> <mi>H</mi> <mn>1</mn> <mn>2</mn> </msubsup> </semantics></math> (dashed lines) and of the Cascaded, Shifted, Fractional-order, LEad Compensators (CS-FLEC) <math display="inline"><semantics> <msub> <mi>H</mi> <mn>12</mn> </msub> </semantics></math> (solid lines) for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ν</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>3</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>3</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mn>0.5</mn> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>5</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>5</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> <mspace width="4.pt"/> <mn>0.7</mn> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>7</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>7</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Bode phase diagrams of <math display="inline"><semantics> <msubsup> <mi>H</mi> <mn>1</mn> <mn>2</mn> </msubsup> </semantics></math> (dashed lines) and of the CS-FLEC <math display="inline"><semantics> <msub> <mi>H</mi> <mn>12</mn> </msub> </semantics></math> (solid lines) for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ν</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>3</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>3</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> <mspace width="4.pt"/> <mn>0.5</mn> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>5</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>5</mn> </msub> <mrow> <mo>)</mo> <mo>,</mo> </mrow> <mspace width="4.pt"/> <mn>0.7</mn> <mspace width="4.pt"/> <mrow> <mo>(</mo> <mi>curves</mi> </mrow> <mspace width="4.pt"/> <mi mathvariant="normal">a</mi> <msub> <mrow/> <mn>7</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mi mathvariant="normal">b</mi> <msub> <mrow/> <mn>7</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Frequency response by the first plant <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msub> </semantics></math> (dash-dotted lines), the compensator <span class="html-italic">H</span> with the CS-FLEC (dashed lines) and the compensated system <math display="inline"><semantics> <msub> <mi>H</mi> <mrow> <mi>O</mi> <mi>L</mi> </mrow> </msub> </semantics></math> (solid lines).</p> Full article ">Figure 8
<p>Frequency response by the second plant <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mi>p</mi> <mn>2</mn> </mrow> </msub> </semantics></math> (dash-dotted lines), the compensator <span class="html-italic">H</span> with the CS-FLEC (dashed lines) and the compensated system <math display="inline"><semantics> <msub> <mi>H</mi> <mrow> <mi>O</mi> <mi>L</mi> </mrow> </msub> </semantics></math> (solid lines).</p> Full article ">Figure 9
<p>Unit step response in the first example for different values of <math display="inline"><semantics> <msub> <mi>K</mi> <mi>c</mi> </msub> </semantics></math>.</p> Full article ">Figure 10
<p>Unit step response in the first example for different plant time constants.</p> Full article ">Figure 11
<p>Unit step response in the first example for different fractional orders.</p> Full article ">Figure 12
<p>Control variable in the first example for different values of <math display="inline"><semantics> <msub> <mi>K</mi> <mi>c</mi> </msub> </semantics></math>.</p> Full article ">Figure 13
<p>Step disturbance rejection in the first example for different values of <math display="inline"><semantics> <msub> <mi>K</mi> <mi>c</mi> </msub> </semantics></math>.</p> Full article ">Figure 14
<p>Left: frequency response by the first plant <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msub> </semantics></math> (dash-dotted lines), the compensator <span class="html-italic">H</span> with two identical FLECs (dashed lines), and the compensated system <math display="inline"><semantics> <msub> <mi>H</mi> <mrow> <mi>O</mi> <mi>L</mi> </mrow> </msub> </semantics></math> (solid lines). Right: unit step response for different values of <math display="inline"><semantics> <msub> <mi>K</mi> <mi>c</mi> </msub> </semantics></math>.</p> Full article ">Figure 15
<p>Unit step response in the second example for different fractional orders.</p> Full article ">
15 pages, 474 KiB  
Article
Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective
by Agneta M. Balint and Stefan Balint
Fractal Fract. 2020, 4(3), 36; https://doi.org/10.3390/fractalfract4030036 - 21 Jul 2020
Cited by 13 | Viewed by 2969
Abstract
In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of [...] Read more.
In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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<p>Aquifers and wells.</p> Full article ">
22 pages, 904 KiB  
Article
Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate
by Mehmet Yavuz and Ndolane Sene
Fractal Fract. 2020, 4(3), 35; https://doi.org/10.3390/fractalfract4030035 - 16 Jul 2020
Cited by 112 | Viewed by 5535
Abstract
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral [...] Read more.
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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<p>Phase diagram of fractional P–PM with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Dynamical behavior of fractional P–PM under no harvesting rate with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Dynamical behavior of fractional P–PM with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Dynamical behavior of fractional P–PM with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Dynamics of the predator and prey in the model for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Dynamics of the predator in the model for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Dynamics of the prey in the model for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Dynamics of the predator and prey in the model for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Dynamics of the predator in the model for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Dynamics of the prey in the model for <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>.</p> Full article ">
11 pages, 1131 KiB  
Article
Hardware Implementation and Performance Study of Analog PIλDμ Controllers on DC Motor
by Dina A. John, Saket Sehgal and Karabi Biswas
Fractal Fract. 2020, 4(3), 34; https://doi.org/10.3390/fractalfract4030034 - 15 Jul 2020
Cited by 5 | Viewed by 3238
Abstract
In this paper, the performance of an analog PI λ D μ controller is done for speed regulation of a DC motor. The circuits for the fractional integrator and differentiator of PI λ D μ controller are designed by optimal pole-zero interlacing algorithm. [...] Read more.
In this paper, the performance of an analog PI λ D μ controller is done for speed regulation of a DC motor. The circuits for the fractional integrator and differentiator of PI λ D μ controller are designed by optimal pole-zero interlacing algorithm. The performance of the controller is compared with another PI λ D μ controller—in which the fractional integrator circuit employs a solid-state fractional capacitor. It can be verified from the results that using PI λ D μ controllers, the speed response of the DC motor has improved with reduction in settling time ( T s ), steady state error (SS error) and % overshoot (% M p ). Full article
(This article belongs to the Special Issue Fractional Calculus in Control and Modelling)
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<p>Circuit diagram of DC motor emulator.</p> Full article ">Figure 2
<p>Block diagram of the controlled DC motor emulator.</p> Full article ">Figure 3
<p>Circuit diagrams of fractional integrator (<math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>i</mi> </msub> <msup> <mi>s</mi> <mrow> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> </semantics></math>) for Type A and Type B controllers with <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.4.</p> Full article ">Figure 4
<p>Comparison of results. <span class="html-italic">V<sub>ref</sub></span> = reference signal, <span class="html-italic">V<sub>ω</sub></span> = DC motor response, <span class="html-italic">V<sub>ωA</sub></span> = response from Type A controlled DC motor and <span class="html-italic">V<sub>ωB</sub></span> = response from Type B controlled DC motor</p> Full article ">Figure 5
<p>Error voltage (e) and <math display="inline"><semantics> <msub> <mi>K</mi> <mi>i</mi> </msub> </semantics></math>I<math display="inline"><semantics> <msup> <mrow/> <mi>λ</mi> </msup> </semantics></math> voltage (<math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>I</mi> <mi>A</mi> </mrow> </msub> </semantics></math>) of Type A controller.</p> Full article ">Figure 6
<p>Error voltage (e) and <math display="inline"><semantics> <msub> <mi>K</mi> <mi>d</mi> </msub> </semantics></math>D<math display="inline"><semantics> <msup> <mrow/> <mi>μ</mi> </msup> </semantics></math> voltage (<math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>D</mi> <mi>A</mi> </mrow> </msub> </semantics></math>) of Type A controller.</p> Full article ">Figure 7
<p>Gain stage output (<math display="inline"><semantics> <msub> <mi>V</mi> <mi>G</mi> </msub> </semantics></math>), Integrator-1 output (<math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>I</mi> <mn>1</mn> </mrow> </msub> </semantics></math>) and Integrator-2 output (<math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>ω</mi> <mi>A</mi> </mrow> </msub> </semantics></math>) of a DC motor with Type A controller.</p> Full article ">
11 pages, 279 KiB  
Article
On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions
by Yudhveer Singh, Vinod Gill, Jagdev Singh, Devendra Kumar and Kottakkaran Sooppy Nisar
Fractal Fract. 2020, 4(3), 33; https://doi.org/10.3390/fractalfract4030033 - 9 Jul 2020
Cited by 2 | Viewed by 2532
Abstract
In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce [...] Read more.
In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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 - 8 Jul 2020
Cited by 12 | Viewed by 2516
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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Figure 1

Figure 1
<p>Plots of the the fundamental solution <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">G</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> versus <span class="html-italic">x</span> (<math display="inline"><semantics> <mrow> <mi>x</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>) for different values of <span class="html-italic">t</span>; (<b>a</b>) diffusion regime (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>q</mi> </msub> <mo>&lt;</mo> <msub> <mi>τ</mi> <mi>T</mi> </msub> </mrow> </semantics></math>); (<b>b</b>) propagation regime (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>q</mi> </msub> <mo>&gt;</mo> <msub> <mi>τ</mi> <mi>T</mi> </msub> </mrow> </semantics></math>).</p> Full article ">Figure 2
<p>Plots of the the fundamental solution <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">G</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> versus <span class="html-italic">x</span> (<math display="inline"><semantics> <mrow> <mi>x</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>) for fixed <span class="html-italic">t</span> and different values of <math display="inline"><semantics> <mi>α</mi> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>05</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>25</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>75</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>95</mn> </mrow> </semantics></math>, compared to <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (dashed line); (<b>a</b>) diffusion regime (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>q</mi> </msub> <mo>&lt;</mo> <msub> <mi>τ</mi> <mi>T</mi> </msub> </mrow> </semantics></math>); (<b>b</b>) propagation regime (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>q</mi> </msub> <mo>&gt;</mo> <msub> <mi>τ</mi> <mi>T</mi> </msub> </mrow> </semantics></math>).</p> Full article ">Figure 3
<p>Plots of the fundamental solution <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">G</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> versus <span class="html-italic">x</span> (<math display="inline"><semantics> <mrow> <mi>x</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>) for different values of the relaxation times <math display="inline"><semantics> <msub> <mi>τ</mi> <mi>q</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>τ</mi> <mi>T</mi> </msub> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>q</mi> </msub> <mo>/</mo> <msub> <mi>τ</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> </mrow> </semantics></math>—diffusion regime; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>T</mi> </msub> <mo>/</mo> <msub> <mi>τ</mi> <mi>q</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> </mrow> </semantics></math>—propagation regime.</p> Full article ">
19 pages, 364 KiB  
Article
Existence Results for Fractional Order Single-Valued and Multi-Valued Problems with Integro-Multistrip-Multipoint Boundary Conditions
by Sotiris K. Ntouyas, Bashir Ahmad and Ahmed Alsaedi
Fractal Fract. 2020, 4(3), 31; https://doi.org/10.3390/fractalfract4030031 - 5 Jul 2020
Cited by 4 | Viewed by 2090
Abstract
We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is [...] Read more.
We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is discussed for convex valued as well as non-convex valued multi-valued map. Examples are also constructed to illustrate the main results. The results presented in this paper are not only new in the given configuration but also provide some interesting special cases. Full article
9 pages, 307 KiB  
Article
Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative
by Esra Karatas Akgül, Ali Akgül and Dumitru Baleanu
Fractal Fract. 2020, 4(3), 30; https://doi.org/10.3390/fractalfract4030030 - 3 Jul 2020
Cited by 34 | Viewed by 4553
Abstract
In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag–Leffler [...] Read more.
In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag–Leffler functions. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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