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Fractal Fract
Volume 2
Issue 3
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Fractal Fract., Volume 2, Issue 3 (September 2018) – 3 articles

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12 pages, 270 KiB  
Article
Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs
by Kamal Ait Touchent, Zakia Hammouch, Toufik Mekkaoui and Fethi B. M. Belgacem
Fractal Fract. 2018, 2(3), 22; https://doi.org/10.3390/fractalfract2030022 - 7 Sep 2018
Cited by 29 | Viewed by 2844
Abstract
In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is [...] Read more.
In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is also discussed and presented. The obtained results prove that the employed method is very simple and effective for treating analytically various kinds of problems comprising local fractional derivatives. Full article
17 pages, 464 KiB  
Article
Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations
by Guy Joseph Eyebe, Gambo Betchewe, Alidou Mohamadou and Timoleon Crepin Kofane
Fractal Fract. 2018, 2(3), 21; https://doi.org/10.3390/fractalfract2030021 - 5 Aug 2018
Cited by 20 | Viewed by 4021
Abstract
In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler–Pasternak foundation is studied using nonlocal elasticity theory. The D’Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution [...] Read more.
In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler–Pasternak foundation is studied using nonlocal elasticity theory. The D’Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations. Full article
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Figure 1

Figure 1
<p>Boundary conditions for different beam supports. (<b>a</b>) Simple-simple case and (<b>b</b>) clamped-clamped case.</p> Full article ">Figure 2
<p>First three vibration mode shapes for the simple-simple case boundary condition.</p> Full article ">Figure 3
<p>First three vibration modes shapes for the clamped-clamped case boundary condition.</p> Full article ">Figure 4
<p>First three modes of the fractional nonlinear frequency versus nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>).</p> Full article ">Figure 5
<p>First three modes of the fractional nonlinear frequency versus amplitude (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 6
<p>Fractional nonlinear frequency versus amplitude for different values of <math display="inline"><semantics> <mrow> <mi>K</mi> <mi>p</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 7
<p>Fractional nonlinear frequency versus amplitude for different values of <span class="html-italic">K</span> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 8
<p>Fractional nonlinear frequency versus amplitude for different values of <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 9
<p>Frequency-response curves versus amplitude for different values of <math display="inline"><semantics> <mi>χ</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>=</mo> <mn>0.025</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> <mo>=</mo> <mn>0.025</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>w</mi> <mi>o</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>F</mi> <mo stretchy="false">¯</mo> </mover> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 10
<p>Fractional contribution frequency versus stiffness <span class="html-italic">K</span> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 11
<p>Fractional contribution frequency versus stiffness <span class="html-italic">K</span> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 12
<p>Fractional contribution frequency versus stiffness <span class="html-italic">K</span> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>).</p> Full article ">Figure 13
<p>Fractional contribution frequency versus stiffness <math display="inline"><semantics> <mrow> <mi>K</mi> <mi>p</mi> </mrow> </semantics></math> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 14
<p>Fractional contribution frequency versus stiffness <math display="inline"><semantics> <mrow> <mi>K</mi> <mi>p</mi> </mrow> </semantics></math> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 15
<p>Fractional contribution frequency versus fractional damping coefficient <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> </mrow> </semantics></math> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 16
<p>Fractional contribution frequency versus fractional damping coefficient <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> </mrow> </semantics></math> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 17
<p>Fractional contribution frequency versus fractional damping coefficient <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>p</mi> </mrow> </semantics></math> and nonlocality <math display="inline"><semantics> <mi>η</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p> Full article ">
15 pages, 965 KiB  
Article
Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels
by Maike A. F. Dos Santos
Fractal Fract. 2018, 2(3), 20; https://doi.org/10.3390/fractalfract2030020 - 29 Jul 2018
Cited by 36 | Viewed by 5033
Abstract
The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion [...] Read more.
The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and Mittag-Leffler memory-kernels in the context of Caputo-Fabrizio and Atangana-Baleanu fractional operators on Caputo sense. Thus, exact expressions for the probability distributions are obtained, in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class of diffusive behaviour. Moreover, I propose a generalised model to describe the random walk process with resetting on memory kernel context. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
►▼ Show Figures

Figure 1

Figure 1
<p>This curves illustrate the changes of distribution <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> caused by the exponential kernel (Equation (<a href="#FD3-fractalfract-02-00020" class="html-disp-formula">3</a>) in Equation (<a href="#FD1-fractalfract-02-00020" class="html-disp-formula">1</a>)), and the usual random walk (Brownian motion) case <math display="inline"><semantics> <mrow> <mi mathvariant="script">K</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (or <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> in Equation (<a href="#FD3-fractalfract-02-00020" class="html-disp-formula">3</a>)), considering <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a family of characteristic times <math display="inline"><semantics> <mrow> <mo>{</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics></math> (in context Caputo-Fabrizio operator <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> <msup> <mi>α</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p> Full article ">Figure 2
<p>This curves illustrate the changes of distribution <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> caused by the exponential kernel (Equation (<a href="#FD3-fractalfract-02-00020" class="html-disp-formula">3</a>) in Equation (<a href="#FD1-fractalfract-02-00020" class="html-disp-formula">1</a>)), considering <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> to different times <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>0.1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo>}</mo> </mrow> </semantics></math> (in context of Caputo-Fabrizio operator <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p> Full article ">Figure 3
<p>This curves illustrate the changes of distribution <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> caused by the Mittag-Leffler kernel (Equation (<a href="#FD4-fractalfract-02-00020" class="html-disp-formula">4</a>) in Equation (<a href="#FD1-fractalfract-02-00020" class="html-disp-formula">1</a>)), and the usual (Brownian motion) case <math display="inline"><semantics> <mrow> <mi mathvariant="script">K</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (or <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> in Equation (<a href="#FD4-fractalfract-02-00020" class="html-disp-formula">4</a>)), considering <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <msup> <mi>b</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and different values to index <math display="inline"><semantics> <mi>α</mi> </semantics></math>.</p> Full article ">Figure 4
<p>This curves illustrate the changes of distribution <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> caused by the Mittag-Leffler kernel (Equation (<a href="#FD4-fractalfract-02-00020" class="html-disp-formula">4</a>) in Equation (<a href="#FD1-fractalfract-02-00020" class="html-disp-formula">1</a>)), and the usual (Brownian motion) case <math display="inline"><semantics> <mrow> <mi mathvariant="script">K</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (or <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> in Equation (<a href="#FD4-fractalfract-02-00020" class="html-disp-formula">4</a>)). Considering ,<math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>δ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <msup> <mi>b</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> and different values of parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math>.</p> Full article ">Figure 5
<p>These curves illustrate the changes of MSD caused by a set of fractional <math display="inline"><semantics> <mi>α</mi> </semantics></math>-index in Mittag-Leffler kernel, considering <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <msup> <mi>b</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">
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