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Fractal Fract, Volume 2, Issue 3 (September 2018)

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Open AccessArticle Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs
Fractal Fract 2018, 2(3), 22; https://doi.org/10.3390/fractalfract2030022
Received: 28 March 2018 / Revised: 30 July 2018 / Accepted: 31 July 2018 / Published: 7 September 2018
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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
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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
Open AccessArticle Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations
Fractal Fract 2018, 2(3), 21; https://doi.org/10.3390/fractalfract2030021
Received: 12 June 2018 / Revised: 1 August 2018 / Accepted: 3 August 2018 / Published: 5 August 2018
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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
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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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Open AccessArticle Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels
Fractal Fract 2018, 2(3), 20; https://doi.org/10.3390/fractalfract2030020
Received: 13 July 2018 / Revised: 24 July 2018 / Accepted: 27 July 2018 / Published: 29 July 2018
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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
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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)
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