Application of he’s fractional derivative and fractional complex transform for time fractional Camassa-Holm equation

Anjum, Naveed; Ain, Qura Tul

doi:10.2298/tsci190930450a

Cited by 42 publications

(20 citation statements)

References 39 publications

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“…In [7], the famous Lighthill equation and Duffing equations were solved. In [8], the method was found to be powerful to fractional differential equations. In [9], this method was applied to solve the nonlinear damped equation of Mathieu with periodic coefficients, and the behavior of stability at both cases of resonance and non-resonance were studied.…”

Section: Introductionmentioning

confidence: 99%

Periodic Property and Instability of a Rotating Pendulum System

Amer

Elnaggar

et al. 2021

Axioms

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The current paper investigates the dynamical property of a pendulum attached to a rotating rigid frame with a constant angular velocity about the vertical axis passing to the pivot point of the pendulum. He’s homotopy perturbation method is used to obtain the analytic solution of the governing nonlinear differential equation of motion. The fourth-order Runge-Kutta method (RKM) and He’s frequency formulation are used to verify the high accuracy of the obtained solution. The stability condition of the motion is examined and discussed. Some plots of the time histories of the gained solutions are portrayed graphically to reveal the impact of the distinct parameters on the dynamical motion.

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Section: Introductionmentioning

confidence: 99%

Periodic Property and Instability of a Rotating Pendulum System

Amer

Elnaggar

et al. 2021

Axioms

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“…The two-scale fractal theory has been widely used as an effective mathematics tool to analyze various discontinuous problems, for examples, fractional Camassa-Holm equation [36], biomechanism of silkworm cocoon [37], snow's thermal insulation [38], fractal calculus for analysis of wool fiber [39] and polar bear hairs [40,41]. The two-scale fractal dimension is defined as [32][33][34][35]…”

Section: Electronic Propertymentioning

confidence: 99%

Fractal Approach to Mechanical and Electrical Properties of Graphene/Sic Composites

Zuo

Liu

2021

FU Mech Eng

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Graphene and carbon nanotubes have a Steiner minimum tree structure, which endows them with extremely good mechanical and electronic properties. A modified Hall-Petch effect is proposed to reveal the enhanced mechanical strength of the SiC/graphene composites, and a fractal approach to its mechanical analysis is given. Fractal laws for the electrical conductivity of graphene, carbon nanotubes and graphene/SiC composites are suggested using the two-scale fractal theory. The Steiner structure is considered as a cascade of a fractal pattern. The theoretical results show that the two-scale fractal dimensions and the graphene concentration play an important role in enhancing the mechanical and electrical properties of graphene/SiC composites. This paper sheds a bright light on a new era of the graphene-based materials.

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“…After taking the limit p ⟶ 1, we obtain an approximate solution for (26) with boundary conditions (27) as follows:…”

Section: Application Of the Phpm With The End To Obtain Two Approximate Solutions For The Nonlinear Blasius Problemmentioning

confidence: 99%

“…In effect, an exact solution to the proposed nonlinear problem can rarely be obtained [3], and for the same reason, several methods have been added to the best known classical methods. Next, we provide a list with some of most employed analytical methods in accordance with the literature: variational approaches [4][5][6], the tanh method [7], exp-function [8,9], Adomian's decomposition method [10][11][12][13][14][15], parameter expansion [16], the homotopy perturbation method (HPM) [1,6,[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], the perturbation method [34][35][36], the modified Taylor series method [37], the Picard method [38], the PSEM [39][40][41][42], the homotopy analysis method [25,43], the variational iteration method [44], and the differential transform method …”

Section: Introductionmentioning

confidence: 99%

A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem

Filobello-Nino

Vázquez-Leal

Huerta-Chua

et al. 2021

Discrete Dynamics in Nature and Society

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This work studies the nonlinear differential equation that models the Blasius problem (BP) which is of great importance in fluid dynamics. The aim is to obtain an approximate analytical expression that adequately describes the phenomenon considered. To find such approximation, we propose a new method denominated powered homotopy perturbation (PHPM). Unlike HPM algorithm, the successive integration process generated by PHPM will consider zero the constants of integration in each approximation, except the last one. In the same way, PHPM will propose an adequate initial trial function provided of some unknown parameters in such a way that it will not evaluate the initial conditions in the iterations of the process; therefore, this set of parameters will be employed with the purpose of adjusting in the best accurate way the proposed approximation at the final part of the process. As a matter of fact, we will note from this analysis that the proposed solution is compact and easy to evaluate and involves a sum of five exponential functions plus a linear part of two terms, which is ideal for practical applications. With the purpose to get a better approximation, we find useful to combine PHPM with the power series extender method (PSEM) which implies to add to the PHPM solution one rational function with parameters to adjust. From this proposal, we find an approximate solution competitive with others from the literature.

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Application of he’s fractional derivative and fractional complex transform for time fractional Camassa-Holm equation

Cited by 42 publications

References 39 publications

Periodic Property and Instability of a Rotating Pendulum System

Periodic Property and Instability of a Rotating Pendulum System

Fractal Approach to Mechanical and Electrical Properties of Graphene/Sic Composites

A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem

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