Homotopy perturbation method with three expansions

He, Ji‐Huan; El‐Dib, Yusry O.

doi:10.1007/s10910-021-01237-3

Cited by 82 publications

(64 citation statements)

References 29 publications

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“…However, in future research, we propose exploring alternative ways to construct other homotopy schemes. For example, the homotopy perturbation method with two expanding parameters [28], homotopy perturbation method with three expansions [29], Li-He homotopy perturbation method [30], homotopy perturbation with an auxiliary parameter [31], and He Laplace method [32,33].…”

Section: Discussionmentioning

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%

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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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Section: Discussionmentioning

confidence: 99%

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

View full text Add to dashboard Cite

show abstract

“…Due to the variation of the stiffness and damping in the movement of the leaf spring system, there are damping force and elastic force with nonlinear characteristics in the system. At present, the homotopy perturbation method [30][31][32] is popular for solving nonlinear oscillator, which does not need ε as a small parameter 33,34 and can effectively solve the high-order approximate solution. 35 Meanwhile, frequency-amplitude relationship is solved in nonlinear vibration by He's frequency formulation.…”

Section: Object Description and Mechanics Analysismentioning

confidence: 99%

Study on dynamic characteristics of leaf spring system in vibration screen

Zhou

Wang

et al. 2021

Journal of Low Frequency Noise, Vibration and Active Control

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By studying dynamic characteristics of the leaf spring system, a new elastic component is designed to reduce the working load and to a certain extent to ensure the linearity as well as increase the amplitude in the vertical and horizontal directions in vibration screen. The modal parameters, amplitudes, and amplification factors of the leaf spring system are studied by simulation and experiment. The modal results show that the leaf spring system vibrates in horizontal and vertical directions in first and second mode shapes, respectively. It is conducive to loosening and moving the particles on the vibration screen. In addition, it is found that the maximum amplitude and amplification factor in the horizontal direction appear at 300 r/min (5 Hz) while those in the vertical direction appear at 480 r/min (8 Hz), which are higher than those in the disc spring system. Moreover, the amplitude of the leaf spring system increases proportionally with the increase of exciting force while the amplification factors are basically the same under different exciting forces, indicating the good linearity of the leaf spring system. Furthermore, the minimum exciting force occurs in the leaf spring system under the same amplitude by comparing the exciting force among different elastic components. The above works can provide guidance for the industrial production in vibration screen.

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“…Therefore, researchers from different fields have explored the nature of nonlinear oscillators for decades. 1 Currently, more and more analytical and approximate methods have been developed to find approximate solutions of nonlinear oscillators, for instance, He's variational iteration method [2][3][4][5][6][7][8] ; the homotopy perturbation method [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] ; the variational method 25 ; the parameter-expanding method 26,27 and other methods. 28 Researchers worldwide have sought a concise and practical approach to solve various nonlinear oscillators for a long time.…”

Section: Introductionmentioning

confidence: 99%

A simplified He’s frequency–amplitude formulation for nonlinear oscillators

Ren

2021

Journal of Low Frequency Noise, Vibration and Active Control

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Derived from an ancient Chinese algorithm, He’s frequency–amplitude formulation is an effective approach to finding an approximate solution of a nonlinear oscillator. In this article, based on He’s formulation, a simplified formulation is proposed. Some nonlinear oscillators are adopted as examples to demonstrate the solving process using this simplified formulation. Through the demonstration, it can be seen that the solving process is simplified.

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Homotopy perturbation method with three expansions

Cited by 82 publications

References 29 publications

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

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

Study on dynamic characteristics of leaf spring system in vibration screen

A simplified He’s frequency–amplitude formulation for nonlinear oscillators

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