A Good Initial Guess for Approximating Nonlinear Oscillators by the Homotopy Perturbation Method

He, Ji‐Huan; He, Chun‐Hui; Alsolami, Abdulrahman Ali

doi:10.22190/fume230108006h

Cited by 23 publications

(14 citation statements)

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“…This semi-analytical method offers a series representation of the solution, facilitating the computation of solution components. To ensure the effectiveness of the HPM, two key features are crucial: a suitable homotopy equation and a good initial guess [33]. The HPM has been applied to address diverse problems, including heat transfer equations [34], Lane-Emden type equations [35], and strongly nonlinear oscillators [36].…”

Section: (A) Literature Review and Motivationmentioning

confidence: 99%

Homotopy perturbation method and its convergence analysis for nonlinear collisional fragmentation equations

Yadav,

Das,

Singh

et al. 2023

Proc. R. Soc. A.

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The exploration of collisional fragmentation pheno-mena remains largely unexplored, yet it holds considerable importance in numerous engineering and physical processes. Given the nonlinear nature of the governing equation, only a limited number of analytical solutions for the number density function corresponding to empirical kernels are available in the literature. This article introduces a semi-analytical approach using the homotopy perturbation method to obtain series solutions for the nonlinear collisional fragmentation equation. The method presented here can be readily adapted to solve both linear and nonlinear integral equations, eliminating the need for domain discretization. To gain deeper insights intothe accuracy of the proposed method, a convergence analysis is conducted. This analysis employs the concept of contractive mapping within the Banach space, a well-established technique universally acknowledged for ensuring convergence. Various collisional kernels (product and polymerization kernels), breakage distribution functions (binary and multiple breakage) and various initial particle distributions are considered to obtain the new series solutions. The obtained results are successfully compared against finite volume method [26] solutions in terms of number density functions and their moments. The error between the exact and obtained series solutions is shown in plots and tables to confirm the applicability and accuracy of the proposed method.

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Section: (A) Literature Review and Motivationmentioning

confidence: 99%

Homotopy perturbation method and its convergence analysis for nonlinear collisional fragmentation equations

Yadav,

Das,

Singh

et al. 2023

Proc. R. Soc. A.

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“…Equation ( 8) can be effectively solved by the homotopy perturbation method. 17,[32][33][34][35][36] The previous studies are all based on a continuous space, but in reality, roes are located in discontinuous space, [37][38][39][40] so it is necessary to consider the fractal agglomeration to study discontinuous nonlinear vibrations. In a fractal space, equation (8) can be described as follows…”

Section: Math Modelmentioning

confidence: 99%

Pull-down plateau of a Toda-like fractal-fractional oscillator and its application in grass carp’s roes’ agglomeration

Niu

2023

Journal of Low Frequency Noise, Vibration and Active Control

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Grass carp’s roes should be agglomerated together for maximizing their survival rate against various predators. Any vibration induced by any environmental perturbation should be attenuated immediately. A Toda-like fractal-fractional oscillator is established, which shows a low-frequency property for most cases; however, the grass carp has evolved a very ability to attenuate the perturbated vibration by sticky adhesion. The pull-down stability of the roes’ vibration is discovered through the results of phase diagrams. The mathematical analysis reveals that there is a pull-down plateau for the attenuating process, the plateau’s height and width are discussed graphically, and the main factors affecting the plateau’s properties are elucidated. The paper offers a totally new window for biomechanics, especially for biomimicking design of chatter vibration systems inspired by the agglomerated roes.

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“…However, if B > 2A, we have |A-B| > A, under such case, the periodic motion is forbidden and its motion is pulled down. So equation (26) needs to have a positive real root greater than two for pull-down instability to occur.…”

Section: à Amentioning

confidence: 99%

“…17 El-Dib studied the properties of complex Helmholtz-Duffing oscillator arising in the fluid mechanics. 18,19 More recently, there are many analytical and numerical methods to find an approximate solution of a nonlinear oscillator, such as the variational iteration method, [20][21][22] the homotopy perturbation, [23][24][25][26] the Hamiltonian approach, [27][28][29] Adomian decomposition method, 30 and others. 31,32 However, the previous research focused on searching for its approximate solution, and the pulldown instability and asymmetric period have not yet studied in literature.…”

Section: Introductionmentioning

confidence: 99%

Symmetry-breaking and pull-down motion for the Helmholtz–Duffing oscillator

Niu,

He,

Alsolami

2023

Journal of Low Frequency Noise, Vibration and Active Control

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An accurate frequency of the Helmholtz–Duffing oscillator is obtained by a sophisticated modification of He’s frequency formulation. The pull-down instability existing in the symmetric breaking phenomenon is a newly discovered dynamic motion for oscillators with even nonlinearities. A criterion for predicting the asymmetrical amplitude motion and the pull-down instability is built by measuring the amplitude change. The good matching performance between the analytic results and numerical ones indicates that the criterion offers a starting point for future research into the pull-down phenomenon, and opens a new path for studying more complex nonlinear oscillators with even nonlinearities.

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A Good Initial Guess for Approximating Nonlinear Oscillators by the Homotopy Perturbation Method

Cited by 23 publications

References 0 publications

Homotopy perturbation method and its convergence analysis for nonlinear collisional fragmentation equations

Homotopy perturbation method and its convergence analysis for nonlinear collisional fragmentation equations

Pull-down plateau of a Toda-like fractal-fractional oscillator and its application in grass carp’s roes’ agglomeration

Symmetry-breaking and pull-down motion for the Helmholtz–Duffing oscillator

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