He’s multiple scales method for nonlinear vibrations

Ren, Zhong-Fu; Yao, Shao-Wen; He, Ji‐Huan

doi:10.1177/1461348419861450

Cited by 48 publications

(31 citation statements)

References 29 publications

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“…Equation (1) can be easily solved by a numerical method (Li and He, 2019; Alizadeh et al , 2018; Liu, 2019; Pirmohammadi, 2019; Bilige and Han, 2018), and it can be solved by many analytical methods, for example, the variational method, the variational iteration method (He and Latifizadeh, 2020; Anjum and He, 2019; He et al , 2014), the homotopy perturbation method (Yu et al , 2019; Ren et al , 2019; Ban and Cui, 2018; Wu and He, 2018) and Taylor series method (He and Ji, 2019b; He et al , 2019; He, 2019a; Khan, 2018). In this paper, we will apply the exp-function method (He, 2012; He, 2013; He and Wu, 2006; Jia et al , 2015) to study the solution properties at different scales.…”

Section: Kdv – Burgers Equation and Its Solution Propertymentioning

confidence: 99%

Difference equation vs differential equation on different scales

Sedighi

2020

HFF

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Purpose The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used to elucidate the fact. Design/methodology/approach A variational principle is established for the generalized KdV – Burgers equation by the semi-inverse method, and the equation is solved analytically by the exp-function method, and some exact solutions are obtained, including blowup solutions and discontinuous solutions. The solution morphologies are studied by illustrations using different scales. Findings Solitary solution is the basic property of nonlinear wave equations. This paper finds some new properties of the KdV–Burgers equation, which have not been reported in open literature and cannot be effectively elucidated by numerical methods. When the solitary solution or the blowup solution is observed on a much small scale, their discontinuous property is first found. Originality/value The variational principle can explain the blowup and discontinuous properties of a nonlinear wave equation, and the exp-function method is a good candidate to reveal the solution properties.

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Section: Kdv – Burgers Equation and Its Solution Propertymentioning

confidence: 99%

Difference equation vs differential equation on different scales

Sedighi

2020

HFF

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“…The fractal diffusion-reaction equation can be also solved by the homotopy perturbation method, variational iteration method or Taylor series method [30][31][32][33][34][35][36][37][38][39]. This paper gives an ample evidence that the fractal variational approach to amperometric enzymatic reactions is not only simple, but also rigorous.…”

Section: Fractal Variational Principlementioning

confidence: 98%

Fractal diffusion-reaction model for a porous electrode

2021

THERM SCI

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Fractal modifications of Fick?s laws are discussed by taking into account the electrode?s porous structure, and a fractal derivative model for diffusion-reaction process in a thin film of an amperometric enzymatic reaction is established. Particular attention is paid to giving an intuitive grasp for its fractal variational principle and its solution procedure. Extremely fast or extremely slow diffusion process can be achieved by suitable control of the electrode?s surface morphology, a sponge-like surface leads to an extremely fast diffusion, while a lotus-leaf-like uneven surface predicts an extremely slow process. This paper sheds a bright light on an optimal design of an electrode?s surface morphology.

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“…This new modification works particularly well for the nonlinear oscillators. Ren et al [31] made a couple of the multiple time scales with the HPM to become a powerful mathematical tool for various nonlinear equations. It provides solutions to nonlinear equations, while the classical perturbation method became unsuccessful.…”

Section: Introductionmentioning

confidence: 99%

The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

El‐Dib

2020

Numerical Methods Partial

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In the current work, we apply a nonstandard scheme to solve the third‐order Duffing equation. This equation is produced from the strong damped Klein–Gordon equation under the traveling wave transformation. The solution and the stability conditions for the third‐order Duffing equation have been discussed for the first time. The present analysis is new and used the reducing rank method with the homotopy perturbation method. A nonoscillator solution with the oscillating solutions is derived individually and frequency formula is obtained. Besides, the stability analysis is discussed.

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He’s multiple scales method for nonlinear vibrations

Cited by 48 publications

References 29 publications

Difference equation vs differential equation on different scales

Difference equation vs differential equation on different scales

Fractal diffusion-reaction model for a porous electrode

The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

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