Homotopy perturbation method for Fangzhu oscillator
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Homotopy perturbation method for Fangzhu oscillator Ji‑Huan He1,2,3 · Yusry O. El‑Dib4 Received: 22 May 2020 / Accepted: 4 September 2020 © Springer Nature Switzerland AG 2020
Abstract An accurate frequency-amplitude relationship is very needed to elucidate the properties of the oldest device of Fangzhu for collecting water from the air. The Fangzhu oscillator was derived and solved approximately (He et al. in Math Methods Appl Sci, 2020, https://doi.org/10.1002/mma.6384), here we show that the singular Duffing-like oscillator can be more effectively solved by the homotopy perturbation method and a criterion is obtained for the existence of a periodic solution for the singular differential equation. The results obtained in this paper are helpful for the optimal design of the Fangzhu device. Keywords Homotopy perturbation method · Frequency expansion method · Periodic solution · Fangzhu oscillator Mathematics Subject Classification 34-K13 · 34-K27 · 34-L30 · 37-J25 · 37-K45 · 41-A10 · 42-A15 · 42-B20
1 Introduction The Fangzhu device was considered as the oldest nanotechnology used in ancient China for collecting water from the air, its nano-scale surface morphology plays an important role in water collection efficiency, the super-hydrophobic surface is designed to attract water molecules from the air, and its super-hydrophilic partner is used to deliver the * Yusry O. El‑Dib [email protected] Ji‑Huan He [email protected] 1
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China
2
School of Science, Xi’an University of Architecture and Technology, Xi’an, China
3
National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren‑Ai Road, Suzhou, China
4
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt
13
Vol.:(0123456789)
Journal of Mathematical Chemistry
attracted water molecules to the water collector. Its mechanism was fully elucidated by a singular differential equation by He et al. [1]: (1)
ÿ + 𝜔20 y + Qy−𝛼 = g(t).
where y is the distance of the attracted molecule from its equilibrium position. The understanding of each parameter is given in Ref. [1]. A low frequency is beneficial for the attracted molecule to be transmitted from the super-hydrophobic surface of the super-hydrophilic surface, so an accurate estimation of its solution is very needed. The nano-scale surface has a high surface energy or geometric potential, different geometric patterns result in different wetting property of the surface [2–11]. Equation (1) is a Duffing-like oscillator, and we call it as Fangzhu oscillator, the periodic property or the instability property of the absorbed water on the Fangzhu’s surface plays an important role [1]. There are many analytical methods to solve such an equation [12–22], and this paper adopts the homotopy perturbation method [23–28] to reveal the periodic property of Eq. (1). The approximate analytical solutions derived by HPM for the en
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