Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method

Manimegalai, K.; F, Sagar Zephania C; Bera, P.; Das, Subir; Sil, Tapas

doi:10.1140/epjp/i2019-12824-6

Cited by 25 publications

(8 citation statements)

References 39 publications

(51 reference statements)

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“…We compare the solution of equation (3) obtained from WG, HPM and HW methods with the exact solution in figure 1. The first-order HPM solution of (3) is obtained as (following the work in [42,43]),…”

Section: Resultsmentioning

confidence: 99%

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Zephania

Harisankar²,

Sil³

2023

Phys. Scr.

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Shear waves find applications in several branches of science, such as geophysics, earth science, medical science etc. The Haar wavelet (HW) scheme is employed to solve the governing equation of the horizontal component of the shear wave (SH). The solutions of SH waves obtained from HW are compared with the exact solutions and some of the available results from approximation methods, such as the homotopy perturbation method (HPM) and wavelet Galerkin method with Daubechies wavelet (WG). HW solutions are found to be more accurate than WG at points away from the resonance and at the proximity of the resonance. HW yields solutions with higher accuracy than HPM solutions. The SH wave equation is also studied using the concept of fractional calculus by introducing arbitrary parameter $\alpha$, especially in the vicinity of the resonance with the values of $\alpha$ around one. The solutions are found to be damped oscillatory for $ \alpha <1 $, and diverging oscillatory for $\alpha > 1 $, respectively. The solutions are insensitive to small variations $ \alpha $ at and around the resonance point corresponding to the ODE. At a point far from the resonance, the solution with $\alpha \approx 1 $ matches nicely with those for $ \alpha \ne 1 $. The amplitude of the solution for $ \alpha= 1 $ becomes very large at a point very close to the resonance. In contrast, amplitudes of the solutions for $\alpha \ne 1$ remain the same in the vicinity of the resonance, including it. Therefore, if necessary, the parameter $ \alpha $ may be the control to avoid resonance.

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Section: Resultsmentioning

confidence: 99%

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Zephania

Harisankar²,

Sil³

2023

Phys. Scr.

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“…Therefore, the frequency from the zeroth order approximation is, which is same as the frequency from the first order approximation in ATHPM [1] and HT [28]. The remaining part of x 1 (τ ) in equation (28) gives the first order approximation of x,…”

Section: Applicationsmentioning

confidence: 99%

“…The percentage error in ω are 2.385 and 0.584 for AT HP M and LT HP M h, respectively, for a = 1 which reduce to 0.167 if the second order term in LT HP M h is considered. We plot the displacements obtained from HT (x HT , green cross) [28] , AT HP M (x AT , red diamond) [1] and LT HP M h (x L , blue circles) expressed in equation ( 35) respectively, with increasing t for four sets of parameters A(a = 0.2, OP = 0.2), B(0.2, 1.0), C(1.0, 0.2) and D(1.0, 1.0) in the left column of figure 2, and compared with the same obtained by numerical solution of equation ( 21) employing the fourth order Runge-Kutta (RK4) method (x RK4 (t), black solid line). It is seen that for small values of the parameters, approximate displacements match well with x RK4 ) but a significant deviations for x AT and x HT from x RK4 are noticed for large values of force parameters.…”

Section: Applicationsmentioning

confidence: 99%

“…

In a recent article [1], Aboodh transform based homotopy perturbation method (AT HP M ) has been found to produce approximate analytical solutions in a simple way but with better accuracy in comparison to those obtained from some of the established approximation methods [2,3] for some physically relevant anharmonic oscillators such as, autonomous conservative oscillator (ACO). In this article, expansion of frequency (ω) and an auxiliary parameter (h) are introduced in the framework of homotopy perturbation method (HPM) to improve the accuracy by retaining its simplicity.

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Study of autonomous conservative oscillator using an improved perturbation method

Zephania,

Sil

2020

Preprint

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“…In the past decades, several techniques have been proposed to get the approximate analytic solution of N/ MEMS problems such as the homotopy perturbation method (HPM), higher-order HPM [8], Taylor series [9], energy balance technique [10], spreading residual harmonic balance method [11], higher-order Hamiltonian method [12], Adomian decomposition method (ADM) [13], Li-He modified HPM [14], modified ADM [15], variational approach [16], Galerkin decomposition method [17], and so on. It is also noted that, besides these methods, there are various analytical techniques for getting the approximate solution to the nonlinear equations, for example, the He-Laplace method [18], global residual harmonic balance method [19], integral transform-based methods [20][21][22], max-min approach [23], frequency-amplitude formulation method [24], Hamiltonian approach [25], and others [26][27][28][29]. Moreover, there have been several review articles that have appeared on the analytical methods for oscillatory problems during the past decade [30][31][32].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Analytical Approach for the Periodicity of Nano/Microelectromechanical Systems’ Oscillators

Anjum

Rahman

et al. 2022

Mathematical Problems in Engineering

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Periodic behavior analysis of nano/microelectromechanical systems (N/MEMS) is an essential field owing to their many promising applications in microinstruments. The interesting and unique properties of these systems, particularly, small size, batch fabrication, low power consumption, and high reliability, have fascinated researchers and industries to implement these structures for the production of different microdevices. The dynamic oscillatory behavior of N/MEMS is very intricate due to the various types of nonlinearities present in these structures. The foremost objective of this study is to explore the periodicity of oscillatory problems from N/MEMS. The variational iteration method (VIM), which has been considered as an effective approach for nonlinear oscillators, is coupled with the Laplace transform to obtain the approximate analytic solution of these nonlinear vibratory systems with fewer computations. This coupling of VIM and Laplace transform not only helps in the identification of the Lagrange multiplier without getting into the details of the cryptic theory of variations, but also finds the frequency-amplitude relationship and the analytic approximate solution of N/MEMS. A generalized vibratory equation for N/MEMS is followed by three examples as special cases of this generalized equation are given to elucidate the effectivity of the coupling. The solution obtained from the Laplace-based VIM not only exhibits good agreement with observations numerically but also higher accuracy yields when compared to other established techniques in the open literature.

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Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method

Cited by 25 publications

References 39 publications

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Study of autonomous conservative oscillator using an improved perturbation method

An Efficient Analytical Approach for the Periodicity of Nano/Microelectromechanical Systems’ Oscillators

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