A heuristic review on the homotopy perturbation method for non-conservative oscillators

2021 ◽  
pp. 146134842110592
Author(s):  
Chun-Hui He ◽  
Yusry O El-Dib
Keyword(s):  
Exact Solution ◽  
Perturbation Method ◽  
Analytical Methods ◽  
Homotopy Perturbation Method ◽  
Duffing Oscillator ◽  
Nonlinear Problems ◽  
Nonlinear Oscillators ◽  
Complex Problem ◽  
Review Article ◽  
Homotopy Perturbation

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

Laplace–Pade Parametric Homotopy Perturbation Method for Uncertain Nonlinear Oscillators

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
S. Chakraverty ◽  
N. R. Mahato
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Nonlinear Oscillator ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Parametric Form ◽  
Homotopy Perturbation

Nonlinear oscillators have wide applicability in science and engineering problems. In this paper, nonlinear oscillator having initial conditions varying over fuzzy numbers has been initially taken into consideration. Here, the fuzziness in the uncertain nonlinear oscillators has been handled using parametric form. Using parametric form in terms of r-cut, the nonlinear uncertain differential equations are reduced to parametric differential equations. Then, based on classical homotopy perturbation method (HPM), a parametric homotopy perturbation method (PHPM) is proposed to compute solution enclosure of such uncertain nonlinear differential equations. A sufficient convergence condition of parametric solution obtained using PHPM is also proved. Further, a parametric Laplace–Pade approximation is incorporated in PHPM for retaining the periodic characteristic of nonlinear oscillators throughout the domain. The efficiency of Laplace–Pade PHPM has been verified for uncertain Duffing oscillator. Finally, Laplace–Pade PHPM is also applied to solve other uncertain nonlinear oscillator, viz., Rayleigh oscillator, with respect to fuzzy parameters.

A new modification of the homotopy perturbation method

2021 ◽  
pp. 095745652199987
Author(s):  
Magaji Yunbunga Adamu ◽  
Peter Ogenyi
Keyword(s):  
Comparative Analysis ◽  
Error Analysis ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Oscillators ◽  
Absolute Error ◽  
New Method ◽  
Optimal Analysis ◽  
Homotopy Perturbation ◽  
Accuracy And Precision

This study proposes a new modification of the homotopy perturbation method. A new parameter alpha is introduced into the homotopy equation in order to improve the results and accuracy. An optimal analysis identifies the parameter alpha, aimed at improving the solutions. A comparative analysis of the proposed method reveals that the new method presents results with higher degree of accuracy and precision than the classic homotopy perturbation method. Absolute error analysis shows the convenience of the proposed method, providing much smaller errors. Two examples are presented: Duffing and Van der pol’s nonlinear oscillators to demonstrate the efficiency, accuracy, and applicability of the new method.

EXACT SOLITARY-WAVE SOLUTIONS FOR THE NONLINEAR DISPERSIVE K(2,2,1) and K(3,3,1) EQUATIONS BY THE HOMOTOPY PERTURBATION METHOD

2009 ◽  
Vol 23 (30) ◽  
pp. 3667-3675 ◽  
Author(s):  
AHMET YILDIRIM
Keyword(s):  
Exact Solution ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Convergent Series ◽  
Perturbation Series ◽  
Solitary Wave Solutions ◽  
Homotopy Perturbation ◽  
Wave Solutions ◽  
Explicit Exact Solution

We implemented homotopy perturbation method for approximating the solution to the nonlinear dispersive K(m,n,1) type equations. By using this scheme, the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. To illustrate the application of this method, numerical results are derived by using the calculated components of the homotopy perturbation series.

ADDENDUM: NEW INTERPRETATION OF HOMOTOPY PERTURBATION METHOD

2006 ◽  
Vol 20 (18) ◽  
pp. 2561-2568 ◽  
Author(s):  
JI-HUAN HE
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Proper Understanding ◽  
Homotopy Perturbation ◽  
New Interpretation ◽  
Guided Tour

The present work constitutes a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems. It gives a new interpretation of the concept of constant expansion in the homotopy perturbation method.

scholarly journals Solving Heat and Wave-Like Equations Using He's Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Syed Tauseef Mohyud-Din
Keyword(s):  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Iterative Scheme ◽  
Nonlinear Problems ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Clear Advantage ◽  
Suggested Technique ◽  
Adomian’S Polynomials

We use He's polynomials which are calculated form homotopy perturbation method (HPM) for solving heat and wave-like equations. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that suggested technique solves nonlinear problems without using Adomian's polynomials is a clear advantage of this algorithm over the decomposition method.

Sign in / Sign up

Export Citation Format

Share Document