A Coupled Homotopy-Variational Method and Variational Formulation Applied to Nonlinear Oscillators With and Without Discontinuities

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
M. Akbarzade ◽  
J. Langari ◽  
D. D. Ganji
Keyword(s):  
Periodic Solution ◽  
Variational Method ◽  
Perturbation Method ◽  
Variational Formulation ◽  
Homotopy Perturbation Method ◽  
Nonlinear Oscillators ◽  
High Accuracy ◽  
Coupled Method ◽  
Homotopy Perturbation ◽  
Approximate Frequency

In this paper, two novel and different methods are applied to nonlinear oscillators. It has been found that the coupled method of homotopy perturbation method and variational formulation and amplitude-frequency formulation work very well for the whole range of initial amplitudes. The analytical approximate frequency and the corresponding periodic solution are valid for small as well as large amplitudes of oscillation. Contrary to the conventional methods, only one iteration leads to high accuracy of the solutions. Some examples are given to illustrate the accuracy and effectiveness of these methods.

Higher-order homotopy perturbation method for conservative nonlinear oscillators generally and microelectromechanical systems’ oscillators particularly

2020 ◽  
Vol 34 (32) ◽  
pp. 2050313
Author(s):  
Naveed Anjum ◽  
Ji-Huan He
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Microelectromechanical Systems ◽  
Nonlinear Oscillators ◽  
High Accuracy ◽  
Higher Order ◽  
Basic Properties ◽  
Homotopy Perturbation ◽  
Special Cases

A modification of the homotopy perturbation method is proposed by taking advantage of the enhanced perturbation method and the parameter expanding technology. A generalized oscillatory equation and some nonlinear oscillators as the special cases of this equation are considered as examples to outline the basic properties of the modification, and the result is of high accuracy.

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.

scholarly journals Spectral-Homotopy Perturbation Method for Solving Governing MHD Jeffery-Hamel Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Nonlinear Boundary Value Problems ◽  
Pseudospectral Methods ◽  
New Approach ◽  
Homotopy Perturbation ◽  
Chebyshev Pseudospectral

We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the MHD Jeffery-Hamel flow and the effect of MHD on the flow has been discussed. Comparisons are made between the proposed technique, the previous studies, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the presented approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method at small orders. The MATLAB software has been used to solve all the equations in this study.

scholarly journals An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
B. M. Ikramul Haque ◽  
M. M. Ayub Hossain
Keyword(s):  
Perturbation Method ◽  
Iteration Method ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Homotopy Perturbation Method ◽  
Harmonic Balance Method ◽  
Percentage Error ◽  
Cube Root ◽  
Homotopy Perturbation ◽  
Approximate Frequency

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.

Homotopy perturbation method with three expansions for Helmholtz-Fangzhu oscillator

2021 ◽  
Vol 35 (24) ◽  
Author(s):  
Ji-Huan He ◽  
Yusry O. El-Dib
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
High Accuracy ◽  
Singular Terms ◽  
Homotopy Perturbation ◽  
Damping Rate

This paper shows that the Toda oscillator can be converted to a new Helmholtz-Fangzhu equation, which is solved by the homotopy perturbation method with three expansions. The solution, the negative frequency and the damping rate are expanded into power forms in the homotopy parameter, the solution process is given step by step, and the results are of high accuracy. This paper gives a general approach to damp vibrations with singular terms,especially to the Helmholtz-Fangzhu oscillator.

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.

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