Determination of natural frequencies by coupled method of homotopy perturbation and variational method for strongly nonlinear oscillators

2011 ◽  
Vol 52 (2) ◽  
pp. 023518 ◽  
Author(s):  
M. Akbarzade ◽  
J. Langari
Keyword(s):  
Variational Method ◽  
Natural Frequencies ◽  
Nonlinear Oscillators ◽  
Coupled Method ◽  
Strongly Nonlinear ◽  
Homotopy Perturbation ◽  
Strongly Nonlinear Oscillators

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.

Green’s Function Iterative Approach for Solving Strongly Nonlinear Oscillators

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Marwan Abukhaled
Keyword(s):  
Fixed Point ◽  
Green’S Function ◽  
Green's Function ◽  
Nonlinear Oscillators ◽  
Iterative Approach ◽  
Numerical Examples ◽  
Strongly Nonlinear ◽  
Homotopy Perturbation ◽  
Linear Integral ◽  
Strongly Nonlinear Oscillators

In this paper, a Green’s function based iterative algorithm is proposed to solve strong nonlinear oscillators. The method’s essential part is based on finding an appropriate Green’s function that will be incorporated into a linear integral operator. An application of fixed point iteration schemes such as Picard’s or Mann’s will generate an iterative formula that gives reliable approximations to the true periodic solutions that characterize these kinds of equations. The applicability and stability of the method will be tested through numerical examples. Since exact solutions to these equations usually do not exist, the proposed method will be tested against other popular numerical methods such as the modified homotopy perturbation, the modified differential transformation, and the fourth-order Runge–Kutta methods.

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