Periodic solution of the Duffing-Van der Pol oscillator by homotopy perturbation method

2010 ◽  
Vol 87 (12) ◽  
pp. 2688-2696 ◽  
Author(s):  
Aiyong Chen ◽  
Guirong Jiang
Keyword(s):  
Periodic Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Homotopy Perturbation

scholarly journals Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes ◽  
Yasir Khan ◽  
Uriel Filobello-Nino ◽  
Alejandro Diaz-Sanchez
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Homotopy Perturbation ◽  
Physical Phenomena

The fact that most of the physical phenomena are modelled by nonlinear differential equations underlines the importance of having reliable methods for solving them. This work presents the rational biparameter homotopy perturbation method (RBHPM) as a novel tool with the potential to find approximate solutions for nonlinear differential equations. The method generates the solutions in the form of a quotient of two power series of different homotopy parameters. Besides, in order to improve accuracy, we propose the Laplace-Padé rational biparameter homotopy perturbation method (LPRBHPM), when the solution is expressed as the quotient of two truncated power series. The usage of the method is illustrated with two case studies. On one side, a Ricatti nonlinear differential equation is solved and a comparison with the homotopy perturbation method (HPM) is presented. On the other side, a nonforced Van der Pol Oscillator is analysed and we compare results obtained with RBHPM, LPRBHPM, and HPM in order to conclude that the LPRBHPM and RBHPM methods generate the most accurate approximated solutions.

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.

scholarly journals Rational Homotopy Perturbation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Héctor Vázquez-Leal
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Homotopy Perturbation ◽  
Solution Methods ◽  
Rational Version ◽  
Physical Phenomena

The solution methods of nonlinear differential equations are very important because most of the physical phenomena are modelled by using such kind of equations. Therefore, this work presents a rational version of homotopy perturbation method (RHPM) as a novel tool with high potential to find approximate solutions for nonlinear differential equations. We present two case studies; for the first example, a comparison between the proposed method and the HPM method is presented; it will show how the RHPM generates highly accurate approximate solutions requiring less iteration, in comparison to results obtained by the HPM method. For the second example, which is a Van der Pol oscillator problem, we compare RHPM, HPM, and VIM, finding out that RHPM method generates the most accurate approximated solution.

scholarly journals Linearized Homotopy Perturbation Method for Two Nonlinear Problems of Duffing Equations

2021 ◽  
Vol 13 (6) ◽  
pp. 10
Author(s):  
Chein-Shan Liu
Keyword(s):  
Periodic Solution ◽  
Perturbation Method ◽  
Trivial Solution ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Analytic Solutions ◽  
Higher Order ◽  
Duffing Equation ◽  
Duffing Equations ◽  
Homotopy Perturbation

In the paper, we solve two nonlinear problems related to the Duffing equations in space and in time. The first problem is the bifurcation of Duffing equation in space, wherein a critical value of the parameter initiates the bifurcation from a trivial solution to a non-trivial solution. The second problem is an unconventional periodic problem of Duffing equation in time to determine period and periodic solution. To save computational cost and even enhance the accuracy in seeking higher order analytic solutions of these two problems, a modified homotopy perturbation method is invoked after a linearization technique being exerted on the Duffing equation, whose nonlinear cubic term is decomposed at two sides via a weight factor, such that the Duffing equation is linearized as the Mathieu type differential equation. The constant preceding the displacement is expanded in powers of homotopy parameter and the coefficients are determined to avoid secular solutions appeared in the derived sequence of linear differential equations. Consequently, after setting the homotopy parameter equal to unity and solving the amplitude equation, the higher order bifurcated solutions can be derived explicitly. For the second problem, we can determine the period and periodic solution in closed-form, which are very accurate. For the sake of comparison the results obtained from the fourth-order Runge-Kutta numerical method are used to evaluate the presented analytic solutions.

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