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.

scholarly journals Analytical Study of Nonlinear Vibrations of the Tympanic Membrane by Homotopy Perturbation Method

2020 ◽  
Vol 2 (5) ◽  
pp. 187-192
Author(s):  
Masoud Rahmani ◽  
Amin Moslemi Petrudi ◽  
Ionut Cristian Scurtu
Keyword(s):  
Perturbation Method ◽  
Tympanic Membrane ◽  
Harmonic Balance ◽  
Homotopy Perturbation Method ◽  
Harmonic Balance Method ◽  
Engineering Research ◽  
Homotopy Perturbation ◽  
The Common ◽  
Vibration Problems ◽  
Human Tympanic Membrane

Nonlinear vibration problems are generally of great importance in physics, mechanical structures, and other engineering research. This type of equation is very difficult and time-consuming. Researchers, therefore, focus on analytical and numerical methods; In this paper, using the new Newton-Harmonic Balance analytical method, which includes a combination of Newton methods and the harmonic balance method, the common Homotopy Perturbation method is used to solve the nonlinear equation of human tympanic membrane vibration and are compared and validated.

scholarly journals The Approximate Solution of Fractional Fredholm Integrodifferential Equations by Variational Iteration and Homotopy Perturbation Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Abdelouahab Kadem ◽  
Adem Kilicman
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Constant Coefficients

Variational iteration method and homotopy perturbation method are used to solve the fractional Fredholm integrodifferential equations with constant coefficients. The obtained results indicate that the method is efficient and also accurate.

scholarly journals Variational Homotopy Perturbation Method for Solving Fractional Initial Boundary Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yanqin Liu
Keyword(s):  
Perturbation Method ◽  
Boundary Value Problems ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Initial Boundary Value

A variational homotopy perturbation method (VHPM) which is based on variational iteration method and homotopy perturbation method is applied to solve the approximate solution of the fractional initial boundary value problems. The nonlinear terms can be easily handled by the use of He's polynomials. It is observed that the variational iteration method is very efficient and easier to implements; illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

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