Abstract
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.
Highlights
Nonlinear systems are widely involved in science, engineering, medical science, etc
E development of the theorems of dynamical systems has been derived by the modeling and formulating of nonlinear oscillators. us, nonlinear oscillators are one of the most important parts of nonlinear dynamical systems
To show the validity of the obtained solutions, we have compared these with the existing results determined by Mickens iteration method [25], Mickens HB method [25], He’s amplitude-frequency formulation [34], modified He’s homotopy perturbation method [15], improved harmonic balance method [6] and homotopy perturbation method [9] in Table 1, He’s energy balanced method [21] in Table 3, Mickens iteration method [25], Mickens HB method [25] in
Summary
Nonlinear systems are widely involved in science, engineering, medical science, etc. Research on the nonlinear systems has enriched science, engineering, medical science, etc. E development of the theorems of dynamical systems has been derived by the modeling and formulating of nonlinear oscillators. Us, nonlinear oscillators are one of the most important parts of nonlinear dynamical systems. Many scientists have made significant improvement in finding a new mathematical tool which would be related to nonlinear dynamical systems whose understanding will rely not on analytic techniques and on numerical and asymptotic methods. Ey set up many fruitful and potential methods to operate the nonlinear systems. The proposed method is more easy, simple, and
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