The simplest approach to solving the cubic nonlinear jerk oscillator with the non‐perturbative method

Abstract

Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third-order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high-order nonlinear conservative or non-conservative oscillators. The current work sheds light on the approximate proposition to the damped third-order nonlinear differential equations.The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the nonperturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature.

Immediate solution for fractional nonlinear oscillators using the equivalent linearized method

“…At this end, equation ( 13) can be linearized and solved with the non-perturbative approach. [31][32][33] In the simplest approach, the equivalent linearized form of equation ( 10) can be sought in the form…”

“…Figure 9 illustrates the solution of equation (24) relative to the numerical solution of equation (30) in the case of α ¼ 1. The variation of the parameter α of the quasi-exact solution (32) is displayed in Figure 10. As can be seen from Fig.…”

“…The solution will be obtained based on the equivalent approach established in the present paper. (32) with the numerical solution of (28) for the same system given in Figure 1 in the case of α ¼ 1.…”

“…The approach depends on replacing the original fractional equation with an equivalent linear oscillation with the integer derivatives. Based on the awareness of the non-perturbative method, [30][31][32] one could obtain a good asymptotic solution for a fractional-order dynamical system, and then analyze some important properties such as the amplitude-frequency equation. Accordingly, it may present more information about fractional-order differential equations.…”

“…Figure 9. Comparison of the solution(32) with the numerical solution of (28) for the same system given in Figure1in the case of α ¼ 1.…”