Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

Abstract

In this investigation, some different approaches including the hybrid homotopy perturbation method (H-HPM) which sometimes is called Krylov-Bogoliubov-Mitropolsky (KBM) method and the multiple scales method (MSM), are implemented for analyzing a generalized forced damped complex Duffing oscillator. All mentioned methods are applied to obtain some accurate and stable approximations to the proposed problem without decoupling for the original problem. All obtained approximations are discussed graphically using different numerical values to the relevant parameters. Moreover, all obtained approximate solution are compared with the 4^{th}-order Runge-Kutta (RK4) numerical approximation. Also, the maximum residual distance error (MRDE) is estimated in order to verify the high-accuracy to the obtained analytic approximations.

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

“…The Multiple Scales Method (MSM) and Krýlov-Bogoliúbov-Mitropólsky method (KBMM) were employed to provide approximate solutions for a time Delay Duffing-Helmholtz equation [25]. Furthermore, both KBMM and MSM were used for analyzing and solving several nonlinear oscillators with strong nonlinearity [26][27][28][29][30][31][32].…”

The Krylov–Bogoliubov–Mitropolsky method for modeling a forced damped quadratic pendulum oscillator

“…35 Many researchers proved that the KBM technique is better than many techniques mentioned in the literature because it is characterized by high accuracy and ease of application. [35][36][37][38][39][40][41] In addition, it does not require to solve a system of differential equations but only solves a system of algebraic equations.…”

“…In the present investigation, we follow the recent studies [35][36][37][38][39][40][41] for analyzing the QFDP Eq. (1) using both ansatz and KBM methods.…”