Approximate solution to a generalized Van der Pol equation arising in plasma oscillations

Abstract

Motivated by some published theoretical investigations and based on the two-fluid model, nonlinear plasma oscillations are analyzed and discussed in the framework of the generalized Van der Pol equation. This equation is analyzed and solved using two different analytical approaches. In this first approach, the ansatz method is carried out for deriving an approximation in the form of a trigonometric function. In the second approach, the Krylov–Bogoliubov–Mitropolsky (KBM) technique is applied for obtaining a high-accurate approximation. The obtained approximations are compared with the numerical approximation using the Runge–Kutta (RK) method. Moreover, the distance error between the obtained approximations (using the ansatz method and the KBM technique) and the RK numerical approximation is estimated. In our investigation, both the proposed methods and obtained approximations can help many authors investigate several nonlinear oscillations in different plasma models and fluid mechanics.

Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

“…The analytical solution (13) and the approximate solution ( 16) to the i.v.p. ( 1) and the exact solution (20) to the i.v.p. (19) (linearized form to the i.v.p.…”

“…The model of nonlinear oscillators that describes the motion of different types of pendulum oscillators with different rotations and directions is one of the most successful models for describing and modeling many physical and engineering applications such as the wind vibration [11][12][13][14][15][16][17][18][19][20]. To find an exact solution to the dynamical system of the pendulum oscillation is not an easy task, and sometimes it can be impossible due to its nonlinearity.…”

“…However, some numerical methods or analytical approximation methods could be more suitable to get some approximate solutions that may be close enough to the exact solutions [21]. Many authors derived in details the rotating pendulum equation of motion and solved this equation by using different approaches [11][12][13][14][15][16][17][18][19][20]. For instance, He et al [11] used the HPM to derive an analytic solution to the rotating pendulum oscillator.…”

Dynamic and steady-state performance analysis of a linear solenoid parallel elastic actuator (LSPEA) with nonlinear stiffness

“…( 20) considers the influence of current, the result is still a cosine function. Therefore, the KBM method [43] is used to further improve the solution accuracy. In the KBM method, the displacement x, the amplitude a, and the phase  in Eq.…”

On Perturbative Methods for Analyzing Third-Order Forced Van-der Pol Oscillators

“…They found that both the analytical and numerical solutions were completely identical, which confirmed the high efficiency of the KBM approximations. Moreover, the KBMM was implemented to find a highly accurate analytic approximation to the generalized VdP oscillatory equation [16]. Both forced and unforced damped/undamped parametric pendulum oscillatory equations were analyzed to obtain approximate solutions using certain effectiveness, and more accurate, analytical and numerical techniques, including He's frequency-amplitude formulation, He's HPM, the KBMM, and many others [17].…”