A new approach for modelling the damped Helmholtz oscillator: applications to plasma physics and electronic circuits

Abstract

In this paper, a new approach is devoted to find novel analytical and approximate solutions to the damped quadratic nonlinear Helmholtz equation (HE) in terms of the Weiersrtrass elliptic function. The exact solution for undamped HE (integrable case) and approximate/semi-analytical solution to the damped HE (non-integrable case) are given for any arbitrary initial conditions. As a special case, the necessary and sufficient condition for the integrability of the damped HE using an elementary approach is reported. In general, a new ansatz is suggested to find a semi-analytical solution to the non-integrable case in the form of Weierstrass elliptic function. In addition, the relation between the Weierstrass and Jacobian elliptic functions solutions to the integrable case will be derived in details. Also, we will make a comparison between the semi-analytical solution and the approximate numerical solutions via using Runge–Kutta fourth-order method, finite difference method, and homotopy perturbation method for the first-two approximations. Furthermore, the maximum distance errors between the approximate/semi-analytical solution and the approximate numerical solutions will be estimated. As real applications, the obtained solutions will be devoted to describe the characteristics behavior of the oscillations in RLC series circuits and in various plasma models such as electronegative complex plasma model.

Some Novel Approaches for Analyzing the Unforced and Forced Duffing–Van der Pol Oscillators

“…For instance, Du ng-type equation is one of the most famous and successful equations that has been used for modeling and interpreting many nonlinear oscillations in many different dynamical systems such as electrical circuit, optical stability, the buckled beam, and di erent oscillations in a plasma [16][17][18][19]. In plasma physics, there are many evolution equations that can be reduced to Du ng-type equation, Helmholtz-type equation, Du ng-Helmhlotz equation, and Mathieu equation in order to investigate the various oscillations that occur within complicated plasma systems [20][21][22][23]. ere is another type of equation of motion that was used for modeling the nonlinear oscillations in biology, electronics, engineering, plasma physics, and chemistry which is called Van der Pol-Du ng (VdPD) (sometimes called Du ng-Van der Pol (DVdP)) equation and its family [24,25].…”

Galerkin method, ansatz method, and He’s frequency formulation for modeling the forced damped parametric driven pendulum oscillators

“…[6][7][8][9][10] The simple pendulum has been used as a physical model to several solve problems related to many realistic and physical problems, for example, nonlinear plasma oscillations, [11][12][13] Duffing oscillators, [14][15][16][17] Helmholtz oscillations, 18 the nonlinear equation of wave, 19 and many other oscillators. [20][21][22][23][24][25] It is know that the main objective of the numerical approaches is to find some numerical solutions to various realistic physical, engineering, and natural problems, especially when exact solutions are unavailable or extremely difficult to determine. There are many numerical approaches that were used for analyzing the family of the Duffing oscillator and Duffing-Helmholtz oscillator with constant coefficients.…”

Novel Analytical and Numerical Approximations to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Method

“…e pendulum oscillator and some related equation have been used as a physical model to solve several natural problems related to bifurcations, oscillations, and chaos such as nonlinear plasma oscillations [1][2][3][4][5][6][7][8][9], Du ng oscillators [10][11][12][13][14], and Helmholtz oscillations [12], and many other applications can be found in [15][16][17][18][19][20][21][22][23][24]. ere are few attempts for analyzing the equation of motion of the nonlinear damped pendulum taking the friction forces into account [25].…”