A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

Abstract

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.

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

“…To find an analytical approximation to the i.v.p. (18), the proposed method (ansatz method (AM)) is summarized in the following steps…”

“…Step (7) Now, let us retune to the i.v.p. (18) and suppose that its analytical approximation is given by…”

“…[1][2][3][4][5] In the framework of nonlinear dynamics, there is no doubt that the scenario of dynamic mechanism of the pendulum motion is one of the main objects that have deserved more attention in modeling different kind of (non)linear phenomena related to the nonlinear oscillations, chaos, and bifurcations. [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.…”

On the Approximate Solutions of the Constant Forced (Un)Damping Helmholtz Equation for Arbitrary Initial Conditions

“…e initial value problem (IVP) (1) and its family have many applications in several fields, starting from analyzing the signals that propagate in electrical circuits, plasma physics, general relativity, betatron oscillations, vibrations of shells, vibrations of the acoustically driven human eardrum, solid-state physics, etc. [26][27][28][29][30][31][32][33].…”

The Duffing Oscillator Equation and Its Applications in Physics

“…Since the 1970s, it has become really popular with researchers into chaos, as it is possibly one of the simplest equations that describes chaotic behavior of a system. is equation is also useful in the study of soliton solutions to important physics models such as KdV equation, mKdV equation, sine-Gordon equation, Klein-Gordon equation, nonlinear Schrodinger equation, and shallow water wave equation [8][9][10][11][12][13][14][15][16][17][18].…”