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 in the sense that . 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.
1. Introduction
Early descriptions of many of them date back to, at least, 1892, when the book by Greenhill [1] appeared, presenting a variety of such problems: a simple pendulum, catenaries, and a uniform chain that rotates steadily with a constant angular velocity about an axis to which the chain is fixed at two points. Later applications include nonlinear plasma oscillations [2], Duffing oscillators [3], rigid plates satisfying the Johansen yield criterion [4], nonlinear transverse vibrations of a plate carrying a concentrated mass [5], a beam supported at an axially oscillating mount [6], doubly periodic cracks subjected to concentrated forces [7], surface waves in a plasma column [8], coupled modes of nonlinear flexural vibrations of a circular ring [9], dual-spin spacecrafts [10], spacecraft motion about slowly rotating asteroids [11], nonlinear vibration of buckled beams [12], a nonlinear wave equation [13], deep-water waves with two-dimensional surface patterns [14], oscillations of a body with an orbital tethered system [15], and nonlinear mathematical models of DNA [16, 17]. Numerical studies of phase spaces, stability analysis, solution by means of finite differences, application of the Bernoulli wavelet method for estimating a solution of linear stochastic integral equations, existence of periodic solutions, and numerical simulations can be found in [18–22]. The search for new analytical methods that lead to the exact solution of the Helmholtz equations is the vital importance, since the developed methods can be applied to Schrodinger’s nonlinear differential equation, which is known to have different applications in nonlinear optics, plasma physics, fluid mechanics, and Bose–Einstein condensates [23, 24]. As a contribution to the literature, in this article, we present the exact solution to the Helmholtz Oscillator for the given arbitrary initial conditions by means of the Weierstrass elliptic function.
In this paper, we will derive the exact solution to the Helmholtz oscillator:where denotes the displacement of the system, , , is the natural frequency, is a nonlinear system parameter, and is a system parameter independent of the time.
Equation (1) is applied for mathematical modeling in physics and engineering like general relativity, betatron oscillations, vibrations of shells, vibrations of the acoustically driven human eardrum, and solid-state physics [24–28]. The Helmholtz oscillator can be interpreted as a particle moving in a quadratic potential field, and it has also been studied in a nonlinear circuit theory.
One of the possible interesting interpretations of equation (1) is given by a simple electrical circuit.
2. The Physical Models and Helmholtz Equation
2.1. Alternation the LC Series Circuit
Let us consider alternating LC series circuit consisting of a linear inductor and a capacitor of two terminals as a dipole as shown in Figure 1. Also, it is known that a functional relationship between the electric charge , the capacitor voltage , and the time charging has the form as follows:
The relationship between the charge of the nonlinear capacitor and the voltage drop across it may be approximated by the following quadratic equation [29]:where gives the potential across the plates of the nonlinear capacitor and and are constants related to the capacitor.
Now, Kirchhoff’s voltage law for the LC series circuit could be written as follows:where , is the inductor inductance measured by Henry unit, and gives the voltage of the battery which is constant. The relation between the current and charge reads , and by rearrange equation (4), the Helmholtz equation is obtained aswhere , , , where represents the resonant angular frequency, and , where donates the capacitance of the capacitor in farad and represent the initial value of charge on the capacitor plates which has minimum or zero value for the charging capacitor and for discharge, it has maximum values. The LC circuit is considered as an oscillating circuit which stores energy to oscillate at the resonant frequency of the circuit where could be calculated from the resonance condition: inductor impedance = capacitor impedance .
2.2. Electronegative Plasmas and the KdV–Helmholtz Equation
Let us consider the propagation of electrostatic nonlinear ion-acoustic structures in a collisionless electronegative plasma consisting of thermal particles (including Maxwellian electrons and light negative ions) and fluid cold positive ions. The dynamics of nonlinear electrostatic structures are governed by the following dimensionless fluid equations [30–33]:where and are the normalized positive ion number density and fluid velocity, respectively, and gives the normalized electrostatic potential [30]. Here, gives the temperature ratio of electron-to-negative ion, gives the electron concentration, and refers to the negative concentration, where , , and are the unperturbed equilibrium number densities of the positive ion, electron, and negative ion, respectively. Accordingly, the neutrality condition at equilibrium reads , with and , where .
By applying the RPT, the independent variables are given by and and the dependent quantities are expanded as , , and . Here, is the normalized wave phase velocity and is a real and small parameter . Inserting stretching and expansion values of the independent and dependent quantities into equations (6)–(8), we get a system of reduced equations with different orders of . The first-order dependent quantities could be obtained from solving the system of the lowest orders in as , , and . By solving the system of reduced equations to the next order in , the following KdV equation is obtained:where the coefficient of the nonlinear term and the coefficient of the dispersion term , where and . Note that, for , the positive (negative) pulses can exist and propagate in the present plasma model. The sign and the values of are related to the relevant plasma parameters.
Using the travelling wave transform and into equation (9) and integrating the obtained result over , we getwhere , , , and represents the normalized velocity of the moving frame. Here, , where is the integrating constant, , and .
3. Exact Solution
In this section, we are going to solve, in general, the equation:
To do that, the following ansatz is suggested:where , stands for the Weierstrass elliptic function. This function satisfies the following ordinary differential equation (ODE):
Inserting ansatz (13) into (12) and making use of (15) gives , where
Equating the coefficients of to zero gives an algebraic system, and by solving it, we obtainand thus, we get
Now, for determining the values of and , the initial conditions must be applied in addition to the following formula:where and for our case. According to this relation, we getwhere and .
The initial conditions givewhere the values of and are defined in equation (17).
Finally, the value of can be estimated from the conditionwhich gives
It is clear that must obey the cubic equation:
We will choose the first root of equation (24), i.e.,with
For , i.e., and , the following solution is obtained:
In the case, when , the solution to initial value problem (1) is given byor, in a more compact form,where .
Expression stands for the inverse of the Weierstrass elliptic function which is defined asfor real and .
The obtained solution (29) is periodic with periodwhere is the greatest real root of the cubic
Taking (30) into account, the period in (29) may also be expressed as
Remark 1. The Weierstrass elliptic function is related to the Jacobian elliptic functions as follows: whereIn view of (35), the fundamental period of the Weierstrass function may also be expressed in the following form:Further properties of the Weierstrass function with applications in quantum theory of Lame potential may be found in [26].
4. Results and Discussions
Let us consider a nonlinear series circuit with , , , and let the circuit without external battery, i.e., . Thus, the exact solution to the initial value problemaccording to formula (27), is given by
This solution is periodic, and to find its period according to equation (32), the following cubic equation is solved:which is given the following three real roots:
The greatest root is .
Using formula (31), we obtain
Also, we can use equations (34)–(38) to obtainand then, the period of the solution is given by
From equation (43) and (45), the period is obtained as
In Figure 2, the analytical solution (40) is compared with the Runge–Kutta (RK) numerical solution, and the agreement between the two solutions is found to be very good. If the initial current value of the LC circuit is not equal to zero, i.e, , but has a specified value , in this case, we can consider solution (28) or (29) instead of the first one (27). It is observed that the results obtained by the exact solution are in an excellent agreement with the RK numerical solution, as shown in Figure 3. For the plasma application, our analyses are based on the experimental observations data where electron and negative ion temperatures are, respectively, given by and and gives the electron number density. According to these data, we get , 11.5, and 17.25 and the negative ions concentration [26, 27, 30]. Using these data, we get , and by plotting these data according to solution (27), we note, from Figure 4, the excellent matching between our solution and the RK numerical solution.
5. Conclusions
The quadratic nonlinear Helmholtz differential equation is solved analytically for any arbitrary initial conditions through the Weierstrass elliptic function. Accordingly, some new analytical solutions are obtained for the first time using ansatz (12). The proposed methodology is of great importance in solving several physics problems in plasma physics and electronic circuits. For instance, our obtained solutions are devoted for describing the dynamics of nonlinear oscillations that propagate in electronegative complex plasmas. Moreover, our solutions are applied for studying the characteristic behavior of signal oscillations in the LC series circuits.
Data Availability
No data were used to support this research.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors thank the University Francisco Jose de Caldas for the support to carry out this work. Also, the authors thank Associate Prof. S.A. El-Tantawy, the chief of Research Center for Physics (RCP), Faculty of Science and Arts, Al-Baha University, Saudi Arabia, and Faculty of Science, Port Said University, Egypt, for fruitful discussions, careful reading, and improving their manuscript very well.