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

Authors: Alvaro H. Salas; Castillo H. Jairo E; M. R. Alharthi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper presents some novel solutions to the family of the Helmholtz equations (including the constant forced undamping Helmholtz equation (equation (1)) and the constant forced damping Helmholtz equation (equation (2))) which have been reported. In the beginning, equation (1) is solved analytically using two different techniques (direct and indirect solutions): in the first technique (direct solution), a new assumption is introduced to find the analytical solution of equation (1) in the form of the Weierstrass elliptic function with arbitrary initial conditions. In the second case (indirect solution), the solution of the undamping (standard) Duffing equation is devoted to determine the analytical solution to equation (1) in the form of Jacobian elliptic function with arbitrary initial conditions. Moreover, equation (2) is solved using a new ansatz and with the help of equation (1) solutions. Also, the evolution equations (equations (1) and (2)) are solved numerically via the Adomian decomposition method (ADM). Furthermore, a comparison between the approximate analytical solution and approximate numerical solutions using the fourth-order Runge–Kutta method (RK4) and ADM is reported. Furthermore, the maximum distance error for the obtained solutions is estimated. As a practical application, the Helmholtz-type equation will be derived from the fluid governing equations of quantum plasma particles with(out) taking the ionic kinematic viscosity into account for investigating the characteristics of (un)damping oscillations in a degenerate quantum plasma model.


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## Body

## 1. Introduction

The ordinary and partial differential equations have played an important role in explaining many natural phenomena, in addition to their applications in many engineering and physical problems. Due to the great role played by these equations, many authors focused their efforts on finding some solutions to these equations [1–10]. The undamping Helmholtz equation q¨t+∑i=1nkiqit=0,whereiis the odd number,i.e.,i=1,3,5,… and undamping Duffing equation q¨t+∑i=1nkiqit=0,wherei=1,2,3,… in addition to their family (including friction/damping force in addition to excitation/perturbation force) are among the most famous differential equations in dynamic, electrical, and engineering systems [11–16]. A lot of physical and engineering problems such as the human eardrum oscillations, the dynamics of the ships, the electrical circuits signal oscillations, heavy symmetric gyroscope, and microperforated panel absorber [17–21] have been investigated using different solutions of the Helmholtz-type oscillator. The Helmholtz-type oscillator is a second-order differential equation with a quadratic nonlinear term in addition to some other terms. For realistic physical situations, we cannot ignore both the friction/dissipation force and excitation/perturbation forcing. Accordingly, the general form of the Helmholtz equation reads [22–24](1)q¨+2γq˙+αq+βq2=F,q0=q0&q′0=q˙0,where q denotes the displacement of the system, α is the natural frequency, β is a nonlinear system parameter, γ represents the damping factor, and F is a constant/excitation force. The first equation in system (1) is called the constant forced damping Helmholtz equation (equation (2)). If the coefficient γ of the damping term 2γq˙ is neglected, equation (2) reduces to the constant forced Helmholtz equation q¨+αq+βq2=F (equation (1)). Also, if both the coefficient γ of the damping term and the constant force F are neglected, the traditional form of Helmholtz equation is covered q¨+αq+βq2=0. The 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. [26–33].It is well known that, in the absence of both friction and the excitation forces from the IVP (1), the unforced and undamping Helmholtz equations are covered. The exact analytic solutions to the unforced and undamping Helmholtz equation have been derived in detail in the literature in terms of the Weierstrass elliptic function [34–38] and Jacobi elliptic functions [38–41]. Generally, to solve any quadratic or cubic nonlinear second-order differential equation, firstly, we should transform it to an elliptic integral and then we solve it [24]. It is known that the unforced and undamping Helmholtz equations are completely integrable, so they have exact solutions, but if the friction force (damping term) is included, then the unforced damping Helmholtz equation becomes nonintegrable and cannot support an exact solution for arbitrary values of its coefficients γ,α,β. Thus, under certain condition, the unforced damping Helmholtz equation has been solved analytically in terms of the Jacobi elliptic functions by Johannessen [24]. Also, Almendral and Sanjuán [27] derived an exact solution to the unforced damping Helmholtz equation using the Lie theory under certain conditions for the coefficients γ,α,β.In many realistic physical models, both the damping and the excitation/external terms are very important to be included, and thus the problem becomes more complicated to find its analytical solutions. In this paper, we will derive some analytical solutions to equation (1) in the terms of the Weierstrass and Jacobian elliptic functions. Also, an approximate analytical solution to equation (2) for arbitrary values to the coefficients and the initial conditions will be derived in detail. Moreover, the problem under consideration will be solved numerically via using the RK4 and ADM to make a comparison between the obtained solutions and the approximate numerical solutions. Furthermore, the maximum distance error between the approximate analytical solution and the approximate numerical solutions will be estimated. Also, the dynamics of nonlinear oscillations that can be generated in the RLC electronic circuits and quantum plasma will be investigated using the solution of equation (2).The rest of this work is organized in the following manner: in Section2, we will introduce in detail our methodology for solving the family of the Helmholtz-type equations. Also, we will introduce some new approaches for solving equation (1) as well as the exact solution of the undamped and unforced Duffing equation, which will be devoted to finding an approximate analytical solution to equation (2). In Section 3, a comparison between the obtained solutions and the approximate numerical solution using the ADM will be investigated. In Section 4, some realistic applications will be introduced. Finally, our results will be summarized in Section 5.


## 2. Our Methodology for Solving the Family of the Helmholtz Equations

Before proceeding in solving the IVP (1), it is necessary to refer to two fundamental equations and their solutions: the first one is called the Duffing equation and the other is called the constant forced Helmholtz equation.


### 2.1. Duffing Equation and Its Solution

The analytical solution to the following IVP, which is called Duffing equation [30],(2)η″t+Rηt+Sη3t=0,η0=η0&η′0=η˙0,is given by the following formula:(3)ηt=c1cnR+Sc12t+c2,m.By inserting this relation into the IVP (2) and after several tedious but simple math operations, we finally get the values of c1 and c2 as follows:(4)c1=±−R+ΔSt,c2=cn−1η0c1,Sc122R+Sc12,where R, S, η0, and η˙0 are real numbers and Δ is called the discriminant to Duffing (2):(5)Δ=R+η02S2+2η˙02S>0.Solution (3) could be expressed as(6)ηt=η0cnωt|m+η˙0/ωdnωt|msnωt|m1−1/21−R+Sη02/Δsnωt|m2,where(7)ω=Δ4,m=12−R2Δ.For negative discriminantΔ<0, the solution may be written in the following form:(8)ηt=ρ−2ρ1+κscωt+sc−1ρ+η0/κρ−η0,m,where(9)m=4ρ2Sρ2S−R2ρ2Sρ2S−R+R−3ρ2S,ω=142ρ2Sρ2S−R+R−3ρ2S,κ=2ρ2Sρ2S−R−R+3ρ2SR+ρ2S,ρ=±2Rη02+Sη04+2η˙02−S4.For a zero discriminantΔ=0, the solution of Duffing equation (2) will be(10)ηt=c1tanhc1−S2t+c2,with(11)c1=±−Sη04−2η02η˙0−s4Sη0,c2=12logη˙0−2η02−S−−S42η04−S−22η02η˙0η˙0,where η˙0η0≠0.WhenR+η02S2+2η˙02S=η0=0, the solution reads(12)ηt=−S2η˙0tanh−S2η˙0t.In caseR+η02S2+2η˙02S=η˙0=0, the solution becomes the constant function ηt=η0.


### 2.2. The Analytical Solution to the Constant Forced Helmholtz Equation

The solution of the constant forced Helmholtz equation(13)ξ″t+a+bξt+cξ2t=0,ξ0=ξ0&ξ′0=ξ˙0,may be expressed in either one of the following forms. Note that a=−F.


#### 2.2.1. First Formula

Suppose that the solution of system (13) has the following form:(14)ξt=A+B1+C1/4d2−℘′t;g2,g3/d1−℘t;g2,g32−d1−℘t;g2,g3,as well as(15)ξt=A+B1+C℘t±℘−1d1;g2,g3;g2,g3,where a represents the constant force and ℘t gives the Weierstrass elliptic function, which satisfies the following relations:(16)℘′x;g2,g32=4℘3x;g2,g3−g1℘x;g2,g3−g3,℘″x;g2,g3=−g22+6℘2x;g2,g3.Inserting equation (14) into system (13) and after tedious straightforward calculations, we can estimate the values of B, C, g2, g3, d1, and d2 as follows:(17)B=−6a+AAc+b2Ac+b,C=122Ac+b,g2=112b2−4ac,g3=12162Ac+bb2−2c3a+A2c−2Abc,d1=A−ξ02Ac+b6a+AAc+b,d2=−ξ˙02Ac+b6a+AAc+b.The value of parameterA represents a root to the following quartic equation:(18)4a2b+a2b2−4a3c+4ab2ξ0−4b2ξ02+4abcξ02−28bcξ02−2ab2−ab3−4a2c+4a2bc−4b2ξ0−2b3ξ0−4abcξ0−2b2cξ02−4ac2ξ02A+b4−4b2+12abc−2ab2c−8a2c2+32bcξ0+12b2cξ0−16c2ξ02+12bc2ξ02A2+2cb3−8b+4ac−4abc+16cξ0+4bcξ0+4c2ξ02A3−c216−b2+4acA4=0.


#### 2.2.2. Second Formula

As another form for the solution of system (13), let us assume that its solution is given by the following relationship:(19)ξt=A+B1−ηt,where η=ηt is the solution to the following Duffing equation:(20)η″t+Rηt+Sη3t=0,η0=η0&η′0=η˙0.By following the same procedures in the above sections, we finally obtain(21)R=3a+b3A+B+Ac3A+2BB,S=−a+AAc+bB,η0=BA−ξ0+1,η˙0=Bξ˙0A−ξ02.The value of parameterA is a solution to the following equation:(22)0=A6c3+3bc2A5+15ac2A4−10c26aξ0+3bξ02+2cξ03+3ξ˙02A3−15c3a2+6abξ0+3b2ξ02+2bcξ03+3bξ˙02A2−39a2b+18ab2ξ0−12a2cξ0+9b3ξ02−6abcξ02+6b2cξ03−4ac2ξ03+9b2ξ˙02−6acξ˙02A+−27a3−54a2bξ0−27ab2ξ02−72a2cξ02−90abcξ03−18b2cξ04−48ac2ξ04−24bc2ξ05−8c3ξ06−27abξ˙02−72acξ0ξ˙02−36bcξ02ξ˙02−24c2ξ03ξ˙02−18cξ˙04.Also, the value of parameterB is a solution to the following equation:(23)0=c4B6−9c2b2−4acB4−432−3a2b2+16a3c−6ab3ξ0+36a2bcξ0−3b4ξ02+18ab2cξ02+36a2c2ξ02−2b3cξ03+48abc2ξ03+9b2c2ξ04+24ac3ξ04+12bc3ξ05+4c4ξ06−3b3ξ˙02+18abcξ˙02+36ac2ξ0ξ˙02+18bc2ξ02ξ˙02+12c3ξ03ξ˙02+9c2ξ˙04.


### 2.3. The Approximate Analytical Solution for the Constant Forced and Damped Helmholtz Equation

Let us rewrite system (1) in the following traditional form:(24)x¨+2εx˙+αx+βx2=F,x0=x0x′0=x˙0,with β≠0.Also, let us assume that(25)α2+4Fβ≥0.Now, suppose that the solution of system (24) is given by(26)xt=d+exp−εtyt,where y=yt is a solution to the following Helmholtz equation:(27)y″+py+βy2=0,y0=y0=x0−d,y′0=y˙0=εx0+x˙0−dε.Inserting (26) into the first equation in system (24), ℝt≡x¨+2εx˙+αx+βx2−F=0, we obtain(28)ℝt=βd2+αd−F+e−2εtytβyt−eεt−α−2βd+ε2+p+βyt.Expression (28) suggests the following choices:(29)−α−2βd+ε2+p=0,βd2+αd−F=0,giving us the values of p and d as follows:(30)p==α+2βd−ε2,d=−α+α2+4Fβ2β.The solution to the following IVP,(31)y″+α+2βd−ε2y+βy2=0,y0=y0=x0−d,y′0=y˙0=εx0+x˙0−dε,may be expressed in different forms as we explained in the previous section.


## 2.1. Duffing Equation and Its Solution

The analytical solution to the following IVP, which is called Duffing equation [30],(2)η″t+Rηt+Sη3t=0,η0=η0&η′0=η˙0,is given by the following formula:(3)ηt=c1cnR+Sc12t+c2,m.By inserting this relation into the IVP (2) and after several tedious but simple math operations, we finally get the values of c1 and c2 as follows:(4)c1=±−R+ΔSt,c2=cn−1η0c1,Sc122R+Sc12,where R, S, η0, and η˙0 are real numbers and Δ is called the discriminant to Duffing (2):(5)Δ=R+η02S2+2η˙02S>0.Solution (3) could be expressed as(6)ηt=η0cnωt|m+η˙0/ωdnωt|msnωt|m1−1/21−R+Sη02/Δsnωt|m2,where(7)ω=Δ4,m=12−R2Δ.For negative discriminantΔ<0, the solution may be written in the following form:(8)ηt=ρ−2ρ1+κscωt+sc−1ρ+η0/κρ−η0,m,where(9)m=4ρ2Sρ2S−R2ρ2Sρ2S−R+R−3ρ2S,ω=142ρ2Sρ2S−R+R−3ρ2S,κ=2ρ2Sρ2S−R−R+3ρ2SR+ρ2S,ρ=±2Rη02+Sη04+2η˙02−S4.For a zero discriminantΔ=0, the solution of Duffing equation (2) will be(10)ηt=c1tanhc1−S2t+c2,with(11)c1=±−Sη04−2η02η˙0−s4Sη0,c2=12logη˙0−2η02−S−−S42η04−S−22η02η˙0η˙0,where η˙0η0≠0.WhenR+η02S2+2η˙02S=η0=0, the solution reads(12)ηt=−S2η˙0tanh−S2η˙0t.In caseR+η02S2+2η˙02S=η˙0=0, the solution becomes the constant function ηt=η0.


## 2.2. The Analytical Solution to the Constant Forced Helmholtz Equation

The solution of the constant forced Helmholtz equation(13)ξ″t+a+bξt+cξ2t=0,ξ0=ξ0&ξ′0=ξ˙0,may be expressed in either one of the following forms. Note that a=−F.


### 2.2.1. First Formula

Suppose that the solution of system (13) has the following form:(14)ξt=A+B1+C1/4d2−℘′t;g2,g3/d1−℘t;g2,g32−d1−℘t;g2,g3,as well as(15)ξt=A+B1+C℘t±℘−1d1;g2,g3;g2,g3,where a represents the constant force and ℘t gives the Weierstrass elliptic function, which satisfies the following relations:(16)℘′x;g2,g32=4℘3x;g2,g3−g1℘x;g2,g3−g3,℘″x;g2,g3=−g22+6℘2x;g2,g3.Inserting equation (14) into system (13) and after tedious straightforward calculations, we can estimate the values of B, C, g2, g3, d1, and d2 as follows:(17)B=−6a+AAc+b2Ac+b,C=122Ac+b,g2=112b2−4ac,g3=12162Ac+bb2−2c3a+A2c−2Abc,d1=A−ξ02Ac+b6a+AAc+b,d2=−ξ˙02Ac+b6a+AAc+b.The value of parameterA represents a root to the following quartic equation:(18)4a2b+a2b2−4a3c+4ab2ξ0−4b2ξ02+4abcξ02−28bcξ02−2ab2−ab3−4a2c+4a2bc−4b2ξ0−2b3ξ0−4abcξ0−2b2cξ02−4ac2ξ02A+b4−4b2+12abc−2ab2c−8a2c2+32bcξ0+12b2cξ0−16c2ξ02+12bc2ξ02A2+2cb3−8b+4ac−4abc+16cξ0+4bcξ0+4c2ξ02A3−c216−b2+4acA4=0.


### 2.2.2. Second Formula

As another form for the solution of system (13), let us assume that its solution is given by the following relationship:(19)ξt=A+B1−ηt,where η=ηt is the solution to the following Duffing equation:(20)η″t+Rηt+Sη3t=0,η0=η0&η′0=η˙0.By following the same procedures in the above sections, we finally obtain(21)R=3a+b3A+B+Ac3A+2BB,S=−a+AAc+bB,η0=BA−ξ0+1,η˙0=Bξ˙0A−ξ02.The value of parameterA is a solution to the following equation:(22)0=A6c3+3bc2A5+15ac2A4−10c26aξ0+3bξ02+2cξ03+3ξ˙02A3−15c3a2+6abξ0+3b2ξ02+2bcξ03+3bξ˙02A2−39a2b+18ab2ξ0−12a2cξ0+9b3ξ02−6abcξ02+6b2cξ03−4ac2ξ03+9b2ξ˙02−6acξ˙02A+−27a3−54a2bξ0−27ab2ξ02−72a2cξ02−90abcξ03−18b2cξ04−48ac2ξ04−24bc2ξ05−8c3ξ06−27abξ˙02−72acξ0ξ˙02−36bcξ02ξ˙02−24c2ξ03ξ˙02−18cξ˙04.Also, the value of parameterB is a solution to the following equation:(23)0=c4B6−9c2b2−4acB4−432−3a2b2+16a3c−6ab3ξ0+36a2bcξ0−3b4ξ02+18ab2cξ02+36a2c2ξ02−2b3cξ03+48abc2ξ03+9b2c2ξ04+24ac3ξ04+12bc3ξ05+4c4ξ06−3b3ξ˙02+18abcξ˙02+36ac2ξ0ξ˙02+18bc2ξ02ξ˙02+12c3ξ03ξ˙02+9c2ξ˙04.


## 2.2.1. First Formula

Suppose that the solution of system (13) has the following form:(14)ξt=A+B1+C1/4d2−℘′t;g2,g3/d1−℘t;g2,g32−d1−℘t;g2,g3,as well as(15)ξt=A+B1+C℘t±℘−1d1;g2,g3;g2,g3,where a represents the constant force and ℘t gives the Weierstrass elliptic function, which satisfies the following relations:(16)℘′x;g2,g32=4℘3x;g2,g3−g1℘x;g2,g3−g3,℘″x;g2,g3=−g22+6℘2x;g2,g3.Inserting equation (14) into system (13) and after tedious straightforward calculations, we can estimate the values of B, C, g2, g3, d1, and d2 as follows:(17)B=−6a+AAc+b2Ac+b,C=122Ac+b,g2=112b2−4ac,g3=12162Ac+bb2−2c3a+A2c−2Abc,d1=A−ξ02Ac+b6a+AAc+b,d2=−ξ˙02Ac+b6a+AAc+b.The value of parameterA represents a root to the following quartic equation:(18)4a2b+a2b2−4a3c+4ab2ξ0−4b2ξ02+4abcξ02−28bcξ02−2ab2−ab3−4a2c+4a2bc−4b2ξ0−2b3ξ0−4abcξ0−2b2cξ02−4ac2ξ02A+b4−4b2+12abc−2ab2c−8a2c2+32bcξ0+12b2cξ0−16c2ξ02+12bc2ξ02A2+2cb3−8b+4ac−4abc+16cξ0+4bcξ0+4c2ξ02A3−c216−b2+4acA4=0.


## 2.2.2. Second Formula

As another form for the solution of system (13), let us assume that its solution is given by the following relationship:(19)ξt=A+B1−ηt,where η=ηt is the solution to the following Duffing equation:(20)η″t+Rηt+Sη3t=0,η0=η0&η′0=η˙0.By following the same procedures in the above sections, we finally obtain(21)R=3a+b3A+B+Ac3A+2BB,S=−a+AAc+bB,η0=BA−ξ0+1,η˙0=Bξ˙0A−ξ02.The value of parameterA is a solution to the following equation:(22)0=A6c3+3bc2A5+15ac2A4−10c26aξ0+3bξ02+2cξ03+3ξ˙02A3−15c3a2+6abξ0+3b2ξ02+2bcξ03+3bξ˙02A2−39a2b+18ab2ξ0−12a2cξ0+9b3ξ02−6abcξ02+6b2cξ03−4ac2ξ03+9b2ξ˙02−6acξ˙02A+−27a3−54a2bξ0−27ab2ξ02−72a2cξ02−90abcξ03−18b2cξ04−48ac2ξ04−24bc2ξ05−8c3ξ06−27abξ˙02−72acξ0ξ˙02−36bcξ02ξ˙02−24c2ξ03ξ˙02−18cξ˙04.Also, the value of parameterB is a solution to the following equation:(23)0=c4B6−9c2b2−4acB4−432−3a2b2+16a3c−6ab3ξ0+36a2bcξ0−3b4ξ02+18ab2cξ02+36a2c2ξ02−2b3cξ03+48abc2ξ03+9b2c2ξ04+24ac3ξ04+12bc3ξ05+4c4ξ06−3b3ξ˙02+18abcξ˙02+36ac2ξ0ξ˙02+18bc2ξ02ξ˙02+12c3ξ03ξ˙02+9c2ξ˙04.


## 2.3. The Approximate Analytical Solution for the Constant Forced and Damped Helmholtz Equation

Let us rewrite system (1) in the following traditional form:(24)x¨+2εx˙+αx+βx2=F,x0=x0x′0=x˙0,with β≠0.Also, let us assume that(25)α2+4Fβ≥0.Now, suppose that the solution of system (24) is given by(26)xt=d+exp−εtyt,where y=yt is a solution to the following Helmholtz equation:(27)y″+py+βy2=0,y0=y0=x0−d,y′0=y˙0=εx0+x˙0−dε.Inserting (26) into the first equation in system (24), ℝt≡x¨+2εx˙+αx+βx2−F=0, we obtain(28)ℝt=βd2+αd−F+e−2εtytβyt−eεt−α−2βd+ε2+p+βyt.Expression (28) suggests the following choices:(29)−α−2βd+ε2+p=0,βd2+αd−F=0,giving us the values of p and d as follows:(30)p==α+2βd−ε2,d=−α+α2+4Fβ2β.The solution to the following IVP,(31)y″+α+2βd−ε2y+βy2=0,y0=y0=x0−d,y′0=y˙0=εx0+x˙0−dε,may be expressed in different forms as we explained in the previous section.


## 3. A Comparison between Our Solutions and ADM Solution

There are many numerical methods that could be used to find approximate solutions to IVP (24). Here, we make use of the ADM [42–44] to solve IVP (24) with(out) forcing term F. According to this method, the first iteration/approximation for the unforced case F reads(32)xADM0t=e−εtℱεx0+x˙0Θ1+x0ℱΘ2,where Θ1=sintℱ, Θ2=costℱ, and ℱ=α−ε2.For the second iteration/approximation, we have(33)xADM1t=βe−εtαℱ3−eεtℱ+εΘ1+ℱΘ2αx0ℱΘ2+εx0ℱ2+x˙0ℱ2+ε3x0+ε2x˙0Θ12.The approximate Adomian approximate solution reads(34)xt=xADM0t+xADM1t+⋯.Forε,α,β,F=0.1,4,1,0 and x0=0&x′0=0.2, we can compare between the approximate/semianalytical solution (26) (for F=0) and the ADM approximate numerical solution (34) and the RK4 approximate numerical solution as shown in Figure 1. Also, the maximum distance error according to the formula LD=maxti≤t≤tfxRK4t−xmethodt has been estimated:(35)LD=max0≤t≤40xRK4t−xsemi−analyt=0.000504825,LD=max0≤t≤40xRK4t−xADMt=0.0000997768.Figure 1
A comparison between approximate analytical solution (26) (dotted curve) for F=0 and both RK4 (dashed curve) and ADM (dashed-dot curve) numerical solutions is plotted against the time.It is clear that the semianalytical solution (26) (for F=0) gives good results as compared to both the RK4 and ADM approximate numerical solutions.Now, let us find an approximate solution for IVP (24) in the presence of the forcing term, using the ADM. Accordingly, the first approximation is given by(36)xADM0t=e−εtℱ˜εx0−d+x˙0Θ˜1+x0−dℱ˜Θ˜2,where Θ˜1=sintℱ˜, Θ˜2=costℱ˜, ℱ˜=Λ−ε2, Λ=α+2βd, and d=−α±α2+4Fβ/2β.The second approximation to IVP (24) according to the ADM reads(37)xADM1t=βe−2tε2Λℱ˜2d−x02−2dβ−α+2x˙0εd−x0−x˙02+βe−tεΛℱ˜ℕεd−x0213Λ−8ε2−6Λx˙0d−x0+2x˙02εΘ˜1+βe−2tεΛℱ˜ℕεd−x028ε2−5Λ+x˙0d−x03Λ−8ε2+2x˙02εΘ˜1−βe−tεΛℕd−x023Λ+8ε2+16x˙0εx0−d+6x˙02Θ˜2−βe−2tε2Λℕℱ˜22x˙0εd−x07Λ−8ε2+x˙02−3Λ+4ε2+d−x02−18ε2Λ+3Λ2+16ε4Θ˜2,where ℕ=8ε2−92dβ+α.The approximate Adomian approximate solution is given by(38)xt=xADM0t+xADM1t+⋯.Forε,α,β,F=0.1,2,1,1 and x0=0&x′0=0.2, a comparison between the semianalytical solution (26) (for F≠0) and the ADM approximate numerical solution (38) and the RK4 approximate numerical solution has been investigated as shown in Figure 2. Furthermore, the maximum distance error has been calculated as follows:(39)LD=max0≤t≤40xRK4t−xsemi−analyt=0.00332902,LD=max0≤t≤40xRK4t−xADMt=0.00132421.Figure 2
A comparison between approximate analytical solution (26) (dotted curve) for F≠0 and both RK4 (dashed curve) and ADM (dashed-dot curve) numerical solutions is plotted against the time.Also, the semianalytical solution (26) for F=0 and F≠0 gives excellent results as compared to the ADM approximate numerical solution (38).


## 4. Quantum Plasma Oscillations

In this section, we will reduce the fluid governing equations of a quantum plasma model to an evolution equation using the RPT [45–49]. After a suitable transformation, we will be able to convert the obtained evolution equation to a Helmholtz-type equation in order to investigate the characteristics behavior of the damping oscillations in the model under consideration. Now, let us assume that we have a collisionless and unmagnetized electron-ion quantum plasma consisting of inertialess degenerate trapped electrons which obey the Fermi–Dirac distribution and classical fluid cold positive nondegenerate ion. Thus, the basic normalized fluid equations that govern the nonlinear dynamics of various structures could be presented as [51, 52](40)∂tn+∂xnu=0,∂tu+u∂xu+∂xϕ=η∂x2u,∂x2ϕ=ne−n,where ne and n represent the normalized electron and ion number densities, respectively, u gives the normalized ion speed, η is the normalized kinematic viscosity of the ions, and ϕ indicates the normalized electrostatic wave potential. Therefore, we shall adopt the adiabatic trapped degeneracy for electrons, by relying on notations similar to those in [52], wherein the fundamental algebra is expressed in detail. The electron normalized number density according to Fermi–Dirac distribution reads(41)ne=1+ϕ3+T21+ϕ≈s0+s1ϕ+s2ϕ2+s3ϕ3,where T expresses the normalized temperature of the degenerate electron, s0=1+T2, s1=3−T2/2, s2=31+T2/8, and s3=−1+5T2/6. Note that expression (41) is obtained under the approximation ϕ≪1 for small wave amplitude.For investigating the nonlinear structures and oscillations in the present model, the RPM will be employed for this purpose. Accordingly, the stretching and expansions for the independent and dependent variables are, respectively, introduced as follows:(42)ξ=ε1/2x−Vpht,τ=ε3/2t,(43)nx,t=1+εn1+ε2n2+ε3n3+⋯,ux,t=εu1+ε2u2+ε3u3+⋯,ϕx,t=εϕ1+ε2ϕ2+ε3ϕ3+⋯,where ε is a small ε≪1 and real parameter and Vph represents the normalized phase velocity of the unmodulated structures. It is assumed that the impact of ionic kinematic viscosity is small according to many lab experiments [53, 54]. Thus, we could set η=ε1/2η˜, where η˜ is O1. By substituting stretching (42) and expansions (43) into system (40) and by collecting the terms of different powers of ε, we could get a system of reduced equations. Solving the system of reduced equations for the first-two orders of ε by following the same procedures in [46–49], we finally obtain the Korteweg–de Vries Burgers (KdVB) equation [50].(44)∂τϕ+Apϕ∂ξϕ+Bp∂ξ3ϕ−Cp∂ξ2ϕ=0,with(45)Ap=Bp3Vph4−2s2,Bp=Vph32,Cp=η˜2,Vph=1s1,where Ap, Bp, and Cp represent the coefficients of the nonlinear, dispersion, and dissipative terms, respectively, and ϕ≡ϕ1.Using the traveling wave transformationϕξ+Vft=qζ, where ζ=ξ+Vft and Vf gives the frame velocity, into the KdVB equation (44) and integrating once over ζ, we get the constant forced and damped Helmholtz equation as follows:(46)q″ζ+2γq′ζ+αqζ+βqζ2=ℂ,where ℂ is the integration constant, γ=−Cp/2Bp, α=Vf/Bp, and β=Ap/2Bp. Now, we can apply the above solution of the constant forced and damped Helmholtz equation that is given in equation (26) to equation (46) for investigating the characteristics of the damped oscillations in a quantum plasma.Note that if the ionic kinematic viscosity is neglected, i.e.,Cp=0, then the KdV and undamped Helmholtz equations could be covered. The KdV equation, i.e., equation (44) for Cp=0, is one of the most popular soliton and cnoidal equations and has been extensively investigated. For the soliton solution, the following conditions must be fulfilled: qζ,q′ζ,q″ζ⟶0 at ζ⟶±∞, so the integration constant ℂ in equation (46) must vanish. Accordingly, equation (46) could be reduced to the undamped Helmholtz equation:(47)q″ζ+αqζ+βqζ2=0.It is well known that equation (47) supports many solutions such as periodic solution (see the solutions to the constant forced Helmholtz equation above) and solitons. The soliton solution to equation (47) in the form of the Weierstrass elliptic function ℘ could be written in the following manner:(48)ϕx,t=2αβ+α2β2+2αCα2−9β/2α1+6℘x−tα2β2+2Cα/α;1/12,−1/216,where x,t≡ζ,τ.Here, the obtained approximate analytical solution (26) to the constant forced and damped Helmholtz equation (46) will be analyzed numerically according to the quantum plasma parameters T,η˜=0.1,0.055, i.e., α,β,γ,F=1.828,2.98,−0.05,F, and T,η˜=0.9,0.18, i.e., α,β,γ,F=1.146,1.7273,−0.1,F. The behavior of the quantum plasma oscillations according to the approximate analytical solution (26) and the approximate numerical solution according to the RK4 method is presented in Figure 3 for different values of quantum plasma parameters. It is clear from Figure 3 that our approximate analytical solution (26) is more accurate than the RK4 numerical solution. On the contrary, the RK4 numerical solution gives poor results and with increasing time this solution becomes unstable.Figure 3
A comparison between the approximate analytical solution (26) (dotted curve) and the RK4 numerical solution (dashed curve) is plotted for different values of quantum plasma parameters T,η˜, i.e., for different values of α,β,γ,F. Here, q0=0.1, and q˙0=0.2.
(a)(b)(c)


## 5. Conclusions

The Helmholtz-type equations including the constant forced undamping Helmholtz equation (equation (1)) and the constant forced damping Helmholtz equation (equation (2)) have been solved analytically and numerically. Two techniques were used to get the analytical solutions to equation (1). In the first technique, we used a new assumption to find an analytical solution to equation (1) in the form of Weierstrass elliptic function. In the second case, the solution of the standard Duffing equation has been utilized to find an analytical solution to equation (1) in the form of Jacobian elliptic function. However, the main goal of this paper is to solve equation (2), using the obtained solutions of equation (1). Moreover, both equation (1) and equation (2) have been solved numerically via the ADM. The analytical and approximate analytical solutions of equations (1) and (2) have been compared to the RK4 and ADM approximate numerical solutions. Furthermore, the maximum distance error between the RK4 approximate numerical solution and the approximate analytical solutions in addition to the approximate numerical solution using the ADM has been estimated. It was found that the obtained solutions are generally consistent with both RK4 and ADM solutions. Moreover, the obtained solutions have been applied for analyzing the oscillations that may arise in the quantum plasma. During the analysis, it was found sometimes that the approximate analytical solution is better than the RK4 numerical solution as shown in the quantum plasma model. Finally, these solutions may help us understand the oscillations that may arise in the different physical and engineering systems.In future work, the similar approaches could be used for analyzing and solving higher-order nonlinear oscillator equations. Also, a damping Helmholtz–Duffing equation with time-dependent forced term is considered one of the most important and vital problems due to its great role in explaining many natural phenomena in different branches of science. Thus, in the next work, some new approaches will be devoted to find some solutions for these problems.



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