On the oscillations in a nonextensive complex plasma by improved differential transformation method: An application to a damped Duffing equation

Abstract

In this study, the nonlinear damping oscillations in a complex non-Maxwellian plasma are investigated. For this purpose, the set of fluid equations of the present plasma model is reduced to the Burger-modified Korteweg De Vries equation (BmKdV) equation using a reductive perturbation technique. Using the traveling wave transformation, the BmKdV equation can be reduced to a damped Duffing equation. The numerical solutions to the damped Duffing equation are obtained using multistage differential transformation method (MsDTM). Also, we compared the obtained results to the semi-analytical approximations using the Padé differential transformation (PDTM) method and numerical solution, by the 4th-order Rung Kutta (RK4) method and analytical solution by He’s frequency method. The impact of relevant plasma parameters, namely, negative dust concentrations and ion kinematic viscosity on the profile of dust ion-acoustic oscillations are examined. The suggested mathematical approaches can help many authors for explaining the mystery of their laboratory results. Moreover, the suggested numerical method can be applied for solving higher order nonlinearity oscillations for a long domain.

Keywords

Damped duffing equation Pad’e differential transformation method multistage differential transformation method plasma oscillations Korteweg De Vries Burgers equation

Introduction

The study of the dynamics of nonlinear systems is still one of the most pressing concerns in engineering and many other fields of science like condensed matter, plasma physics, nonlinear hydrodynamics, nonlinear optics, nonlinear magnetism, and so on. Many researchers have done their best in preparing some accurate mathematical models to describe the dynamics of many nonlinear phenomena that surround all aspects of our life.^1,2,3,2 Among the most successful models that researchers have reached in this regard are Duffing-type equation as well as Helmholtz-type equation.^{4,5,6,7,8,9,10} Due to the achieve great success by Duffing equation in engineering, mechanics, theory of sound, and plasma physics, numerous number of authors devoted most of their efforts to model many physical and engineering problems based on this equation and many other related equations. Many authors have innovated and developed many analytical and numerical methods to find some analytical and numerical solutions for this group of second-order differential equations. For instance, He used gamma function to find a solution for a nonlinear oscillator and compared the obtained solution with both the approximate solution using the homotopy perturbation method (HPM)^11,12 and the exact solution and found that gamma function technique is extremely simple and more accurate.^13,14,15 The two-scale transforms and He-Laplace method were applied for solving a fractal Duffing-Van der Pol oscillator.¹⁶ Also, the HPM was devoted for analyzing and solving Toda and Fangzhu oscillators.¹⁷ Moreover, some strong nonlinear conservative oscillators have been solved using frequency-amplitude formulation.¹⁸

One of the powerful methods in the mathematical computation field is the differential transformation method (DTM).¹⁹ The DTM was used to solve differential equations that are commonly encountered in physics, chemistry, and engineering.²⁰ The solution of differential equations can be in different type such as exact, approximate, and entirely numerical. Because they are trial-and-error in nature or need extensive symbolic computations, the majority of these methods are computationally costly.²¹ DTM is a numerical approach for ordinary (partial) differential equations that uses the shape of polynomials as an approximation to the exact solution.²² Zhou²³ used this method to study electric circuits for the first time in the engineering sector. The boundary value problem,^24,25 structural dynamics, fluid flow difficulties, and other problems have all been solved using this method.^26,27,28,29 The differential transform is an iterative procedure for obtaining Taylor series solutions of differential equations according to Jang.³⁰ The differential transformation method (DTM) differs from the typical high-order Taylor series approach, which necessitates symbolic computation of the data function’s necessary derivatives and is computationally expensive for high-order data. The finite Taylor series is used to evaluate the approximate solution in the DTM. The derivative is not generated directly in the differential transformation method; rather, the relative derivatives are determined through an iteration procedure. The solution by DTM can converge to the exact solution if we know the close form of the infinite series of the solution. In majority of realistic applications, we obtain an unknown close form of the series of the solution and we need to plot the solution as finite series. Since it is based on Taylor expansion which is local convergent, the solution converges only in a small domain about the initial point. However, there are new techniques which have been used to develop DTM.³¹ Her-Moths and Padé were the first researchers who introduced the Padé approximation. This technique has been applied widely in sciences. Recently, the Padé approximation was used to improve some methods and also to develop DTM in order to obtain the solution in long domain.^32,33 Recently, reference [34,35] has used the technique of multistage to develop the DTM for solving ordinary differential equations (ODEs). The authors proved that the multistage DTM (MsDTM) is powerful and accurate more than the Padé-DTM (PDTM) and faster than the 4th-order Rung Kutta (RK4) method. One of the advantages of the MsDTM is that the accuracy of the method can be controlled by the iteration number and the step size. Therefore, we do not need very small step size to obtain high-accurate results. The MsDTM is used to obtain an approximate solution in the long domain.^34,35

In this investigation, we will apply the suggested method for studying the characteristics of the dust ion-acoustic oscillations in a non-Maxwellian complex plasma consisting of inertial cold positive ions and inertialess non-Maxwellian (nonextensive) electrons as well as stationary dust grains with negative charges. The set of fluid equations of the present plasma model is reduced to the nonlinear oscillations damped Duffing equation for investigating the damped oscillations. After that, we can apply the suggested method for analyzing the damped Duffing oscillator. This paper is organized as follows: in Section. II, the DTM and its developed techniques are described in detail. The damped Duffing equation and modeling oscillations are shown in Section III. The analytical solution is shown in Section IV. Numerical simulations have been presented in Section V. The last section is the conclusion of the work.

Numerical Method

The DTM

The definition of Differential transformation technique

Assume z(t) is function in given domain T and it is continuously differentiated, then z(t) can be written as a finite-term Taylor series plus a remainder as follows

z (t) = \frac{1}{u (t)} \sum_{h = 0}^{m} \frac{{(t - t_{0})}^{h}}{N (h)} \frac{d^{h} z (t)}{d t^{h}} + R_{m_{+} 1} (t),

where, R_m+1 is the remains term of Taylor series, N(h) is weighting factor, and u(t) is a kernel corresponding to z(t). In this article, we consider, N(h) =

\frac{1}{h!}

and u(t) = 1. We got the follows

z (t) = \sum_{h = 0}^{m} {(t - t_{0})}^{h} Z (h) + R_{m + 1} (t),

where,

Z (h) = \frac{1}{h!} [\frac{d^{h} z (t)}{d t^{h}}], where h = 0,1, \dots, \infty .

(1)

The Table 1 represents transformation for some operators by DTM.

Table 1.

The differential transformation application for some operators.³⁶

Original function	Transformed function
z(t) ± v(t)	Z(h) ± V(h)
$\frac{d^{m} z (t)}{d t^{m}}$	$\frac{(h + m)!}{h!} Z (h + m)$
z(t)v(t)	$\sum_{l = 0}^{h} Z (l) V (h - l)$

The Multistage DTM

The idea of multistage technique is based on dividing the study domain into sub-domains and after that, we can apply the method of differential transformation in each sub-domain. This method is characterized by the high-accuracy to its obtained approximations depending on the iteration number and a time step. For more details about the algorithm of this method, we advise the reader to see the Refs [34, 35]. The other modification in literature for DTM is by using the Padé approximation, we refer the reader to the reference [37] for more details.

We can apply this method for studying the dust ion-acoustic oscillations in a complex plasma as shown in the next section. In the next section, the fluid equations of the complex non-Maxwellian plasma will be reduced to the nonlinear damped Duffing equation. After that, we can apply all mentioned numerical methods for analyzing the oscillation equation.

Damped duffing equation and modeling oscillations in a nonextensive complex plasma

One of the most important applications to the damped Duffing equation (2) is its ability to describe the nonlinear oscillations in different plasma models. For instance, we can see this realistic application by reducing the basic equations of an unmagnetized and collisionless non-Maxwellian complex plasma consisting of inertial cold positive ions and inertialess non-Maxwellian (nonextensive) electrons as well as stationary dust grains with negative charge. By following the same models in Refs [38,39], the below normalized fluid equations are introduced

{\begin{array}{c} \partial_{t} n + \partial_{x} (n u) = 0, \\ \partial_{t} u + u \partial_{x} u + \partial_{x} ϕ = Γ \partial_{x}^{2} u, \\ \partial_{x}^{2} ϕ + n - n_{e} - μ_{d} = 0, \end{array}

(2)

with

\begin{array}{l} n_{e} = ν {[1 + ϕ (q - 1)]}^{\frac{q + 1}{2 (q - 1)}} \\ = α_{0} + α_{1} ϕ + α_{2} ϕ^{2} + α_{3} ϕ^{3} + \dots, \end{array}

(3)

where,

α_{0} = ν

α_{1} = \frac{ν}{2} (q + 1)

α_{2} = - \frac{α_{1}}{4} (q - 3)

, and

α_{3} = - \frac{α_{2}}{6} (3 q - 5)

Here, n, n_e, u, ϕ, and Γ represent the scaled/normalized following quantities: number density of positive ions, number density of electrons, fluid ion velocity, electrostatic wave potential, and the ion kinematic viscosity, respectively. In equation (3), q denotes the nonextinsive parameter. It is known that the nonextinsive distribution can be recovered the Maxwellian distribution for q = 1, μ_d and ν indicate the dust and electron concentrations which these values must satisfy the neutrality condition, that is, ν = 1 − μ_d.

Now, for investigation the dynamics of nonlinear oscillations in the present plasma model, the reductive perturbation technique (RPT) is employed for reducing equations (2) and (3) to an evolution equation. Before proceeding to apply the RPT to obtain an evolution equation; let us first give a brief summary about the results in Refs [38, 39]. In Refs. 38, the authors reduced the fluid equations of the present plasma model to Burgers equation in order to investigate the dust ion acoustic (DIA) shocks. As well known the Burgers equation has a dissipative term but does not include dispersion term, thus if we use the traveling wave transformation (TWT), the Duffing-type equation cannot recover. However, in Ref. 39, the authors used another stretching for the independent variables and they finally obtained a Zakharov-Kuznetsov-Burgers (ZK-Burgers). This equation can be reduced to a damped Helmholtz equation using the TWT. In Refs [38,39], the authors found that the present plasma model supports both compressive and rarefactive DIA shocks which means that the coefficient of the nonlinear term (A) in Burgers equation (11) in Ref. 38 or in ZK-Burgers (13) in Ref [39] vanishes at a critical value of the nonextensive parameter q. In this case, both Burgers equation (11) in Ref. 38 and in ZK-Burgers (13) in Ref. 39 fails to describe the DIA shock oscillations. Consequently, a higher order nonlinearity should be taken into consideration. To do that the following, new stretching for the independent variables is introduced

X = ε (χ - λ_{p h} t), T = ε^{3} t and Γ = ε η,

(4)

and the dependent quantities are expanded as

{\begin{array}{c} n = 1 + \sum_{i = 1}^{\infty} ε^{i} n^{(i)} (X, T), \\ u = \sum_{i = 1}^{\infty} ε^{i} u^{(i)} (X, T), \\ ϕ = \sum_{i = 1}^{\infty} ε^{(i)} ϕ^{(i)} (X, T), \end{array}

(5)

where, ɛ is a small real number (ɛ ≪ 1).

By inserting both stretching (4) and expansion (5) in the basic equations (2) and (3) and after several long and tedious calculations, we finally obtain the modified KdV-Burgers (mKdVB) equation

\partial_{T} Ψ + C Ψ^{2} \partial_{X} Ψ - D \partial_{X}^{2} Ψ + B \partial_{X}^{3} Ψ = 0,

(6)

with

\begin{array}{l} B = \frac{λ_{p h}^{3}}{2}, C = \frac{λ_{p h}^{3}}{4} [- 6 α_{3} + \frac{15}{λ_{p h}^{6}}], \\ D = \frac{η}{2} and λ_{p h} = \frac{1}{{\sqrt{α}}_{1}} . \end{array}

(7)

Remember that equation (6) is valid only at the critical value of the nonextensive parameter q, that is, at q = q_c which this value can be obtained by equating the coefficient of the quadratic nonlinear term in equation (13) given in Ref. 39 to zero. Accordingly, we get

q_{c} = - 1 + \frac{4}{1 + 3 ν},

(8)

Now, for investigating the shock oscillations, the following TWT is introduced

ψ (X, T) = y (t) and t = X + λ T,

(9)

in the KdVB equation (6), we get

y^{(3)} - \frac{D}{B} y^{″} + \frac{λ}{B} y^{'} + \frac{C}{B} y^{2} y^{'} = 0,

(10)

where, the prime represents the derivative with respect to t. Integrating equation (10) once over t, the following damped Duffing equation is obtained

\ddot{y} + 2 μ \dot{y} + ω_{0}^{2} y + Q y^{3} = 0,

(11)

where, μ = −D/(2B),

ω_{0}^{2} = λ / B

, and Q = C/(3B).

Now, for investigating the characteristics of the damping oscillation in the present plasma model, we should solve the damped Duffing equation (11) using the above suggested numerical methods. For the values of the physical parameters related to the model under consideration, we follow the same data in Refs [38,39].

He’s frequency technique for analyzing oscillator equation

One of the great contribution in the mathematical computation methods of non-dumping equation is using perturbation methods as following relationship:

ω^{2} (A) = ω_{0}^{2} + \frac{3}{4} Q A^{2},

(12)

ω^{2} (A) = 0.13 + \frac{3}{4} (1.55) {(0.1)}^{2} = 0.1416, ω (A) = 0.3762,

(13)

Ω^{2} (A) = ω_{0}^{2} - \frac{2}{4} μ^{2} + \frac{3}{4} Q A^{2} = 0.13 - \frac{2}{4} {(- 0.065)}^{2} + \frac{3}{4} (1.55) A^{2},

(14)

\begin{array}{c} Ω^{2} (A) & = 0.1279 + 1.1625 A^{2}, \end{array}

(15)

\begin{array}{c} Ω^{2} (A) & = 0.139525, \end{array}

(16)

where, A is the initial amplitude of oscillation. The exact solution is,

y (t) = A \cos (Ω t) .

(17)

Numerical simulation

The numerical solutions using the MsDTM are compared with the PDTM numerical solutions as shown in Figure 1. It is shown that the PDTM solution does not converge to the solution at long domain. Also, the numerical solution by MsDTM is compared with the analytical solution using He’s frequency method as illustrated Figure 2. We noticed that the numerical solution converges to the analytical solution and this confirms the high-accuracy of the MsDTM. On the other hand, the numerical solutions for different values to the plasma parameters as given in table (2) are obtained by MsDTM and presented in Figure 3. We observed that the amplitude of the plasma oscillations increases with the enhancement of the dust concentration μ_d while the ion kinematic viscosity η have an opposite effect on the amplitude of the plasma oscillations. Thus, by controlling both the dust concentration and the ion kinematic viscosity, we can control the amplitude of the generated oscillations in order to use them in industrial and medical purposes.

Figure 1.

The plot of the numerical solution using PDTM is compared to the solution by MsDTM for μ = 0.

Figure 2.

The plot of the numerical solutions using MsDTM is compared to the analytical solution by He’s frequency method for μ = −0.065 and $ω_{0}^{2} = 0.13, Q = 1.55$ , A = 0.1.

Figure 3.

The plot of the solutions by MsDTM for μ ≠ 0.

We also observed that the MsDTM solutions are characterized more stable, convergence, and accurate than many other numerical approximations such as the numerical approximation using PDTM over a long time. This is one of the most important features that distinguishes MsDTM from other numerical methods as shown in Figure 1. Also, we noted that the PDTM cannot converge to the solutions while MsDTM is able to reach the solution in long domain which confirms the powerful of the MsDTM.

It is important to study of vibrating mechanical systems, thus a practical resonance is found as follows:

R = \sqrt{w_{0}^{2} - 2 μ^{2}} .

The results in Table 2 and the pure resonance is the value of ω₀.

Table 2.

The values of the physical parameters of the present plasma model.

The physical parameter	μ	$ω_{0}^{2}$	Q	R
μ_d = 0.4	−0.065	0.13	1.55	0.34864
μ_d = 0.6	−0.03535	0.07071	0.445	0.26117
μ_d = 0.8	−0.0125	0.025	0.047	0.157123
η = 0.2	−0.03535	0.07071	0.445	0.26117
η = 0.4	−0.07071	0.07071	0.445	0.24639
η = 0.6	−0.1060	0.07071	0.445	0.21963

Also, modified homotopy perturbation method was also used successfully to solve damped third-order oscillator that exhibits the nonlinearity of the Duffing equation.^40,41,42 Thus, in future work, we can use these techniques for studying the proprieties of nonlinear oscillations in different plasma models.

Conclusion

The dust ion-acoustic oscillations have been investigated in a collisionless non-Maxwellian complex plasma consisting of inertial cold positive ions and inertialess non-Maxwellian (nonextensive) electrons as well as stationary dust grains with negative charges. For this purpose, the fluid equations of the present plasma model are reduced to the nonlinear damped Duffing equation. This equation has been solved and analyzed numerically using both the hybrid Padé transformation with differential transformation method (Padé-DTM) and multistage DTM (MsDTM). The comparison between the suggested methods has been considered which found that the numerical approximations using MsDTM are more accurate and convergence than the Padé-DTM. Also, we made a comparison between the suggested methods and Runge-Kutta numerical approximations to confirm the high efficiency of MsDTM. Moreover, the comparison between the analytical solution using He’s frequency formula and the numerical approximations using the MsDTM was examined. The impact of related plasma parameters, namely, negative dust concentration and ion kinematic viscosity on the profile of damped Duffing oscillations was examined. It was found that the increase of dust concentrations would lead to the enhancement of the plasma oscillation amplitude. On the contrary, with respect to the ion kinematic viscosity, that is, the amplitude of plasma oscillations decreases with increasing the ion kinematic viscosity.

Footnotes

Acknowledgement

The Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia has founded this project, under grant No.(KEP-MSc: 35-665-1443). The authors, therefore, acknowledge with thanks DSR technical and financial support.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia has founded this project, under grant No.(KEP-MSc: 35-665–1443).

ORCID iDs

Noufe H Aljahdaly

S A EL-Tantawy

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