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Indian Journal of Physics

, Volume 92, Issue 12, pp 1643–1650 | Cite as

Investigation of cylindrical shock waves in dusty plasma

  • A. Nazari-Golshan
Original Paper
  • 66 Downloads

Abstract

Electronegative dusty plasma composed of Boltzmann electrons, Boltzmann negative ions, inertial positive ions and charge fluctuating dust has been considered. The fractional modified Burgers’ (FMB) equation, which is derived using Euler–Lagrange variational technique, is analytically obtained and solved for studying the cylindrical geometry effect on the propagation of the dust ion acoustic shock wave. The Laplace homotopy perturbation method, the so-called LHPM is applied to solve the FMB equation. The effect of the fractional parameter, positive ion number density at equilibrium, the number of equilibrium electrons residing on the dust grain surface and shock velocity on the behavior of the shock waves in the dusty plasma has been investigated.

Keywords

Dust ion acoustic waves Shock wave Laplace transforms homotopy perturbation method Fractional modified Burgers’ equation Negative ions 

PACS Nos.

52.27.Lw 52.35.Fp 52.35.Mw 52.35.Tc 

1 Introduction

The existence of ion acoustic (IA), dust ion acoustic (DIA) [1, 2, 3, 4, 5, 6, 7, 8], electron acoustic (EA) and dust acoustic (DA) [9, 10, 11, 12] waves has been investigated both theoretically and experimentally in dusty plasma. A number of the authors have rigorously investigated the linear structure of the DIA waves [1, 4, 13, 14]. They have found that the presence of the negative ions, which occur in space and laboratory dusty plasma [15, 16], significantly has modified the charging of dust particles. On the other hand, the negative ion significantly reduces the magnitude of the charge of the dust [15] which can cause to occur a transition from negative to positive dust [15, 16, 17]. Accordingly, the presence of the negative ions can be proved in nonlinear structure, space and astrophysical plasma systems, which is relevant for interstellar space, planetary atmospheres, cometary tails, nebula and ionosphere of Saturn’s moon Titan [16, 18, 19, 20, 21]. Therefore, the shock structure which is a nonlinear structure, has received considerable interest in the last few decades. On the other hand, depending upon whether the dust particles are considered to be static or mobile in plasma, solitary or shock waves can be appeared. Mamun et al. [22] have considered a multi-ion dusty plasma composed of electrons, positive ions, heavy negative ions, and stationary massive dust grains and have studied the formation of DIA shock waves. Based on their findings, in the lower speed limit small amplitude shock waves are formed, while in the larger speed limit large amplitude shock waves are formed. The nonlinear propagation of the DIA shock waves in a dusty electronegative plasma has been studied by Mamun and Tasnim [23]. Based on their findings, the dust charge fluctuation is a source of dissipation, and is responsible for the formation of DIA shock structures. Zobaer et al. [24] have generalized the work of Mamun and Tasnim [23] to nonplanar cylindrical and spherical geometries. They have showed the cylindrical DIA shock waves travel faster than the planner ones, but slower than the spherical ones. They have also found the amplitude of the cylindrical DIA shock waves is larger than that of the planar ones, but smaller than that of the spherical ones. Recently, some researchers have rigorously investigated fractional derivatives to plasma physics [25, 26, 27, 28, 29, 30]. El-Wakil et al. [25] have studied the time fractional derivatives to the KdV equation for plasma with two different electron temperature and stationary ion. They found that the time fractional parameters significantly change the soliton amplitude of the electron acoustic solitary waves. Guo et al. [26] have investigated the effect of the plasma parameters on the compressive and rarefactive ion acoustic solitary waves by using fractional derivatives. Nazari-Golshan and Nourazar [27] have applied the time fractional derivatives to the modified KdV equation for studying the nonlinear propagation of small but finite amplitude dust ion-acoustic solitary waves in un-magnetized dusty plasma with trapped electrons.

The aim of this paper is to study the propagation of the cylindrical DIA shock waves by using the fractional modified Burgers’ (FMB) equation in unmagnetized plasma consisting of Boltzmann electrons, Boltzmann negative ions, inertial positive ions and charge fluctuating dusts. The paper has been arranged as: Sec. I containing the literature survey, Sec. II with the FMB equation derived by using Euler–Lagrange variational technique, and Sec. III showing the results of our work.

2 Governing equations

2.1 Basic equations

We consider un-magnetized dusty electronegative plasma composed of Boltzmann electrons, Boltzmann negative ions, inertial positive ions and charge fluctuating dusts. The one-dimensional equations describing the nonlinear propagation of the cylindrical DIA shock wave in such dusty electronegative plasma are given as follows:
$$ \frac{{\partial n_{i} }}{\partial t} + \frac{1}{r}\frac{\partial }{\partial r}\left( {rn_{i} v_{i} } \right) = 0, $$
(1)
$$ \frac{{\partial v_{i} }}{\partial t} + v_{i} \frac{{\partial v_{i} }}{\partial r} + \frac{\partial \varphi }{\partial r} = - \frac{{g_{i} }}{{n_{i} }}\frac{{\partial n_{i} }}{\partial r}, $$
(2)
$$ \frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial \varphi }{\partial r}} \right) = - n_{i} + \eta_{d} \left( {1 + Z_{d} } \right) + \eta_{e} \exp \left( \varphi \right) + \eta_{n} { \exp }\left( {g_{n} \varphi } \right), $$
(3)
$$ \frac{{\partial Z_{d} }}{\partial t} + I_{e} + I_{n} + I_{i} = 0 $$
(4)
where \( n_{i} \) is the positive ion number density normalized by its equilibrium value \( n_{i0} \), \( v_{i} \) is the positive ion fluid speed normalized by \( C_{i} = \left( {T_{e} /m_{i} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \), \( \varphi \) is the electrostatic wave potential normalized by \( T_{e} /e \), \( Z_{d} \) is perturbed part of the number of electrons residing on the dust grain surface normalized by its equilibrium value \( Z_{d0} \), \( I_{e} \), \( I_{n} \) and \( I_{i} \) are the electron, negative ion and positive ion currents respectively normalized by \( Z_{d0} e/\omega_{i} \), \( \omega_{i}^{ - 1} = \left( {m_{i} /4\pi n_{i0} e^{2} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \) is the ion plasma period, \( m_{i} \) is the ion mass, The time, t, and cylindrical space , r, are variables normalized by \( \omega_{pi}^{ - 1} \) and \( \lambda_{Dm} \) respectively, \( \eta_{e} = 1/\left( {1 + \alpha_{1} + \alpha_{2} } \right) \), \( \eta_{d} = \alpha_{2} /\left( {1 + \alpha_{1} + \alpha_{2} } \right) \), \( \eta_{n} = \alpha_{1} /\left( {1 + \alpha_{1} + \alpha_{2} } \right) \), \( g_{i} = T_{i} /T_{e} \), \( g_{n} = T_{e} /T_{n} \), \( \alpha_{1} = n_{n0} /n_{e0} \), \( \alpha_{2} = Z_{d0} n_{d0} /n_{e0} \); \( n_{e0} , n_{n0} \) and \( n_{d0} \) are electron, negative ion and dust number density at equilibrium respectively, \( T_{i} \left( {T_{n} } \right) \) is the positive (negative) ion thermal energy, the normalized electron, negative ion and positive ion currents (\( I_{e} \), \( I_{n} \) and \( I_{i} \)) are given by [23]
$$ I_{n} = - 2\sqrt {2\pi } r_{d}^{2} \lambda_{Dm} n_{n0} \sqrt {\frac{{m_{i} }}{{g_{n} m_{n} }}} exp\left( {g_{n} \varphi - \gamma_{n} Z_{d} - \gamma_{n} } \right) $$
(5)
$$ I_{e} = - 2\sqrt {2\pi } r_{d}^{2} \lambda_{Dm} n_{e0} \sqrt {\frac{{m_{i} }}{{m_{e} }}} exp\left( {\varphi - \gamma_{e} Z_{d} - \gamma_{e} } \right) $$
(6)
$$ I_{i} = n_{i} I_{i0} \left( {1 + \gamma_{i} Z_{d} + \gamma_{i} } \right),\quad I_{i0} = \sqrt {2\pi } r_{d}^{2} \lambda_{Dm} n_{i0} \sqrt {\frac{{T_{i} }}{{T_{e} }}} $$
(7)
where \( \gamma_{e} = Z_{d0} e^{2} /r_{d} T_{e} \), \( \gamma_{n} = Z_{d0} e^{2} /r_{d} T_{n} \), \( \gamma_{i} = Z_{d0} e^{2} /r_{d} m_{i} v_{i0}^{2} \) and \( v_{i0} \) is the positive ion streaming speed which is assumed to be much longer than its thermal speed.

2.2 Derivation of fractional shock wave equation

To study the nonlinear propagation of dust ion acoustic (DIA) shock waves in dusty electronegative plasma according to reductive perturbation method [31, 32], we first introduce the stretched coordinates \( \upxi = \varepsilon \left( {r + V_{0} t} \right),\,\tau = \varepsilon^{2} t \) where, \( \upvarepsilon \) is the small dimensionless parameter measuring the weakness of the nonlinearity and \( V_{0} \) is the wave velocity normalized by \( C_{i} \). The dimensionless variables \( n_{i} \), \( \varphi , \, Z_{d} \) and \( v_{i} \) may be expanded in power series of \( \upvarepsilon \) as:
$$ \begin{aligned} n_{i} & = 1 + \varepsilon n_{i1} + \varepsilon^{2} n_{i2} + \cdots , \\ \varphi & = \varepsilon \varphi_{1} + \varepsilon^{2} \varphi_{2} + \cdots , \\ v_{i} & = \varepsilon v_{i1} + \varepsilon^{2} v_{i2} + \cdots \\ Z_{d} & = \varepsilon Z_{d1} + \varepsilon^{2} Z_{d2} + \cdots \\ \end{aligned} $$
(8)
By substituting Eq. (8) into Eqs. (14) and neglecting terms of higher order than \( \varepsilon \) lead to:
$$ \begin{aligned} n_{i1} & = \frac{{\varphi_{1} }}{{V_{0}^{2} - g_{i} }}, \quad v_{i1} = \frac{{V_{0} \varphi_{1} }}{{V_{0}^{2} - g_{i} }}, Z_{d1} = \left( {\beta_{1} + \beta_{2} \left( {g_{n} - \frac{1}{{V_{0}^{2} - g_{i} }}} \right)} \right)\varphi_{1} , \\ V_{0}^{2} & = g_{i} + \frac{{1 + \eta_{d} \beta_{2} }}{{C_{1} + \eta_{d} \beta_{1} + \eta_{d} \beta_{2} g_{n} }} \\ \beta_{1} & = \frac{{2\sqrt {2\pi } r_{d}^{2} \lambda_{Dm} n_{e0} \sqrt {\frac{{m_{i} }}{{m_{e} }}} exp\left( { - \gamma_{e} } \right)\left( {1 - g_{i} } \right)}}{{\left( {\gamma_{e} - \gamma_{n} } \right)2\sqrt {2\pi } r_{d}^{2} \lambda_{Dm} n_{e0} \sqrt {\frac{{m_{i} }}{{m_{e} }}} exp\left( { - \gamma_{e} } \right) + \gamma_{n} I_{i0} \left( {1 + \gamma_{i} } \right) + \gamma_{i} I_{i0} }} \\ \beta_{2} & = \frac{{I_{i0} \left( {1 + \gamma_{i} } \right)}}{{\left( {\gamma_{e} - \gamma_{n} } \right)2\sqrt {2\pi } r_{d}^{2} \lambda_{Dm} n_{e0} \sqrt {\frac{{m_{i} }}{{m_{e} }}} exp\left( { - \gamma_{e} } \right) + \gamma_{n} I_{i0} \left( {1 + \gamma_{i} } \right) + \gamma_{i} I_{i0} }} \\ C_{1} & = \eta_{e} + \eta_{n} g_{n} \\ \end{aligned} $$
(9)
Preserving terms of up to the second order in \( \varepsilon \) and eliminating the second-order variables, (\( v_{i2} \), \( n_{i2} , Z_{d2} \) and \( \varphi_{2} \)) and using Eq. (9), the modified Burgers’ equation may be obtained as:
$$ \frac{{\partial \varphi_{1} }}{\partial \tau } + A\varphi_{1} \frac{{\partial \varphi_{1} }}{{\partial\upxi}} - B\frac{{\partial^{2} \varphi_{1} }}{{\partial\upxi^{2} }} + \frac{{\varphi_{1} }}{2\tau } = 0. $$
(10)
The nonlinear coefficient \( \left( A \right) \) and dissipation coefficient \( \left( B \right) \) are given by:
$$ \begin{aligned} A & = \frac{{f_{2}^{2} }}{{2V_{0} }}\left( {b_{1} + b_{2} + b_{3} + b_{4} + b_{5} } \right) \\ B & = \frac{{\eta_{d} V_{0} \beta_{2} f_{2}^{2} f_{1} }}{{2V_{0} I_{i0} \left( {1 + \gamma_{i} } \right)f_{3} }} \\ \end{aligned} $$
(11)
where, \( b_{1} = p/f_{2}^{3} , p = 3V_{0}^{2} - g_{i} , b_{2} = 2I_{i0} \beta_{2} \gamma_{i} \eta_{d} f_{1} /\left( {1 + \gamma_{i} } \right)I_{i0} f_{2} f_{3} , b_{3} = - c_{2} /f_{3} , c_{2} = \eta_{e} - \eta_{n} g_{n}^{2} , b_{4} = - f_{4} \left( {g_{n} - \gamma_{n} f_{1} } \right)/f_{2} , b_{5} = - \eta_{d} \beta_{1} \left( {1 - \gamma_{e} f_{1} } \right)/f_{2} f_{1} , f_{1} = \beta_{1} + \beta_{2} \left( {g_{n} - \frac{1}{{f_{2} }}} \right), f_{2} = V_{0}^{2} - g_{i} , f_{3} = 1 + \eta_{d} \beta_{2} , f_{4} = \eta_{d} \beta_{2} - \left( {\eta_{d} \beta_{1} } \right)/\left( {1 - g_{n} } \right) , f_{5} = 1 - g_{n} . \) In Eq. (10) \( \varphi_{1} \) is a potential variable, \( \upxi \) and \( \tau \) are space and time coordinates respectively.
Then, the Euler–Lagrange variational technique [33, 34, 35] can be used to derive the Lagrangian of the modified Burgers’ equation, which leads to give the fractional form of the Lagrangian of the modified Burgers’ equation as (see “Appendix B”):
$$ H\left( {{}_{0}^{{}} D_{\tau }^{\alpha }\Upsilon ,\Upsilon _{\xi } ,\Upsilon } \right) = - \frac{1}{2}\left[ {{}_{0}^{{}} D_{\tau }^{\alpha }\Upsilon } \right]\Upsilon _{\xi } - \frac{1}{6}A\Upsilon _{\xi }^{3} + B\Upsilon _{\xi } \varphi_{1\xi \xi } + \frac{{\varphi_{1} }}{2\tau },\quad 0 < \alpha \le 1, $$
(12)
where \( {}_{a}^{{}} D_{\tau }^{\alpha } \) is the left Riemann–Liouville fractional derivative [36, 37]. Then, Euler–Lagrange variational technique [33, 34] is used to derive the FMB equation in the form (see “Appendix B”)
$$ {}_{0}^{R} D_{\tau }^{\alpha } \varphi_{1} + A\varphi_{1} \frac{{\partial \varphi_{1} }}{\partial \xi } - B\frac{{\partial^{2} \varphi_{1} }}{{\partial \xi^{2} }} + \frac{{\varphi_{1} }}{2\tau } = 0.\quad 0 < \alpha \le 1,\quad \tau \in\left[ {0,T_{0} } \right] $$
(13)
where the fractional operator \( {}_{0}^{R} D_{\tau }^{\alpha } \) is called Riesz fractional derivative. It is represented by [36, 37]
$$ {}_{0}^{R} D_{\tau }^{\alpha } \varphi_{1} = \frac{1}{2}\frac{1}{{\varGamma \left( {k - \alpha } \right)}}\frac{{d^{k} }}{{dt^{k} }}\left[ {\mathop \int \limits_{a}^{b} d\tau \left| {t - \tau } \right|^{k - \alpha - 1} \varphi_{1} } \right] $$
(14)

Equation (13) represents the fractional modified Burgers’ (FMB) equation that is formulated by using Euler–Lagrange variational technique.

The partial differential equations and nonlinear fractional differential equations have been solved using different methods such as: homotopy perturbation method [38, 39, 40], Adomian decomposition method [41, 42], variational iteration method [43, 44], the trigonometric function series method [45, 46], the bifurcation method [47, 48, 49], The extended \( \left( { \frac{G}{G0} } \right) \)-expansion method [50, 51, 52], tanh-coth expansion method and Jacobi elliptic function expansion method [53, 54], the exact traveling wave solutions [55, 56, 57], First integral method and the mapping method [58, 59]. In this paper, the hybrid of Laplace transform and homotopy perturbation method (LHPM) has been used to solve the fractional modified Burgers’ (FMB) equation (See “Appendix A”).

In solution of the Eq. (13), The Laplace transform is applied to both sides of the FMB equation. By using LHPM, all conditions are satisfied over the entire range of time domain and the sequence of recursive equations can be written in the following manner for \( k = 1,2,3, \ldots ,n \)
$$ \begin{aligned} & p^{1} : s^{\alpha } \left( {{\mathcal{L}}\left( {u_{2} } \right)} \right) - \left[ {{}_{0}^{R} D_{\tau }^{\alpha - 1} \left( {u_{1} } \right)} \right]_{\tau = 0} + A\left( {{\mathcal{L}}\left( {H_{1} } \right)} \right) - B\frac{{\partial^{2} }}{{\partial \xi^{2} }}\left[ {{\mathcal{L}}\left( {u_{1} } \right)} \right] + \frac{1}{2}{\mathcal{L}}\left( {\frac{{u_{1} }}{\tau }} \right) = 0 \\ & p^{2} : s^{\alpha } \left( {{\mathcal{L}}\left( {u_{3} } \right)} \right) - \left[ {{}_{0}^{R} D_{\tau }^{\alpha - 1} \left( {u_{2} } \right)} \right]_{\tau = 0} + A\left( {{\mathcal{L}}\left( {H_{2} } \right)} \right) - B\frac{{\partial^{2} }}{{\partial \xi^{2} }}\left[ {{\mathcal{L}}\left( {u_{2} } \right)} \right] + \frac{1}{2}{\mathcal{L}}\left( {\frac{{u_{2} }}{\tau }} \right) = 0 \\ & \vdots \\ & p^{k} : s^{\alpha } \left( {{\mathcal{L}}\left( {u_{k} } \right)} \right) - \left[ {{}_{0}^{R} D_{\tau }^{\alpha - 1} \left( {u_{k - 1} } \right)} \right]_{\tau = 0} + A\left( {{\mathcal{L}}\left( {H_{k - 1} } \right)} \right) - B\frac{{\partial^{2} }}{{\partial \xi^{2} }}\left[ {{\mathcal{L}}\left( {u_{k - 1} } \right)} \right] + \frac{1}{2}{\mathcal{L}}\left( {\frac{{u_{k - 1} }}{\tau }} \right) = 0 \\ & 0 < \alpha \le 1.\quad \tau \in \left[ {0,T_{0} } \right],\quad k = 1,2,3, \ldots ,n \\ \end{aligned} $$
(15)
The zero order correction of the stationary shock solution is selected as initial conditions of Eq. (13),
$$ \left. {\varphi_{1} } \right|_{\tau = 0} = u_{0} = \frac{{U_{0} }}{A}\left( {1 - { \tanh }\left( {\frac{{\upxi\,U_{0} }}{2C}} \right)} \right) $$
(16)
where \( U_{0} \) is the DIA shock wave velocity.
Substituting this zero order approximation and \( H_{0} \) into Eq. (15) lead to the first order approximation as:
$$ u_{1} = \frac{{U_{0}^{3} }}{6AC}\left( { - 2tanh\left( {\frac{{\xi U_{0} }}{2C}} \right) + 2tanh^{3} \left( {\frac{{\xi U_{0} }}{2C}} \right) + 2 - 2tanh^{2} \left( {\frac{{\xi U_{0} }}{2C}} \right)} \right)\frac{{\tau^{2\alpha } }}{{\varGamma \left( {1 + 2\alpha } \right)}} + \frac{{U_{0}^{3} }}{12AC}\left( {3tanh\left( {\frac{{\xi U_{0} }}{2C}} \right) - 3tanh^{3} \left( {\frac{{\xi U_{0} }}{2C}} \right)} \right)\frac{{\tau^{\alpha } }}{{\varGamma \left( {1 + \alpha } \right)}} + \frac{{U_{0} }}{6A}\left( {3 - tanh\left( {\frac{{\xi U_{0} }}{2C}} \right)} \right) $$
(17)
By substituting this equation into Eq. (15) and using the Maple package lead to the second order approximation as:
$$ u_{2} = \frac{{U_{0}^{5} }}{{AC^{2} }}\left[ {\left( { - 64tanh^{5} \left( {\frac{{\xi U_{0} }}{2C}} \right) + 64tanh^{3} \left( {\frac{{\xi U_{0} }}{2C}} \right) +\, 96tanh^{4} \left( {\frac{{\xi U_{0} }}{2C}} \right) + 32 - 128tanh^{2} \left( {\frac{{\xi U_{0} }}{2C}} \right)} \right)\frac{{\tau^{4\alpha } }}{{120\varGamma \left( {1 + 4\alpha } \right)}} + \left( { - 720tanh^{5} \left( {\frac{{\xi U_{0} }}{2C}} \right) + 1200tanh^{3} \left( {\frac{{\xi U_{0} }}{2C}} \right) - 480tanh\left( {\frac{{\xi U_{0} }}{2C}} \right)} \right)\frac{{\tau^{2\alpha } }}{{1440\varGamma \left( {1 + 2\alpha } \right)}} + \left( {540tanh^{5} \left( {\frac{{\xi U_{0} }}{2C}} \right) - 870tanh^{3} \left( {\frac{{\xi U_{0} }}{2C}} \right) - 315tanh^{4} \left( {\frac{{\xi U_{0} }}{2C}} \right) + 330tanh\left( {\frac{{\xi U_{0} }}{2C}} \right) - 105 + 420tanh^{2} \left( {\frac{{\xi U_{0} }}{2C}} \right)} \right)\frac{{\tau^{3\alpha } }}{{480\varGamma \left( {1 + 3\alpha } \right)}}} \right] + \frac{{U_{0}^{3} }}{AC}\left[ {\left( {tanh^{3} \left( {\frac{{\xi U_{0} }}{2C}} \right) + 1 - tanh^{2} \left( {\frac{{\xi U_{0} }}{2C}} \right) - tanh\left( {\frac{{\xi U_{0} }}{2C}} \right)} \right)\frac{{7\tau^{2\alpha } }}{{18\varGamma \left( {1 + 2\alpha } \right)}} + \left( { - tanh^{3} \left( {\frac{{\xi U_{0} }}{2C}} \right) + tanh\left( {\frac{{\xi U_{0} }}{2C}} \right)} \right)\frac{{3\tau^{\alpha } }}{{16\varGamma \left( {1 + \alpha } \right)}}} \right] + \frac{{U_{0} }}{4A}\left[ {1 - tanh\left( {\frac{{\xi U_{0} }}{2C}} \right)} \right] $$
(18)
and so on. The higher order approximations can be calculated using the Maple package to the appropriate order. Consequently, the solution of Eq. (13) in a series form is given by
$$ \varphi_{1} = \mathop \sum \limits_{i = 0}^{\infty } u_{i} $$
(19)

3 Results and discussions

The fractional modified Burgers’ (FMB) equation has been derived to describe the nonlinear propagation of the low phase speed cylindrical dust ion acoustic (DIA) shock waves. We have constructed the approximate solution of fractional modified Burgers’ (FMB) equation by the hybrid of Laplace transform and homotopy perturbation method (LHPM).

Calculations have been done for studying the effects of the number of equilibrium electrons residing on the dust grain surface (\( Z_{d0} \)), positive ion number density at equilibrium (\( n_{i0} \)), shock wave velocity (\( U_{0} \)), and the time fractional parameter (\( \alpha ) \) on the amplitude of cylindrical DIA shock wave. All of the results carried out through this study are calculated using the six terms of the series solution of the Eq. (19). We have chosen the parameters corresponding to the laboratory electronegative plasma presented in Ref. [24]. The variation of the DIA shock amplitude with \( n_{i0} , Z_{d0} , U_{0} \) and \( \alpha \) is shown in Figs. 1, 2, 3, 4, 5 and 6. Figure 1 shows the relation between the shock amplitude \( (\varphi_{1} ) \) and time (\( \tau \)) for different values of the fractional parameter \( \alpha = 0.8, \alpha = 0.6 \) and \( \alpha = 0.4 \) with \( \xi = - 2, n_{e0} = 10^{6} \;{\text{cm}}^{ - 3} , n_{n0} = 0.5 \times 10^{6} \;{\text{cm}}^{ - 3} , r_{d} = 50\;\upmu{\text{m}},U_{0} = 0.1, Z_{d0} = 0.5 \times 10^{4} \)). It shows that the shock amplitude decreases with the increase of the time. The amplitude of the shock \( (\varphi_{1} ) \) versus fractional parameter \( \left( \alpha \right) \) for different values of time (\( \tau \)) is displayed in Fig. 2. It can be seen that as fractional parameter \( \left( \alpha \right) \) increases, the shock amplitude decreases. By comparing results obtained in Figs. 1 and 2, the time and the fractional parameter have had the same effect on DIA shock amplitude. This means that the fractional parameter may be used to decrease shock amplitude instead of increasing the time. In other words, this may physically be more advantageous than increasing the time.
Fig. 1

The amplitude \( \varphi_{1} \) versus \( \tau \) with \( \xi = - 2, n_{i0} = 7 \times 10^{8} \;{\text{cm}}^{ - 3} ,n_{e0} = 10^{6} \;{\text{cm}}^{ - 3} , n_{n0} = 0.5 \times 10^{6} \;{\text{cm}}^{ - 3} , r_{d} = 50\;\upmu{\text{m}} \) for different values of \( \alpha = 0.8 \) (dashdot line), \( \alpha = 0.6 \) (solid line), \( \alpha = 0.4 \) (dot line)

Fig. 2

The amplitude \( \varphi_{1} \) versus \( \alpha \) with \( \xi = - 2, n_{i0} = 7 \times 10^{8} \;{\text{cm}}^{ - 3} ,n_{e0} = 10^{6} \;{\text{cm}}^{ - 3} , n_{n0} = 0.5 \times 10^{6} \;{\text{cm}}^{ - 3} , r_{d} = 50\;\upmu{\text{m}} \) for different values of \( \tau = 10 \) (dashdot line), \( \tau = 20 \) (solid line), \( \tau = 30 \) (dot line)

Fig. 3

The amplitude \( \varphi_{1} \) versus \( \tau \) with \( \xi = - 2, n_{i0} = 7 \times 10^{8} \;{\text{cm}}^{ - 3} ,n_{e0} = 10^{6} \;{\text{cm}}^{ - 3} , n_{n0} = 0.5 \times 10^{6} \;{\text{cm}}^{ - 3} , r_{d} = 50\;\upmu{\text{m}}, \alpha = 1 \) for different values of \( Z_{d0} = 0.5 \times 10^{4} \) (dashdot line), \( Z_{d0} = 0.506 \times 10^{4} \) (solid line), \( Z_{d0} = 0.512 \times 10^{4} \) (dot line)

Fig. 4

The amplitude \( \varphi_{1} \) versus \( U_{0} \) with \( \xi = - 2, \tau = 10,n_{i0} = 7 \times 10^{8} \;{\text{cm}}^{ - 3} ,n_{e0} = 10^{6} \;{\text{cm}}^{ - 3} , n_{n0} = 0.5 \times 10^{6} \;{\text{cm}}^{ - 3} , r_{d} = 50\;\upmu{\text{m}}, \alpha = 1 \)

Fig. 5

The amplitude \( \varphi_{1} \) versus \( n_{i0} \) with \( \xi = - 2, \tau = 10,n_{e0} = 10^{6} \;{\text{cm}}^{ - 3} , n_{n0} = 0.5 \times 10^{6} \;{\text{cm}}^{ - 3} , r_{d} = 50\;\upmu{\text{m}}, \) for different values of \( \alpha = 0.4 \) (dashdot line), \( \alpha = 0.6 \) (solid line), \( \alpha = 0.8 \) (dot line)

Fig. 6

The amplitude \( \varphi_{1} \) versus \( \alpha \) with \( \xi = - 2, \tau = 10,n_{e0} = 10^{6} \;{\text{cm}}^{ - 3} , n_{n0} = 0.5 \times 10^{6} \;{\text{cm}}^{ - 3} , r_{d} = 50\;\upmu{\text{m}}, \) for different values of \( n_{i0} = 5 \times 10^{8} \;{\text{cm}}^{ - 3} \) (dashdot line), \( n_{i0} = 7 \times 10^{8} \;{\text{cm}}^{ - 3} \) (solid line), \( n_{i0} = 9 \times 10^{8} \;{\text{cm}}^{ - 3} \) (dot line)

Figures 3 and 4 concern the effects of the number of equilibrium electrons residing on the dust grain surface \( \left( { Z_{d0} } \right) \) and shock velocity \( \left( {U_{0} } \right) \) on the shock amplitude. It is shown from Fig. 3 that the shock amplitude \( (\varphi_{1} ) \) decreases with the increase in the number of equilibrium electrons residing on the dust grain surface \( \left( { Z_{d0} } \right) \) and the time (\( \tau \)). Figure 4 shows that the shock amplitude increases with the increase in the shock velocity (\( U_{0} \)).

Plot of the shock amplitude \( (\varphi_{1} ) \) versus positive ion number density at equilibrium (\( n_{i0} \)) for different values of fractional parameter \( \alpha = 0.4, \alpha = 0.6 \) and \( \alpha = 0.8 \) is displayed in Fig. 5. It is shown that the shock amplitude decreases as the positive ion number density at equilibrium (\( n_{i0} \)) and the fractional parameter increase (\( \alpha \)). Figure 6 shows the variation of shock amplitude with fractional parameter for different values of positive ion number density at equilibrium \( n_{i0} = 5 \times 10^{8} \;{\text{cm}}^{ - 3} ,n_{i0} = 7 \times 10^{8} \;{\text{cm}}^{ - 3} \) and \( n_{i0} = 9 \times 10^{8} \;{\text{cm}}^{ - 3} \). It is obvious from Fig. 6 that the shock amplitude decreases as the positive ion number density at equilibrium increases. These two Figs. 5 and 6 indicate that the variation of the shock amplitude with \( n_{i0} \) is slight. Therefore, the effect of the positive ion number density at equilibrium on the shock amplitude is slightly negligible. In our obtained results we have applied the effect of negative ions parameter, which is relevant for interstellar space, planetary atmospheres and ionosphere of Saturn’s moon Titan.

The existence of negative ions is revealed in Titan’s ionosphere, the ionosphere of Earth and the D-region altitudes. Furthermore, Negative ions were measured in the inner coma of comet Halley, again coexisting with electrons and the coronal plasma discharge of a nitrogen–methane–argon mixture [18, 60]. These negative ions must play a key role in the formation of organic-rich aerosols (tholins) eventually falling to the surface. Recently however, negative ions may be present at low altitudes at night and they may have a role in poly aromatic hydrocarbon (PAH) chemistry and aerosol production [60]. Therefore, our results can be expected to be useful in the cosmic environments and astrophysical objects [18].

4 Conclusions

The nonlinear propagation of the small amplitude dust ion acoustic (DIA) shock waves is investigated by applying the fractional parameter and cylindrical geometry in the uniform and un-magnetized dusty electronegative plasma composed of Boltzmann electrons, Boltzmann negative ions, inertial positive ions and charge fluctuating dusts. The Euler–Lagrange variational technique is applied to derive the fractional modified Burgers’ equation. Hybrid Laplace transforms and homotopy perturbation method (LHPM) results in a simple and compact form of LHPM equation, which indeed reduces the amount of calculations considerably. The effects of the fractional parameter, positive ion number density at equilibrium, the number of equilibrium electrons residing on the dust grain surface and shock velocity on the behavior of the shock waves in the dusty plasma have been investigated. Our present study is valid for small amplitude DIA shock wave in an unmagnetized and uniform dusty electronegative plasma system. Our results show that the effect of the positive ion number density at equilibrium on the shock amplitude is slightly negligible.

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Copyright information

© Indian Association for the Cultivation of Science 2018

Authors and Affiliations

  1. 1.Physics DepartmentShahed UniversityTehranIran

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