1 23
Astrophysics and Space Science An International Journal of Astronomy, Astrophysics and Space Science ISSN 0004-640X
Volume 349 Number 1
Astrophys Space Sci (2014) 349:205-214 DOI 10.1007/s10509-013-1610-3
Investigation of non-isothermal electron effects on the dust acoustic waves in four components dusty plasma
A. Nazari-Golshan, S. S. Nourazar,
P. Parvin & H. Ghafoori-Fard
1 23
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Astrophys Space Sci (2014) 349:205–214 DOI 10.1007/s10509-013-1610-3
O R I G I N A L A RT I C L E
Investigation of non-isothermal electron effects
on the dust acoustic waves in four components dusty plasma
A. Nazari-Golshan·S.S. Nourazar·P. Parvin· H. Ghafoori-Fard
Received: 4 July 2013 / Accepted: 19 August 2013 / Published online: 17 September 2013
© Springer Science+Business Media Dordrecht 2013
Abstract The time fractional modified KdV, the so-called TFMKdV equation is solved to study the nonlinear prop- agation of the dust acoustic (DA) solitary waves in un- magnetized four components dusty plasma. This plasma consists of positively charged warm adiabatic dust, neg- atively charged cold dust, non-isothermal electrons and Maxwellian ions. The TFMKdV equation is derived by us- ing semi-inverse and Agrawal’s method and solved by the Laplace Adomian decomposition method (LADM). The ef- fects of the time fractional order (β), the ratio of dust to ion temperature (δd), the time (τ), the mass and charge ra- tio (α), the non-isothermal parameter (γ) and wave velocity (v) on the DA solitary wave are studied. Our results show that the variations of the amplitude of DA solitary wave versus (γ) are in agreement with the results obtained pre- viously. Moreover, the time fractional order plays a role of higher order perturbation in modulating the soliton shape.
The achievements of this research for the DA solitary waves may be applicable in space plasma environments and labo- ratory plasmas.
A. Nazari-Golshan·S.S. Nourazar (
B
)·P. ParvinDepartment of Physics, Amirkabir University of Technology, Tehran, Iran
e-mail:[email protected] S.S. Nourazar
Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran
H. Ghafoori-Fard
Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
Keywords Dust acoustic waves·Riemann-Liouville fractional derivative·Laplace transform·Adomian decomposition method·Time fractional modified KdV equation
1 Introduction
A dusty plasma is a multi-component system consisting of electrons, ions, charged microscopic particles (dust grains) and neutral atoms or molecules (Shukla and Eliasson2009;
Morfill and Ivlev2009). The collective behavior of dust par- ticles may lead to propagation of either new or modified waves in the dusty plasmas. The dust acoustic (DA) wave is normally categorized amongst low-frequency and longi- tudinal waves. For the excitation of the DA solitary wave, the dynamics of the dust grains has to be considered be- cause inertia is provided by the mass of the dust particles.
Rao et al. (1990) were the first to report theoretically the existence of the low phase velocity DA solitary waves in multi-component dusty plasma. Subsequently, Barkan et al.
(1995) have experimentally observed DA solitary waves in laboratory experiments.
Some attentions have been paid to investigate the elec- trostatic structures in four-component dusty plasmas (Sayed and Mamun2007). The positively and negatively dust par- ticle distribution functions in such plasmas are considered to be Non-Maxwellian by authors (Sayed and Mamun2007;
Shahmansouri and Tribeche 2013). The presence of posi- tively charged dust particles has also been observed in dif- ferent regions of space; viz., cometary tails (Horanyi1996;
Mendis and Horanyi1991), upper mesosphere (Havnes et al.
1996), Jupiter’s magnetosphere (Horanyi et al.1993). More- over, the experimental observations indicate that the astro- physical and space plasmas have non-Maxwellian particle distribution functions (Vasyliunas1968; Leubner1982).
206 Astrophys Space Sci (2014) 349:205–214 The nonlinear features of the DA solitary wave are stud-
ied in space and laboratory dusty plasmas previously (Ma- mun 1999a, 1999b; Sahu 2011; Sahu and Tribeche2012;
Asgari et al.2013; Merlino et al.2012). The higher order solution of dust acoustic (DA) solitary wave in dusty plasma consisting of positively, negatively charged dust and non- isothermally distributed electrons is studied by Gill et al.
(2011). They showed that the amplitude of the fast and slow mode of DA solitary wave increase with the increase of non-isothermal parameter, γ. Asgari et al. (2013) investi- gated the dust acoustic (DA) solitary waves in dusty plas- mas with non-thermal ions and showed that the existence of non-thermal ions would increase the phase velocity of a dust acoustic wave. El-Wakil et al. (2011a) applied the time fractional derivatives to KdV equation for plasma of two electrons with different temperatures and stationary ions.
They found that the time fractional parameters significantly change the soliton amplitude of the electron acoustic solitary waves. El-Wakil et al. (2011b) also applied the time frac- tional derivatives to KdV equation for plasma of warm ions and isothermal electrons. They showed that the amplitude of the ion-acoustic wave, where the solitons may exist is sensi- tive to the time fractional order.
The basic motivation in the present work is to convert the modified KdV equation into the time fractional modified KdV equation, describing four components dusty plasma with non-isothermal electrons. First, by implementing the reductive perturbation method to the basic equations de- scribing the propagation of DA solitary waves we obtain the modified KdV (MKdV) equations. Second, the semi-inverse and Agrawal’s method (Agrawal 2002,2004; Muslih and Agrawal 2010) is applied to derive the time fractional of the MKdV, the so-called TFMKdV equation. Ultimately, a hybrid of Laplace transform and Adomian decomposition method (LADM) is used to solve the TFMKdV equation to investigate the effects of time fractional parameter on the propagation of DA solitary waves. Moreover, the application of hybrid Laplace transform and ADM result in a simple and compact form of LADM equation, which indeed reduces the amount of calculations considerably. The rest of the paper is organized as follows: In Sect.2we describe the formulation of the MKdV equation. The solution of TFMKdV equation is presented in Sect. 3. Sections 4 and 5 are dedicated to results, discussions and conclusions.
2 Governing equations
2.1 The MKdV equation
We consider the un-magnetized four components dusty plasma consisting of positively charged warm adiabatic dust, negatively charged cold dust, non-isothermal electrons and
Maxwellian ions. The positively charged dust particles are of smaller size as compared to that of large sized negatively charged dust particles. So the motion of negatively charged dust particles is very slow and hence their temperature may be neglected. Thus, the positively charged dust particles can be considered as adiabatically charged dust particles with finite dust temperature. The one dimensional equations de- scribing the propagation of the DA wave in such plasma is given in dimensionless variables as follows (Gill et al.
2011):
For negative dust,
∂n−(x, t )
∂t + ∂
∂x
n−(x, t )u−(x, t )
=0,
∂u−(x, t )
∂t +u−(x, t )∂u−(x, t )
∂x =∂ϕ(x, t )
∂x .
(1)
For positive dust,
∂n+(x, t )
∂t + ∂
∂x
n+(x, t )u+(x, t )
=0,
∂u+(x, t )
∂t +u+(x, t )∂u+(x, t )
∂x
= −α
∂ϕ(x, t )
∂x + δd
n+
∂p+(x, t )
∂x
,
∂p+(x, t )
∂t +u+(x, t )∂p+(x, t )
∂x +3p+(x, t )∂u+(x, t )
∂x =0.
(2)
The Poisson equation is written as:
∂2ϕ(x, t )
∂x2 =n−(x, t )−µ+n+(x, t )+µe exp
δ1ϕ(x, t )
−D
δ1ϕ(x, t )
−µiexp
−ϕ(x, t ) , D(δ1ϕ)=
n
k=1
2(k+1)bk(δ1ϕ)(2k+1)/2/Π (2k+1) ,
(3)
where, bk = √1π(1 − γk), for k = 1 we have b1 =
√1
π(1−γ ),γ=TTefet. Where,Tet(Tef)is trapped (free) elec- tron temperature andγ is the trapping parameter (the ratio of the free electron temperature to trapped electron tem- perature). Here n− (n+) is negative (positive) dust num- ber density normalized by its equilibrium valuen−0(n+0).
u− (u+)is negative (positive) dust fluid speed normalized byC=(Z−kBTi/m−)1/2,α=(m−Z+/m+Z−)is the mass and charge ratio, ϕ is the wave potential normalized by (kBTi/e), δ1=(Ti/Te)is the ratio of the ion temperature to electron temperature.Z−(Z+)is the number of electrons (protons) residing on a negative (positive) dust.m−(m+)is the mass of the negative (positive) dust particle.Ti(Te)is the temperature of ion (electron).kB andeare the Boltzmann
Astrophys Space Sci (2014) 349:205–214 207 constant and electron charge respectively. The p+ is ther-
mal pressure of positive dust fluid normalized byn+0kBTd, δd=(Td/Z+Ti),µe=(ne0/Z−n−0), µi =(ni0/Z−n−0), µ+=1+µe−µi. The x and t are normalized by De- bye length λD =(Z−kBTi/4π Z−2e2n−0)1/2 and inverse of plasma frequency ωp−1=(m−/4π Z−2e2n−0)1/2 respec- tively. The dust charging time is much less than the time period of the low-frequency DA waves under consideration and then the effects of the dust charge fluctuations are ne- glected (Shukla and Mamun2002; Varma et al.1993). The four components dusty plasma with non-isothermal elec- trons is introduced through electron density in the following manner (Schamel1971,2000):
ne(x, t )=1+ϕ(x, t )−4
3b1ϕ3/2(x, t )+1
2ϕ2(x, t ). (4) According to reductive perturbation method, we introduce the stretched coordinates (Schamel1973) asχ=ε1/4(x− v0t ),τ =ε3/4t to study the DA solitary wave. Where,εis a small dimensionless parameter and v0 is the phase ve- locity of the DA solitary waves. The dimensionless vari- ablesn+,−(χ , τ ),u+,−(χ , τ ),p+(χ , τ )andϕ(χ , τ )are ex- panded as follows:
n+,−(χ , τ )=1+εn(1)+,−+ε(3/2)n(2)+,−+ · · ·, u+,−(χ , τ )=εu(1)+,−+ε(3/2)u(2)+,−+ · · ·, p+(χ , τ )=1+εp(1)+ +ε(3/2)p(2)+ + · · ·, ϕ(χ , τ )=εϕ(1)+ε(3/2)ϕ(2)+ · · ·.
(5)
Substitute Eq. (5) into Eqs. (1)–(3), and neglecting terms of higher order thanε,
n(1)− =u(1)−
v0 = −ϕ(1)
v02 , n(1)+ =u(1)+
v0 =αϕ(1) δαv02, p+(1)=3u(1)+
v0 =3αϕ(1)
δαv02 , δα=1−3αδd
v02 .
(6)
Using the quantities introduced in Eq. (6) into Eq. (3), the Poisson’s equation, Eq. (3) is expressed as the following dis- persion relation:
v20=1+µ+α+3αδd(µeδ1+µi) 2(µeδ1+µi)
± 1+µ+α+3αδd(µeδ1+µi) 2(µeδ1+µi)
2
− 3αδd
(µeδ1+µi) 12
. (7) Equation (7) is the linear dispersion relation for the DA soli- tary wave, propagating in four components dusty plasma.
The positive and negative signs correspond to the fast and slow DA-modes respectively. Thus, the system supports two
types of DA-modes which propagate with different phase velocities given by Eq. (7). The fast and slow DA-modes propagate with larger and smaller phase velocity respec- tively.
Preserve terms of up to the second order in ε in Eqs. (1)–(3), and using Eq. (6), the MKdV equation is ob- tained as:
∂ϕ(χ , τ )
∂τ +Aϕ12(χ , τ )∂ϕ(χ , τ )
∂χ +B∂3ϕ(χ , τ )
∂χ3 =0. (8) Where,χ andτ are the space and time variables respec- tively. The coefficientsA(nonlinear term) andB(dispersion term) are given by:
A=2b1µeδ
3 2
1B, B= δ2αv30
2(δα2+µ+α). (9)
3 Solution of the TFMKdV equation using the LADM Several semi-analytical methods can be used to solve the time fractional equations such as: the operational method (Podlubny1999; Samko et al.1998), Homotopy perturba- tion method (Nourazar et al.2011a,2011b; Nazari-Golshan et al. 2013), Adomian decomposition method (Adomian 1986, 1994; Nourazar et al. 2012), variational iteration method (He1997a,1997b; Nourazar et al. 2011a,2011b).
In this paper, the resultant TFMKdV equation will be solved using LADM as follows:
The Laplace transform is applied to both sides of the TFMKdV equation and the solution using the LADM is pre- sented. By using the LADM, all conditions are satisfied over the entire range of time domain. Applying the Laplace trans- form to Eq. (33) and implementing the relation given by Eq. (21):
sβL
ϕ(χ , τ )
−R
0Dτβ−1ϕ(χ , τ )
τ=0
+AL
ϕ12(χ , τ )ϕχ(χ , τ ) +BL
ϕχ χ χ(χ , τ )
=0,
0< β≤1, τ∈ [0, T0]. (10)
The compact form of the TFMKdV equation, Eq. (10), is written as:
sβ
∞
i=1
L
ϕi(χ , τ )
−
R 0Dτβ−1
∞
i=0
ϕi(χ , τ )
τ=0
+A
∞
i=0
L(Ai)+B ∂3
∂χ3
∞
i=0
L
ϕi(χ , τ )
=0,
0< β≤1, τ∈ [0, T0]. (11)
Where,ϕ12(χ , τ )ϕχ(χ , τ )=∞
i=0Ai.
208 Astrophys Space Sci (2014) 349:205–214 Alternatively the sequence of recursive equations de-
duced from Eq. (11), can be written in the following manner:
sβL
ϕ1(χ , τ )
−R
0Dτβ−1
ϕ0(χ , τ )
τ=0
+AL(A0) +B ∂3
∂χ3 L
ϕ0(χ , τ )
=0, sβL
ϕ2(χ , τ )
−R
0Dτβ−1
ϕ1(χ , τ )
τ=0
+AL(A1) +B ∂3
∂χ3 L
ϕ1(χ , τ )
=0, sβL
ϕ3(χ , τ )
−R
0Dτβ−1
ϕ2(χ , τ )
τ=0
+AL(A2) +B ∂3
∂χ3 L
ϕ2(χ , τ )
=0, ...
sβL
ϕk(χ , τ )
−R
0Dτβ−1
ϕk−1(χ , τ )
τ=0
+AL(Ak−1) +B ∂3
∂χ3 L
ϕk−1(χ , τ )
=0, 0< β≤1, τ∈ [0, T0].
(12)
The stationary solution of the MKdV equation, Eq. (8), is selected as initial conditions of Eq. (33),
ϕ0(χ , τ )=ϕ(χ , τ )|τ=0=φmsech4(dχ ), (13) where,φmanddare constants that can be expressed as:
φm=
15v/16b1µeδ
3 2
1B2
, d=
v/16B, (14)
wherevis a traveling wave velocity normalized byC.
Substituting the zero order approximation,ϕ0(χ , τ ), and A0into the recursive equations, Eq. (12), gives the first order approximation as:
ϕ1(χ , τ )= 4dτβ Γ (1+β)
tanh(dχ ) A
φmsech4(dχ )3/2
+2Bφmsech4(dχ )d2
15 tanh2(dχ )−7 .
(15) Substituting the first order approximation, ϕ1(χ , τ ), Eq. (15), andA1into the recursive equations, Eq. (12), (ap- plying the Maple symbolic code) gives the second order approximation as:
ϕ2(χ , τ )
= − 1
Γ (1+2β)φmcosh10(dχ )
× 4τ2βd2 376B2d4φm2sech4(dχ )cosh10(dχ )
+308
φmsech4(dχ )3/2
×tanh2(dχ )d2BAφmcosh10(dχ )
−504
φmsech4(dχ )3/2
×tanh4(dχ )d2BAφmcosh10(dχ )
−7280B2d4φm2sech4(dχ )tanh2(dχ )cosh10(dχ ) +21000B2d4φm2sech4(dχ )tanh4(dχ )cosh10(dχ )
−15120B2d4φm2sech4(dχ )tanh6(dχ )cosh10(dχ )
−20
φmsech4(dχ )3/2
d2BAφmcosh10(dχ ) +A2φm3cosh2(dχ )−9A2φ3msinh2(dχ )
−96A
φm
cosh4(dχ ) 3/2
d2Bφmcosh10(dχ )
+352A
φm cosh4(dχ )
3/2
d2Bφmcosh8(dχ )
−270
φm
cosh4(dχ ) 5/2
d2BAcosh10(dχ )
. (16) Ultimately, the solution of the TFMKdV equation, Eq. (33), is elaborated in series expansion as:
ϕ(χ , τ )=
∞
i=0
ϕi(χ , τ ), (17)
4 Results and discussions
We consider a two fluid plasma model consisting of warm adiabatic positively charged dust and negatively charged cold dust grains with non-isothermal electrons. The basic set of fluid equations governing the dynamics of the DA wave propagation which leads to the derivation of modified KdV (MKdV) equation.
To investigate the effect of plasma and time fractional parameters on the nature of DA solitary waves, the equa- tions governing the amplitude and the width of DA solitary waves are solved using the LADM. In our solutions only the first five terms of series solutions are considered in Eq. (17).
Gill et al. (2011) studied the solutions of dust acoustic (DA) waves in dusty plasma consisting of positively, neg- atively charged dust and non-isothermally distributed elec- trons. They showed that the amplitude of the fast and slow mode of DA solitary waves increases with the increase of non-isothermal parameter (γ) and decreases with the in- crease of mass and charge of ratio (α). To compare our re- sults, we take the corresponding parameters similar to those of dusty plasma presented in reference (Gill et al.2011), v=0.025,γ =0.8,δ1=0.5,δd=0.001,α=2,µi=0.8,
Astrophys Space Sci (2014) 349:205–214 209 µe=0.2. The results are demonstrated in Figs.1–5. Fig-
ure1shows that the amplitude of the fast and slow mode of DA solitary wave increases with non-isothermal parameter (γ) and decreases with the increase of mass and charge ratio (α) at time fractional orderβ=1. The variation of the am- plitude of the fast and slow mode of DA solitary wave versus (γ) is in agreement with corresponding results obtained by Gill et al. (2011).
In Fig. 2, the profiles ofϕ(0, τ )against time fractional order (β) are presented for different values of time τ = 2,3,3.5,4. It shows that the amplitude of the fast and slow DA solitary wave mode decreases with the increase of time fractional order (β) and time (τ). This implies that the time fractional order (β) may be used to decrease the ampli- tude of the solitary waves instead of increasing the value of the ion to electron temperature ratio (δ1) (Gill et al.
2011). This may physically be more advantageous than in- creasing the value of the ion to electron temperature ra- tio (δ1).
The effects of the time fractional order (β) on the am- plitude and width of the fast and slow DA solitary wave mode are presented in Figs. 3 and 4. For the fast mode, these figures show that increasing the time fractional or- der (β) decreases the amplitude and increases the width of the DA solitary wave. This feature leads to the fact that the fractional parameter may be used to modify the shape of the solitary wave instead of adding higher order nonlin- earity or dispersion terms to the equations governing the plasma medium. On the other hand, this implies that the time fractional order (β) may be used to modulate the elec- trostatic potential wave instead of adding a higher order dissipation term to the MKdV equation (El-Wakil et al.
2006).
Figure 5describes the effect of the ratio of dust to ion temperature (δd) on the propagation of DA solitary wave for both the fast and slow modes at time fractional orderβ=1.
For the slow mode, the amplitude of DA solitary wave in- creases when the ratio of dust to ion temperature (δd) de- creases. Moreover, it is to be noted that the variations of the ratio of dust to ion temperature (δd) has no significant effect on the amplitude of the DA solitary wave for the fast mode.
5 Conclusions
Time fractional modified KdV (TFMKdV) equation is de- rived by using the semi-inverse and Agrawal’s methods.
A hybrid of Laplace transform and Adomian decomposition method (LADM) is incorporated into the TFMKdV equation which results in a simple and compact form equation. In our research, the fast and slow modes of the DA solitary wave are observed and investigated. It is found that increasing the
time fractional order (β), the ratio of dust to ion tempera- ture (δd), time (τ) and mass and charge ratio (α) results in decreasing the amplitude of the DA solitary wave. While, in- creasing the non-isothermal parameter (γ) and wave veloc- ity (v) results in increasing the amplitude of the DA solitary wave. The applicability of the results obtained in the present research may be found in cometary tails (Horanyi 1996;
Mendis and Horanyi 1991), upper mesosphere (Havnes et al.1996), Jupiter’s magnetosphere, where the positively charged dust particles are present in different regions of space.
Appendix A: The derivation of time fractional equation
A.1 The fractional derivatives and variations
Several definitions of fractional derivatives can be found in literature (Podlubny1999; Samko et al.1998; Agrawal 2007). The most common used fractional derivatives are the Riemann-Liouville, Caputo and Riesz derivatives. The left and right fractional derivatives,aDtβf (t )andtDbβf (t ), in the Riemann-Liouville sense are defined as (Podlubny1999;
Samko et al.1998):
aDβtf (t )= 1 Γ (k−β)
dk dtk
t a
dτ (t−τ )k−β−1f (τ )
, k−1< β≤k, t∈ [a, b],
tDβbf (t )= (−1)k Γ (k−β)
dk dtk
b t
dτ (τ−t )k−β−1f (τ )
, k−1< β≤k, t∈ [a, b].
(18)
The fractional derivative in the Riesz sense,R0Dβτf (t ), is de- fined as (Podlubny1999; Samko et al.1998; Agrawal2007):
R
0Dτβf (t )=1 2
aDtβf (t )+(−1)ktDbβf (t )
=1 2
1 Γ (k−β)
dk dtk
b a
dτ|t−τ|k−β−1f (τ )
, k−1< β≤k, t∈ [a, b]. (19)
Using the integration by parts, the fractional derivative in the Riemann-Liouville sense is given by (Podlubny1999;
Samko et al.1998):
b a
dtf1(t )aDβtf2(t )
= b
a
dtf2(t )tDbβf1(t ), f1(t ), f2(t )∈ [a, b]. (20) The Laplace transform of the fractional derivative,aDtβf (t ), is given by:
210 Astrophys Space Sci (2014) 349:205–214
Fig. 1 The amplitude of the electrostatic potentialϕ(0,2)versusγ forβ=1,v=0.025,γ =0.8,δ1=0.5,δd=0.001,µi=0.8,µe=0.2 at different values of the mass and charge ratio (α). (a) DA fast mode and (b) DA slow mode
Fig. 2 The amplitude of the electrostatic potentialϕ(0, τ )versus time fractional order (β) forv=0.025,γ=0.8,α=2,δ1=0.5,δd=0.001, µi=0.8,µe=0.2 at different values of the time (τ). (a) DA fast mode and (b) DA slow mode
L
aDtβf (t );s
=sβL f (t )
−
n−1
k=0
sk
aDtβ−k−1f (t )
t=a,
k−1< β≤k. (21)
Where the operatorLdenotes the Laplace transform.
The functional based on the definition of fractional derivative, Eqs. (18) and (20), is defined as,
J (u)=
R
dx
T
dt H
0Dβtu, ux, uxx
, (22)
whereH (0Dβtu, ux, uxx)is a function with continuous first and second (partial) derivatives with respect to all its argu- ments.
Taking the first variations of Eq. (22) with respect to the dependent variable, u(x, t ), the following equation is de- duced as:
δJ (u)=
R
dx
T
dt ∂H
∂0Dβt
δ0Dβtu+ ∂H
∂ux
δux
+ ∂H
∂uxx
δuxx
. (23)
Astrophys Space Sci (2014) 349:205–214 211
Fig. 3 The amplitude of the electrostatic potentialϕ(χ ,2)versusχ forv=0.025,γ=0.8,α=2,δ1=0.5,δd=0.001,µi=0.8,µe=0.2 at different values of the time fractional order (β). (a) DA fast mode and (b) DA slow mode
Fig. 4 The amplitude of the electrostatic potentialϕ(0,2)versusvforγ=0.8,α=2,δ1=0.5,δd=0.001,µi=0.8,µe=0.2 at different values of the time fractional order (β). (a) DA fast mode and (b) DA slow mode
Integrating by parts and using Eq. (20) lead to:
δJ (u)=
R
dx
T
dt tDβT
0
∂H
∂0Dtβu
− ∂
∂x ∂H
∂ux
+ ∂2
∂x2 ∂H
∂uxx
δu. (24)
Here we assume that|δu|T = |δu|R= |δux|R=0.
Optimizing Eq. (24), δJ (u) =0, gives the following Euler-Lagrange equation:
tDβT
0
∂H
∂0Dtβu
− ∂
∂x ∂H
∂ux
+ ∂2
∂x2 ∂H
∂uxx
=0. (25)
A.2 The derivation of TFMKdV equation
Introducing a function Ψ (χ , τ ), where, as ϕ(χ , τ ) = Ψχ(χ , τ ), into Eq. (8) gives the following:
Ψχ τ(χ , τ )+AΨ
1
χ2(χ , τ )Ψχ χ(χ , τ )+BΨχ χ χ χ(χ , τ )=0, (26)
212 Astrophys Space Sci (2014) 349:205–214
Fig. 5 The amplitude of the electrostatic potentialϕ(χ ,2)versusχforv=0.025,γ=0.8,α=2,δ1=0.5,β=1,µi=0.8,µe=0.2 at different values of the ratio of dust to ion temperature (δd). (a) DA fast mode and (b) DA slow mode
whereΨχ andΨτ denote the partial derivatives ofΨ with respect toχandτ respectively.
Using the semi-inverse method (He1997a,1997b,2004), the functional relating to Eq. (26) is given by:
J (Ψ )=
dχ
dτ Ψ (χ , τ )
f1Ψχ τ(χ , τ ) +f2AΨ
1
χ2(χ , τ )Ψχ χ(χ , τ ) +f3BΨχ χ χ χ(χ , τ )
. (27)
Wheref1, f2andf3are constants to be determined. By as- sumingΨχ|R=Ψτ|R=Ψχ|T =0. Integrating Eq. (27) by parts:
J (Ψ )=
dχ
dτ −f1Ψτ(χ , τ )Ψχ(χ , τ )
−2 3f2AΨ
5
χ2(χ , τ )+f3BΨχ χ2 (χ , τ )
. (28)
By taking the first variations of Eq. (28) with respect to the dependent variable, Ψ (χ , τ ), and equating to zero,
2f1Ψχ τ(χ , τ )+5 2f2AΨ
1
χ2(χ , τ )Ψχ χ(χ , τ )
+2f3BΨχ χ χ χ(χ , τ )=0. (29)
The constants f1, f2 and f3are determined by comparing Eq. (29) with Eq. (26) as:
f1=1/2, f2=2/5, f3=1/2. (30)
Therefore, the Lagrangian of MKdV equation, Eq. (28), is given by
F (Ψτ, Ψχ, Ψχ χ)= −1
2Ψτ(χ , τ )Ψχ(χ , τ )
− 4 15AΨ
5
χ2(χ , τ )+1
2BΨχ χ2 (χ , τ ).
(31) According to Eq. (31), the Lagrangian of the time fractional derivative of MKdV equation can be written as:
G
0DτβΨ, Ψχ, Ψχ χ
= −1 2
0DβτΨ (χ , τ )
Ψχ(χ , τ )− 4 15AΨ
5
χ2(χ , τ ) +1
2BΨχ χ2 (χ , τ ), 0< β≤1 (32) where the operator0Dβτ denotes the left Riemann-Liouville fractional derivative given by Eq. (18).
In view of the foregoing, Eq. (19), and owing that ϕ(χ , τ )=Ψχ(χ , τ ), by substituting the Lagrangian of the time fractional derivative of MKdV equation, Eq. (32), into Euler-Lagrange formula, Eqs. (25), (32) can be rewritten as:
R
0Dτβϕ(χ , τ )+Aϕ12(χ , τ )ϕχ(χ , τ )+Bϕχ χ χ(χ , τ )=0,
0< β≤1, τ∈ [0, T0]. (33)
Equation (33) is called the TFMKdV equation that describes the nonlinear propagation of DA solitary wave.
Astrophys Space Sci (2014) 349:205–214 213 Appendix B: The basic idea under laying the Laplace
Adomian decomposition method (LADM) Consider the general form of one-dimensional nonlinear partial differential equation as:
L u(x, t )
+N u(x, t )
=g(x). (34)
WhereL andN denote a linear and a nonlinear operators respectively. Taking the Laplace transform from both sides of Eq. (34):
L L
u(x, t ) +L
N
u(x, t )
=L g(x)
. (35)
Where, symbolLdenotes the Laplace transform. Focusing on the linear operatorLin Eq. (34), the concept of Adomian decomposition method is used to generate a series expan- sion forL(u(x, t ))as follow (Adomian1986,1994; Wazwaz 2009):
u=
∞
i=0
ui, L u(x, t )
=L ∞
i=0
ui
, (36)
where the componentsui, i≥0 are to be determined in a recursive manner.
Switching to the non-linear operator N in Eq. (34), we use the Adomian polynomials,Ai, as follow:
N u(x, t )
=
∞
i=0
Ai. (37)
Where, the Adomian polynomials,Ai, are expressed as:
Ai= 1 i!
di dλi
N
n
j=0
λjuj
λ=0
. (38)
Substituting Eqs. (37) and (36) into Eq. (35):
L
L ∞
i=0
ui
+L ∞
i=0
Ai
=L g(x)
. (39)
Where, the Adomian polynomials,Ai, are expressed by:
A0=N (u0), A1=u1N′(u0), A2=u2N′(u0)+ 1
2!u21N′′(u0), A3=u3N′(u0)+u1u2N′′(u0)+ 1
3!u31N′′′(u0), A4=u4N′(u0)+
1
2!u22+u1u3
N′′(u0) + 1
2!u21u2N′′′(u0)+ 1
4!u41N(iv)(u0).
(40)
On the other hand, Eq. (35) can be rewritten in the following form:
∞
i=0
L L
ui(x, t ) +
∞
i=0
L{Ai} =L{g}. (41) Using Eq. (41) we introduce the recursive relation as:
L L(u0)
=L{g}, ∞
i=1
L L(ui)
+ ∞ i=0
L{Ai} =0. (42) Alternatively the recursive relation, Eq. (42), is expressed as:
L L(u0)
=L{g}, L
L(u1)
+L{A0} =0, L
L(u2)
+L{A1} =0, L
L(u3)
+L{A2} =0, ...
L L(uk)
+L{Ak−1} =0.
(43)
Using the Maple symbolic code, the first part of Eq. (43) gives the value ofL(u0). First, applying the inverse Laplace transform to L(u0)gives the value ofu0, that will define the Adomian polynomial,A0using the first part of Eq. (40).
In the second part of Eq. (43), the Adomian polynomialA0
will enable us to evaluateL(u1). Second, applying the in- verse Laplace transform toL(u1)gives the value ofu1, that will define the Adomian polynomialA1 using the second part of Eq. (40) and so on. This in turn will lead to the com- plete evaluation of the components ofuk,k≥0 upon using different corresponding parts of Eqs. (43) and (40).
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