Effect of trapped electron on the dust ion acoustic waves in dusty plasma using time fractional modified Korteweg-de Vries equation

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Effect of trapped electron on the dust ion acoustic waves in dusty plasma using time fractional modified Korteweg-de Vries equation

A. Nazari-Golshan and S. S. Nourazar

Citation: Phys. Plasmas 20, 103701 (2013); doi: 10.1063/1.4823997 View online: http://dx.doi.org/10.1063/1.4823997

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Effect of trapped electron on the dust ion acoustic waves in dusty plasma using time fractional modified Korteweg-de Vries equation

A. Nazari-Golshan¹and S. S. Nourazar^1,2,a)

1Department of Physics, Amirkabir University of Technology, Tehran, Iran

2Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran (Received 29 May 2013; accepted 11 September 2013; published online 2 October 2013)

The time fractional modified Korteweg-de Vries (TFMKdV) equation is solved to study the nonlinear propagation of small but finite amplitude dust ion-acoustic (DIA) solitary waves in un-magnetized dusty plasma with trapped electrons. The plasma is composed of a cold ion fluid, stationary dust grains, and hot electrons obeying a trapped electron distribution. The TFMKdV equation is derived by using the semi-inverse and Agrawal’s methods and then solved by the Laplace Adomian decomposition method. Our results show that the amplitude of the DIA solitary waves increases with the increase of time fractional orderb, the wave velocityv0, and the population of the background free electrons k. However, it is vice-versa for the deviation from isothermality parameterb, which is in agreement with the result obtained previously.^V^C 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4823997]

I. INTRODUCTION

During the last two decades, dusty plasma has received a great deal of attention due to a variety of new phenomena observed and the novel physical mechanisms involved in it.^1,2 In addition to well-known plasma electrostatic waves,³new oscillatory waves arise in a dusty plasma,^1,2among which the dust ion acoustic (DIA)⁴and dust acoustic (DA) waves^5,6are of significant interest in laboratory dusty plasma discharges.

In the DIA wave, the restoring force comes from the pressure of the inertialess electrons, where the inertia is provided by the ion mass, similar to the standard ion-acoustic waves in electron-ion plasma. The equilibrium charge neutrality condition is maintained by the stationary dust particle in the DIA wave.

Shukla and Silin⁴ have reported theoretically the exis- tence of DIA waves. Later, these waves have been observed experimentally in the laboratory.^7–9

The nonlinear features of the DIA waves have recently been investigated in space and laboratory dusty plasmas.^1,10–17 El-Labanyet al.¹⁶have studied the effects of trapped electrons temperature, dust charge variation, and grain radius on the nonlinear DIA solitons in a dusty plasma with trapped electrons. They found that the soliton amplitude of the electrostatic potential decreased with the trapped electron temperatures.

Alinejad¹³ has investigated the properties of DIA solitary structures by reductive perturbation method for small amplitude limits. He found that the amplitude of DIA solitary waves increased with the increase of population of the background free electrons. The effects of dust charge fluctua- tions on DIA solitary waves in dusty plasma have been also investigated by the same author.¹⁴He showed that the amplitude of DIA solitary waves increased with the increase of population of the background free electrons and decreased with increasing the deviation from isothermality.

By applying fractional derivatives to differential equations, the non-conservative physical systems can be easily studied.^18,19 El-Wakil et al.,²⁰ applied 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 changed the soliton amplitude of the electron acoustic solitary waves. They²¹ also applied the time fractional derivatives to the KdV equation for plasma with warm ions and isothermal electrons and showed that the time fractional derivatives could be used to modulate the electrostatic potential wave.

In this study, we convert the basic coupled equations, describing plasma with trapped electrons, into time fractional modified Korteweg-de Vries (TFMKdV) equation and investigate the effect of time fractional parameter on the propagation of DIA solitary waves using plasma parameters. We use a hybrid of Laplace transform and Adomian decomposition method,^22–24(LADM) to solve the TFMKdV equation. The hybrid LADM results 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 Sec. II, we describe the formulation of the TFMKdV equation using the variational Euler-Lagrange method.^25–27In Sec.III, a hybrid of LADM is discussed.^22–24 The solution of TFMKdV equation is presented in Sec. IV.

SectionsVandVIare dedicated to results, discussions, and conclusions.

II. GOVERNING EQUATIONS

A. The fractional derivatives and variations

Several definitions of fractional derivatives may be found in literatures.^28–30 The most common used fractional derivatives are the Riemann-Liouville, Caputo, and Riesz derivatives. The left and right fractional derivatives,_aD^b_tfðtÞ and _tD^b_bfðtÞ, in the Riemann-Liouville sense are defined as^28,29

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

1070-664X/2013/20(10)/103701/8/$30.00 20, 103701-1 ^V^C2013 AIP Publishing LLC

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aD^b_tfðtÞ ¼ 1 Cðk bÞ

d^k dt^k

ðt a

dsðt sÞ^k ^b ¹fðsÞ

k 1<bk; t2 ½a;b;

tD^b_bfðtÞ ¼ ð 1Þ^k Cðk bÞ

d^k dt^k

ðb t

dsðs tÞ^k ^b ¹fðsÞ

" #

k 1<bk; t2 ½a;b:

(1)

The fractional derivative in the Riesz sense, ^R₀D^b_s, is defined as^28–30

R

0D^b_sfðtÞ ¼ 1

2½aD^b_tfðtÞ þ ð 1Þ^ktD^b_bfðtÞ;

¼1 2

1 Cðk bÞ

d^k dt^k

ðb a

dsjt sj^k ^b ¹fðsÞ

" #

k 1<bk; t2 ½a;b: (2) The fractional derivative in the Riemann-Liouville sense using the integration by parts is given by^28,29

ðb a

dtf1ðtÞaD^b_tf2ðtÞ ¼ ðb

a

dtf2ðtÞtD^b_bf1ðtÞ; f1ðtÞ;f2ðtÞ 2 ½a;b: (3) The Laplace transform of the fractional derivative, _aD^b_tfðtÞ, is given by

LðaD^b_tfðtÞ;sÞ ¼s^bLðfðtÞÞ Xⁿ

1

k¼0

s^k½aD^b_t ^k ¹fðtÞt¼a;

k 1<bk; (4)

where the operatorLdenotes the Laplace transform.

The functional corresponding to the definition of fractional derivative, Eqs.(1)–(3), is defined as

JðuÞ ¼ ð

R

dx ð

T

dtHð0D^b_tu;ux;uxxÞ; (5) where Hð⁰D^b_tu;u_x;u_xxÞ is a function with continuous first and second (partial) derivatives with respect to all its arguments.

Taking the first variations of Eq. (5) with respect to the dependent variable, uðx;tÞ, the following equation is obtained as:

dJðuÞ ¼ ð

R

dx ð

T

dt @H

@0D^b_t

!

d0D^b_tuþ @H

@ux

du_x

"

þ @H

@uxx

du_xx

: (6)

Integrating by parts and using Eq.(3)lead to dJðuÞ ¼

ð

R

dx ð

T

dt _tD^b_T

0

@H

@0D^b_tu

! @

@x

@H

@ux

"

þ@²

@x²

@H

@uxx

du: (7)

Here, we assume thatdujT¼dujR¼du_xjR ¼0:

Optimizing Eq. (7), dJðuÞ ¼0, gives the following Euler-Lagrange equation:

tD^b_T

0

@H

@0D^b_tu

! @

@x

@H

@ux

þ @²

@x²

@H

@uxx

¼0: (8)

B. Derivation of TFMKdV equation

We consider un-magnetized dusty plasma consisting of a cold inertia ions, trapped as well as free electrons and nega- tively charged immobile dust particles.

The one-dimensional equations describing the propagation of the DIA wave in such plasma with dimensionless variables are as follows:³¹

@niðx;tÞ

@t þ @

@xðniðx;tÞuiðx;tÞÞ ¼0; (9)

@uiðx;tÞ

@t þuiðx;tÞ@uiðx;tÞ

@x þ@uðx;tÞ

@x ¼0: (10)

The Poisson equation is

@²uðx;tÞ

@x² ¼kn_eðx;tÞ n_iðx;tÞ þ ð1 kÞ; (11) where n_iðx;tÞ is the ion number density normalized by its equilibrium value n_i0, u_iðx;tÞis the ion-fluid speed normalized by the ion-acoustic speedC_i¼ ðjT_ef=miÞ¹⁼², anduðx;tÞ is the electrostatic wave potential normalized by jTef=e.

Here,jis Boltzmann’s constant,Tef is the constant temperature of the free electrons,m_iis the ion mass, andk¼n_e0=ni0. The time and space variables are given in the units of the ion plasma periodx_pi¹¼ ð0mi=ni0e²Þ¹⁼², and the Debye length k_De¼ ð0jT_ef=ni0e²Þ¹⁼², respectively.

From Eqs.(9)–(11), we can introduce two special func- tions,Wand!defined as

@W

@x ¼n_i; ð12Þ

@W

@t ¼ n_iu_i; ð13Þ

8

>>

><

>>

>:

@!

@x ¼u_i ð14Þ

@!

@t ¼ u_i² 2 þu

: ð15Þ

8

>>

><

>>

>:

In order to construct the principle of variations by semi- inverse method^32–34 for Eqs. (9)–(11), we consider the following functional:

103701-2 A. Nazari-Golshan and S. S. Nourazar Phys. Plasmas20, 103701 (2013)

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J1¼ ðui;u;WÞ ¼ ðt2

t1

dt ðx2

x1

Ldx; (16)

where the trial-Lagrangian,L, can be expressed as L¼u_i@W

@t þ u_i² 2 þu

@W

@x þFðn_i;u_iÞ: (17) Many alternative approaches exist to construct the trial- Lagrangian in various different forms. The illustrative examples may be found in the literature.^34–36Taking the first variations from Eq.(16)with respect tougives the following Euler equation:

@W

@x þdF

du¼0; (18)

where^dF_duis called the variational derivative with respect tou defined as^32–34

dF du¼@F

@u

@

@t

@F

@u_t

@

@x

@F

@u_x

þ@²

@x²

@F

@u_xx

þ (19) We look for function F such that Eq.(18)to be equivalent to one of the field equations, as Eq. (12). In view of Eqs.(11) and(12), we get

dF

du¼ @W

@x ¼ n_i¼@²u

@x² kn_e ð1 kÞ: (20) From which we can identify F as

F¼ 1 2

@u

@x 2

ðkn_eþ ð1 kÞÞuþf1ðu_iÞ; (21) wheref1 is newly introduced unknown function ofui and/or its derivative.

SubstitutingFfrom Eq.(21) into the Lagrangian given by Eq.(17), the new Lagrangian is given as

L¼ui

@W

@t þ u_i² 2 þu

@W

@x 1 2

@u

@x 2

ðkneþ ð1 kÞÞuþf1: (22) Taking the first variations from Eq. (22)with respect to ui

gives the following:

@W

@t þu_i@W

@x þdf1

dui¼0: (23) In view of Eqs.(12)and(13), we have

df1

du_i¼ @W

@t þu_i@W

@x

¼ ð n_iu_iþn_iu_iÞ ¼0; (24) which leads to the result f1¼0. Ultimately, we obtain the first following functional for Eqs.(9)–(11)as

L¼ui

@W

@t þ ui2

2 þu

@W

@x 1 2

@u

@x 2

ðkneþ ð1 kÞÞu: (25)

Similarly, we can establish a generalized variational principle with four independent componentsðn_i;u_i;u;!Þ, which is given as

J2¼ ðni;ui;u;!Þ ¼ ðt2

t1

dt ðx2

x1

L1dx; (26) where the trial-Lagrangian,L1, can be written as follow:

L1¼n_i@!

@t þn_iu_i@!

@xþF1ðn_i;u_i;uÞ: (27) The variations of the functional, Eq.(26), with respect toui

is given as ni

@!

@xþdF1

du_i ¼0 ) dF1

du_i ¼ ni

@!

@x ¼ niui

)F1¼ niui2

2 þf2ðn_i;uÞ: (28) Substituting F1 from Eq.(28) into the Lagrangian given by Eq.(27), the new Lagrangian is given as

L1 ¼n_i@!

@t þn_iu_i@!

@x niui2

2 þf2: (29) Taking the first variations from Eq. (29)with respect to n_i gives the followings:

@!

@t þu_i@!

@x u_i²

2 þdf2

dni¼0 )df2

dn_i ui2

2 þu

þu²_i ui2

2 ¼0;

)f2¼n_iuþf3ðuÞ: (30) Substituting f2 from Eq.(30) into the Lagrangian given by Eq.(29), the new Lagrangian is given as

L1 ¼n_i@!

@t þn_iu_i@!

@x niui2

2 þn_iuþf3ðuÞ: (31) Taking the first variations from Eq. (31) with respect to u gives the followings:

niþdf3

du¼0)df3

du¼ ni¼@²u

@x² ðkneþ ð1 kÞÞ; )f3¼ 1

2

@u

@x 2

ðkneþ ð1 kÞÞu: (32) Ultimately, the second following functional for Eqs.(9)–(11) is given as

L1¼ni

@!

@t þniui

@!

@x niui2

2 1 2

@u

@x 2

þ ðni kne ð1 kÞÞu: (33) Equations(25)and(33)are the two functionals for the basic equations, Eqs.(9)–(11). The corresponding Lagrangians of

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the time fractional derivative of basic equations, Eqs.

(9)–(11)are given as

G¼ui0D^b_tWþ u_i² 2 þu

@W

@x 1 2

@u

@x 2

kneþ ð1 kÞÞu; 0<b1;

ð (34)

and

G1¼n_i0D^b_t!þn_iu_i@!

@x niui2

2 1 2

@u

@x 2

þðni kne ð1 kÞÞu; 0<b1; (35) where the operator 0D^b_t denotes the left Riemann-Liouville fractional derivative given by Eq.(1).

Substituting the Lagrangians of the time fractional derivative of the basic equations, Eqs.(34) and(35), into the Euler-Lagrange formula, Eq.(8), and using Eq.(1)and Eq.

(2), the basic time fractional equations are described as

0D^b_tniþ @

@xðniuiÞ ¼0; 0<b1: (36)

0D^b_tu_iþu_i@ui

@x þ@u

@x¼0; 0<b1: (37)

@²u

@x² ¼kn_e n_iþ ð1 kÞ: (38) The non-isothermal plasma is introduced through electron density in the following manner:^37,38

n_eðx;tÞ ¼1þuðx;tÞ þ1

2u²ðx;tÞ 4

3bu^ð³⁼²^Þðx;tÞ: (39) Here, bis a parameter that defines the degree of deviation from isothermal condition and depends on the temperature parameters of resonant electrons, and is given by

b¼ 1 ffiffiffip

p ð1 cÞ; c¼Tef

T_et; (40) whereT_etis trapped electron temperature and cis the trap- ping parameter.

According to reductive perturbation method, we introduce the stretched coordinates as v¼e¹⁼⁴ðx vtÞ; s¼ e¹⁼⁴t to study the small but finite amplitude solitary wave, where e is the small dimensionless parameter and v is the phase velocity. The dimensionless variables n_iðv;sÞ,uðv;sÞ, andu_iðv;sÞmay be expanded as follows:

niðv;sÞ ¼1þen1iþe^ð³⁼²^Þn2iþ ; uðv;sÞ ¼eu₁þe^ð³⁼²^Þu₂þ ; u_iðv;sÞ ¼eu_1iþe^ð³⁼²^Þu_2iþ

(41)

Substituting Eq. (39)and Eq. (41)into Eqs. (36)–(38), and neglecting terms of higher order thane

n1i¼u₁=v²; u1i¼u₁=v; v¼1= ffiffiffi pk

: (42)

Preserving terms of up to the second order in e and eliminating the second-order variables, (^@u_@v²ⁱ,^@n_@v²ⁱ, and^@u_@v²) in Eqs.(36)–(38), and using Eq.(1)and Eq.(2), the TFMKdV equation may be obtained as

R

0D^b_su₁ðv;sÞþAu₁¹²ðv;sÞ@u1ðv;sÞ

@v þB@³u₁ðv;sÞ

@v³ ¼0;

0<b1; s2½0;T0; (43)

wherevandsare the space and time variables, respectively.

The coefficientsAandBare defined as A¼ 2b

kð1þvÞ; B¼ 1

k²ð1þvÞ: (44) Equation(43)is called the TFMKdV equation that describes the nonlinear propagation of DIA solitary wave.

III. THE BASIC UNDER LAYING THE LADM

Consider the general form of one-dimensional nonlinear partial differential equation as

Lðuðx;tÞÞ þNðuðx;tÞÞ ¼gðxÞ; (45) where L and N denote a linear and a nonlinear operators, respectively.

Taking the Laplace transform from both sides of Eq.(45) LfLðuðx;tÞÞg þ LfNðuðx;tÞÞg ¼ LðgðxÞÞ; (46) whereLdenotes the Laplace transform. Focusing on the linear operatorLin Eq.(45), the concept of Adomian decomposition method is used to generate a series expansion for Lðuðx;tÞÞas follow:^22,23,39

u¼X¹

i¼0

ui; Lðuðx;tÞÞ ¼L X¹

i¼0

ui

!

; (47)

where the componentsu_i;i0 are to be determined in a recursive manner.

Switching to the non-linear operator N in Eq.(45), we use the Adomian polynomials,A_i, as follow:

Nðuðx;tÞÞ ¼X¹

i¼0

Ai; (48)

where the Adomian polynomials,Ai, are expressed as Ai¼1

i!

dⁱ

dkⁱ N Xⁿ

j¼0

k^juj

!

" #

k¼0

: (49)

Substituting Eq.(48)and Eq.(47)into Eq.(46) L L X¹

i¼0

u_i

!

( )

þ L X¹

i¼0

A_i

( )

¼ LðgðxÞÞ; (50)

where the Adomian polynomials,A_i, are elaborated as

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A0¼Nðu0Þ; A1¼u1N⁰ðu0Þ; A2¼u2N⁰ðu0Þ þ1

2!u²₁N⁰⁰ðu0Þ; A3¼u3N⁰ðu0Þ þu1u2N⁰⁰ðu0Þ þ1

3!u³₁N⁰⁰⁰ðu0Þ; A4¼u4N⁰ðu0Þ þ 1

2!u²₂þu1u3

N⁰⁰ðu0Þ þ1

2!u²₁u2N⁰⁰⁰ðu0Þ þ1

4!u⁴₁N^ð^iv^Þðu0Þ:

(51)

Equation(46)can be rewritten in the following form:

X¹

i¼0

LfLðuiðx;tÞÞg þX¹

i¼0

LfAig ¼ Lfgg: (52) Using Eq.(52), we introduce the recursive relation as

LfLðu0Þg ¼ Lfgg; X¹

i¼1

LfLðuiÞg þX¹

i¼0

LfAig ¼0: (53) Alternatively, the recursive relation, Eq.(53), is expressed as

LfLðu0Þg ¼ Lfgg; LfLðu1Þg þ LfA0g ¼0;

LfLðu2Þg þ LfA1g ¼0;

LfLðu3Þg þ LfA2g ¼0;

⯗

LfLðu_kÞg þ LfA_k 1g ¼0:

(54)

Using the Maple symbolic code, the first part of Eq. (54) gives the value ofLfu0g. First, applying the inverse Laplace transform toLfu0ggives the value ofu0that will define the Adomian polynomial,A0 using the first part of Eq. (51). In the second part Eq. (54), the Adomian polynomial A0 will enable us to evaluate Lfu1g. Second, applying the inverse Laplace transform toLfu1ggives the value ofu1 that will define the Adomian polynomialA1 using the second part of Eq. (51) and so on. This in turn will lead to the complete evaluation of the components ofu_k;k0 upon using different corresponding parts of Eqs.(51)and(54).

IV. SOLUTION OF THE TFMKdV EQUATION USING THE LADM

Several methods can be used to solve fractional differential equation such as: the operational method,^28,29Homotopy perturbation method,^40,41variational iteration method,⁴²and Adomian decomposition method.^22–24 In this paper, the TFMKdV equation derived by semi-inverse and Agrawal’s methods will be solved using the LADM as follows:

The Laplace transform is applied to both sides of the TFMKdV equation and the solution is presented. By using LADM, all conditions are satisfied over the entire range of

time domain. Applying the Laplace transform to Eq. (43) and using the relation given by Eq.(4)

s^bLð/ðv;sÞÞ ½^R0D^b_s ¹/ðv;sÞ_s¼0þALð/¹²ðv;sÞ/_vðv;sÞÞ þBLð/_vvvðv;sÞÞ ¼0; 0<b1; s 2 ½0;T0: (55) The compact relation of the TFMKdV equation is written as

s^bX¹

i¼1

Lð/_iðv;sÞÞ ^R0D^b_s ¹X¹

i¼0

/_iðv;sÞ

" #

s¼0

þAX¹

i¼0

LðAiÞ

þB @³

@v³ X¹

i¼0

Lð/_iðv;sÞÞ ¼0; 0<b1; s 2 ½0;T0; (56) where/¹²ðv;sÞ/_vðv;sÞ ¼P₁

n¼0A_i:

Alternatively, the sequence of recursive equations deduced from Eq. (56) can be written in the following manner:

s^bðLð/₁ðv;sÞÞÞ ½^R0D^b_s ¹ð/₀ðv;sÞÞs¼0þAðLðA0ÞÞ þB @³

@v³½Lð/₀ðv;sÞÞ ¼0;

s^bðLð/₂ðv;sÞÞÞ ½^R0D^b_s ¹ð/₁ðv;sÞÞ_s¼0þAðLðA1ÞÞ þB @³

@v³½Lð/₁ðv;sÞÞ ¼0;

s^bðLð/₃ðv;sÞÞÞ ½^R0D^b_s ¹ð/₂ðv;sÞÞs¼0

þAðLðA2ÞÞ þB @³

@v³½Lð/₂ðv;sÞÞ ¼0;

⯗

s^bðLð/_kðv;sÞÞÞ ½^R0D^b_s ¹ð/_k ₁ðv;sÞÞs¼0þAðLðAk 1ÞÞ þB @³

@v³½Lð/_k ₁ðv;sÞÞ ¼0;

0<b1; s2 ½0;T0: (57)

The initial condition of Eq.(43)is chosen as

/₀ðv;sÞ ¼/ðv;sÞj_s¼0¼/_msech⁴ðdvÞ; (58) where /_manddare constants that can be expressed in terms of our problem parameters as

/_m¼ ð15v0=8AÞ²; d¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0=16B

p ; (59)

wherev0is a traveling wave velocity normalized byC_i. Substituting the zero order approximation,/₀ðv;sÞ, and A0 into the recursive equation, Eq. (57), we obtain the first order approximation as

/₁ðv;sÞ ¼ 4ds^b

Cð1þbÞ½tanhðdvÞðAð/_msech⁴ðdvÞÞ³⁼²

þ2B/_msech⁴ðdvÞd²ð15 tanh²ðdvÞ 7ÞÞ: (60) Substituting the first order approximation, /₁ðv;sÞ, Eq. (60), and A1 into the recursive equation, Eq. (57), (applying the Maple symbolic code) give the second order approximation as

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/₂ðv;sÞ ¼ 1

Cð1þ2bÞ/_mcosh¹⁰ðdvÞ

"

4s^2bd²

"

376B²d⁴/²_msech⁴ðdvÞcosh¹⁰ðdvÞ

þ308ð/_msech⁴ðdvÞÞ³⁼²tanh²ðdvÞd²BA/_mcosh¹⁰ðdvÞ 504ð/_msech⁴ðdvÞÞ³⁼²tanh⁴ðdvÞd²BA/_mcosh¹⁰ðdvÞ 7280 B²d⁴/²_msech⁴ðdvÞtanh²ðdvÞcosh¹⁰ðdvÞ þ21000B²d⁴/²_msech⁴ðdvÞtanh⁴ðdvÞcosh¹⁰ðdvÞ

15120B²d⁴/²_msech⁴ðdvÞtanh⁶ðdvÞcosh¹⁰ðdvÞ 20ð/_msech⁴ðdvÞÞ³⁼²d²BA/_mcosh¹⁰ðdvÞ þA²/³_mcosh²ðdvÞ 9A²/³_msinh²ðdvÞ 96A /_m

cosh⁴ðdvÞ

!3=2

d²B/_mcosh¹⁰ðdvÞ þ352A /_m cosh⁴ðdvÞ

!3=2

d²B/_mcosh⁸ðdvÞ

270 /_m

cosh⁴ðdvÞ

!5=2

d²BAcosh¹⁰ðdvÞ

##

: (61)

Ultimately, the solution of the TFMKdV equation, Eq.(43), is elaborated in series expansion as

/ðv;sÞ ¼X¹

i¼0

/_iðv;sÞ: (62)

V. RESULTS AND DISCUSSIONS

Numerical calculations are made for small but finite amplitude DIA waves by deriving the TFMKdV equation. To investigate the effect of plasma and time fractional parameters on the nature of the solitary waves, the equations governing the amplitude and the width of the DIA solitary waves are solved using the LADM. The results are demon- strated in Figs.1–6. In our solutions, only the first five terms of the series solutions are considered in Eq.(62). In Fig. 1, the solitary wave solution of the TFMKdV equation is obtained for two values of the time fractional orders,b¼1;

b¼0:8. Fig. 1 shows that, the solitary wave solutions for any time fractional orders are preserved. In other words, implementing the time fractional orders into Eqs. (9)–(11) do not change the solitary wave solution of the TFMKdV equation.

Alinejad^13,14 studied the propagation of DIA solitary waves in dusty plasma with trapped electrons. He found that the amplitude of DIA solitary waves increases with the increase in population of the background free electrons ðkÞ^13,14 and decreases by increasing the deviation from

isothermalityðbÞ.¹⁴To compare our results, we take the corresponding parameters similar to those of dusty plasma presented in Refs.13and14:c¼0:5;k¼0:2;v0¼0:1. Fig. 2 shows that the soliton amplitude of the electrostatic potential increases with the increase in population of the background

FIG. 1. The electrostatic potential / (v;s) versus v and s for c¼0:5;

k¼0:2, v0¼0:1 at different values of the time fractional orders ðbÞ, (a) b¼1 and (b)b¼0:8.

FIG. 2. The amplitude of the electrostatic potential/(2;2) versuskforv¼ 2; s¼2; c¼0:5; v₀¼0:1b¼1 at different values of the deviation from isothermality ðbÞ.

(8)

free electronsðkÞand decreases with increasing the deviation from isothermality ðbÞat time fractional orderb¼1. These results are in agreement with corresponding results obtained by Alinejad.^13,14 In Fig.3, the profiles of/(2;s) against s are presented for different values of time fractional orders b¼0:7;0:8;0:9, and 1. It shows that the soliton amplitude increases as time fractional order increases.

Figs. 4 and 5 show the amplitude of the electrostatic potential / (2;2) against the time fractional orders ðbÞ for different values of the population of the background free electrons ðkÞ and wave velocities ðv0Þ, respectively. It is clear that the amplitude of the DIA solitary wave increases with the increase of time fractional orderðbÞand population

of the background free electronsðkÞand wave velocityðv0Þ. At smaller values of wave velocityðv0Þ, the variations of the amplitude of the DIA solitary wave tend to be flatter. In Fig.6, the profiles of/(v;2) versusvare presented for different values of time fractional orders b¼0:7;0:8;0:9, and 1. It shows that the soliton amplitude increases as the time fractional order increases. 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 nonlinearity or dispersion terms to the equations governing the plasma medium.^43,44

FIG. 3. The amplitude of the electrostatic potential/ (2;s) versuss for v¼2; c¼0:5,k¼0:2; v0¼0:1 at different values of the time fractional ordersðbÞ.

FIG. 4. The amplitude of the electrostatic potential/(2;2) versusbfor v¼2; s¼2,c¼0:5;v₀¼0:1 at different values ofk.

FIG. 5. The amplitude of the electrostatic potential /(2;2) versusbfor v¼2; s¼2,k¼0:2; c¼0:5 at different values ofv₀.

FIG. 6. The amplitude of the electrostatic potential/(2;2) versusvfor s¼2; c¼0:5;k¼0:2;v₀¼0:1 at different values of the time fractional ordersðbÞ.

(9)

VI. CONCLUSIONS

The nonlinear propagation of DIA solitary waves is investigated by applying the time fractional order in the un- magnetized dusty plasma consisting of cold ion fluid, static dust particles, trapped, and free electrons. The TFMKdV equation is derived by using the semi-inverse and Agrawal’s methods. Hybrid LADM results in a simple and compact form of LADM equation, which indeed reduces the amount of calculations considerably. The effects of the time fractional orderðbÞ, population of the background free electrons ðkÞ, deviation from isothermalityðbÞ, and the wave velocity ðv0Þ on the propagation of DIA soliton wave are investigated. It is found that the amplitude of the DIA solitary wave increases with the increase ofb,v0, andk, while it decreases with the increase of b. Moreover, the soliton amplitude of the electrostatic potential decreases with the increase of b and increases with the increase ofkthat is in agreement with the corresponding results obtained previously.^13,14 It is also concluded that the fractional parameter may be used to modify the shape of the solitary wave instead of adding higher order nonlinearity or dispersion terms to the equations that govern the plasma medium.^43,44

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