Abundant Solutions with Distinct Physical Structure for ... · Partial Differential Equations ......

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345

© Research India Publications. http://www.ripublication.com

5332

Abundant Solutions with Distinct Physical Structure for Nonlinear Integro and

Partial Differential Equations

Taher A. Nofal1,2

1Mathematics Department, Faculty of Science, Taif University, Saudi Arabia. 2 Mathematics Department, Faculty of Science, Minia University, Egypt.

Khaled A Gepreel 1,3 1Mathematics Department, Faculty of Science, Taif University, Saudi Arabia. 3 Mathematics Department, Faculty of Science, Zagazig University, Egypt.

Abstract

In this article, we use two direct methods namely the

generalized Kudryashov method and the generalized (G'/G)-

expansion method to discuss the traveling wave solutions to

the nonlinear integro- partial differential equations. In the

generalized (G'/G)-expansion method, we suppose the trial

equation for G satisfies the nonlinear second order

differential equation 0)( 22 GCEGGBGGAG while Q in the generalized Kudryashov method satisfies

Bernoulli first order differential equation BQAQQ 2

We construct the exact solutions for some nonlinear integro -

partial differential equations in mathematical physics via

(3+1)- dimensional Gardner type integro- differential

equation and (2+1) dimensional Sawada- Kotera nonlinear

integro partial differential equation. We obtain the traveling

wave solutions as a rational formula in the hyperbolic

functions, trigonometric functions and rational function,

when G satisfies a nonlinear second order ordinary

differential equation and Q satisfies the Bernoulli first order

differential equation. When the parameters are taken some

special values, the solitary wave are derived from the

traveling waves. This method is reliable, simple and gives

many new exact solutions.

Keywords: Generalized ( GG / )- expansion method,

Generalized Kudryashov method, Traveling wave solutions,

Gardner type integro- differential equation, Sawada- Kotera

nonlinear integro partial differential equation

INTRODUCTION

The study of partial differential equations has a significant

role in identifying some of the physical and natural

phenomena surrounding us and through its knowledge of

predicting some natural problems that may be induced in the

near future. Many natural and physical problems can be

visualized in many nonlinear partial differential equations and

by analyzing their analytical solutions, physicists and

engineers can interpret those. There are many methods for

obtaining exact solutions to nonlinear partial differential

equations such as the inverse scattering method [1], Hirota’s

bilinear method [2], Backlund transformation [3], the first

integral method [4], Painlevé expansion [5], sine–cosine

method [6], homogenous balance method [7], extended trial

equation method [8,9], perturbation method [10,11], variation

method [12], tanh - function method [13,14], Jacobi elliptic

function expansion method [15,16], Exp-function method

[17,18] and F-expansion method [19,20] . Wang etal [21]

suggested a direct method called the ( GG / ) expansion

method to find the traveling wave solutions for nonlinear

partial differential equations (NPDEs) . Zayed etal [22,23]

have used the ( GG / ) expansion method and modified

( GG / ) expansion method to obtain more than traveling

wave solutions for some nonlinear partial differential

equations. Shehata [24] have successfully obtained more

traveling wave solutions for some important NPDEs when

G satisfies a linear differential equations 0 GG .

There are many authors have successively applied the

( )/ GG Iexpansion method to study the exact solutions for

nonlinear evoluation equations see [25-28]. In this paper we

use the generalized ( GG / )- expansion function method

when G satisfies a nonlinear differential equations

,0)( 22 GCEGGBGGAG where ,,, CBA

E are real arbitrary constants to find the traveling wave

solutions for some nonlinear integro- partial differential

equations in mathematical physics . Also we use the

generalized Kudryashov method [29,30] to discuss the

rational traveling wave solutions for some nonlinear integro-

partial differential equations. The solitary wave solutions are

deduced form the traveling wave solutions when the

parameter are taking some special values.

DESCRIPTION OF THE GENERALIZED ( GG / )

EXPANSION FUNCTION METHOD FOR NPDEs

In this part of the manuscript, the generalized ( GG / )

expansion method will be given. In order to apply this method

to nonlinear partial differential equations we consider the



5333

following steps[27,28]

Step 1. We consider the nonlinear partial differential

equation, say in two independent variables x and t is given

by

,0,..),,,,,( xtxxttxt uuuuuuP (1)

where ),( txuu is an unknown function, P is a

polynomial in ),( txuu and its various partial

derivatives, in which the highest order derivatives and

nonlinear terms are involved.

Step 2. We use the following travelling wave transformation:

,),( wtxkUu ii

(2)

where wki , is a nonzero constant. We can rewrite Eq.(1) in

the following form:

0,...),,( UUUP (3)

Step 3. We assume that the solutions of Eq. (3) can be

expressed in the following form:

,

)(/)(1

)(/)()(

m

mii

nn

inni

GG

GGaU

(4)

where ),...,1,0( miai are arbitrary constants , is

nonzero constant to be determined later, m is a positive

integer and )(G satisfies a nonlinear second order

differential equation

,0)( 22 GCEGGBGGAG (5)

where ECBA ,,, are real nonzero constants.

Step 4. Determine the positive integer m by balancing the

highest order nonlinear term(s) and the highest order derivative in

Eq (3).

Step 5. Substituting Eq. (4) into (3) along with (5), cleaning

the denominator and then setting each coefficient of

,..2,1,0,))(/)(( iGG i to be zero, yield a set of

algebraic equations for ),...,1,0( miai , k and .

Step 6. Solving these over-determined system of algebraic

equations with the help of Maple software package to

determine ),...,1,0( miai , k and .

Step 7. The general solution of Eq. (5), takes the following

cases :

(i) When 0B , 0)(42 CAEB ,

CA , we obtain the hyperbolic exact solution of

Eq.(5) takes the following form:

A

A

B

CCe

G )2

sinh()2

cosh(

][

)( 21

2

2

(6)

where 1C and 2C are arbitrary constants. In this case the

ratio between Gand G takes the form

)2

sinh()2

cosh(

)2

cosh()2

sinh(

2221

21

CC

CCB

G

G

(7)

(ii) When 0B , 0)(42 CAEB ,

CA , we obtain the trigonometric exact solution

of Eq.(5) takes the form

)2

sin()2

cos(

)2

cos()2

sin(

2221

21

CC

CCB

G

G

(8)

(iii) When 0B , 0)(42 CAEB , we

obtain the rational exact solution of Eq.(5) takes the form

21

2

2 CC

CB

G

G

(9)

(iv) When 0B , ,0 E , we obtain the

hyperbolic exact solution of Eq.(5) takes the following

form:

)sinh()cosh(

)cosh()sinh(

21

21

CC

CC

G

G (10)

(v) When 0B , ,0 E we obtain the

hyperbolic exact solution of Eq.(5) takes the following

form:

)sin()cos(

)cos()sin(

21

21

CC

CC

G

G

(11)

Step 8. Substituting the constants ),...,1,0( mii , k

and which obtained by solving the algebraic equations in

Step 5, and the general solutions of Eq.(5) in step 6 into



5334

Eq.(4) , we obtain more new exact solutions of Eq. (1)

immediately.

DESCRIPTION OF THE GENERALIZED

KUDRYASHOV METHOD FOR NPDE The basic steps in the application of the GKM detailed in the

following [29,30]:

Step 1. We suppose the exact solution of Eq. (3) to be in the

following rational form:-

M

j

jj

N

i

ii

Qa

Qa

V

0

0

)(

)(

)(

(12)

Where ji ba , are constants to be determined later such that

, . We suppose the trial equation for

satisfies the first order Bernoulli differential equation:

BQAQQ 2 (13)

Step 3. Determine the positive integer numbers and in

Eq. (12) by balancing the highest order derivatives and the

nonlinear terms in Eq. (3).

Step 4. Substituting Eqs. (12) and (13) into Eq. (3), we obtain

a polynomial in , . Setting all

coefficients of this polynomial to be zero, we obtain a system

of algebraic equations which can be solved by the Maple or

Mathematica software package to get the unknown

parameters and .

Consequently, we obtain the exact solutions of Eq. (1).

TRAVELING WAVE SOLUTIONS FOR THE FIRST

EQUATION TO THE GARDNER TYPE INTEGRO-

DIFFERENTIAL EQUATION

In this section, we use two different methods namely the

generalized ( GG / ) expansion method and the generalized

kuderyshov method to discuss the exact solutions for the

nonlinear evolution equations in mathematical physics via the

(3+1) dimensional Gardner type intergro- differential

equations which are very important in the mathematical

science and have been paid attention by many researchers in

physics and engineering. The (3+1) dimensional Gardner type

integro- differential equation takes the following form:

0'3'3

'32

36

2

222

x

zz

x

yx

x

yyxxxxxt

dxudxuu

dxuuuuuuu

(14)

where ,, and are arbitrary constants. Gardner type

intergro- differential equations have many applications in

different branches of physics such as plasma physics,

fluid physics, and quantum field theory [1–7]. We take

the transformation:

,xvu (15)

to convert the Gardner type intergro- differential equations to

the nonlinear partial differential equation:

.033

32

36

2

222

zzyxx

yyxxxxxxxxxxxt

vvv

vvvvvvv

(16)

Traveling wave transformation

,),( 321 wtzkykxkv (17)

permits us to convert the nonlinear partial differential

equation (16) to the following ordinary differential equation

.0333

2

36

23

22

21

22

2

241

2)4(41

311

kkkk

kkkwk (18)

By using the integration equation (18) can be written in the

following form:

.08

1

2

1

)2

1()33(

2

1

2144

1224

1

32

21

31

21

23

222

2

cckk

kkkwkkk

(19)

where 1c and 2c are the integration constants. If , we take

)()( equation (19) can be reduced to the following

ODE’s:

.08

1

2

1

)2

1()33(

2

1

2144

1224

1

32

21

31

21

23

222

2

cckk

kkkwkkk

(20)

Generalized ( GG / ) expansion method to the (3+1)

dimensional Gardner type integro- differential equation :

In this subsection we discuss the solution of Eq.(20) by using

generalized ( GG / ) expansion method. Balancing the

highest order derivative 2 with the nonlinear term

4 ,

we get the solution formula of Eq.(20) has the following

form:

)(/)(

)](/)(1[

)](/)(1[

)(/)()(

1

10

GG

GGb

GG

GGaa

(21)

where 110 ,, baa and are constants to be determined



5335

later. Substituting Eq. (21) along with (5) into Eq. (20) and

cleaning the denominator and collecting all terms with the

same order of ( )(/)( GG ) together, the left hand side of

Eq. (20) are converted into polynomial in ( )(/)( GG ).

Setting each coefficient of these polynomials to be zero , we

derive a set of algebraic equations for

wkkkbaa ,,,,,, 321110 , . Solving the set of algebraic

equations by using Maple or Mathematica , software package

to get the following results:

Case 1.

2

=,2

,2

)(4,

2

21

1

21

2

121

2

210

E

B

kA

Eb

EAk

ACEBa

k

kka

,

)](6312

12) )(4(4[2

1

23

22

22222

22221

2

221

222

21

2

kkAkAkkA

kAACEBAk

w

}8612

)](4[8

)](4[4{2

1

331

221

22

222212

2

32

33221

2

23242

121

kAkkAkkA

kAACEBk

ACEBkkA

c

})2(

)](4[16)](4

[32)](4[8

)](4[32{8

1

412

4

224

221

322422

22

22221

2

441

62

kkA

ACEBACE

BBkkAACEBkA

ACEBkAAk

c

(22)

where ,,,,,, AEBC , 321 ,, kkk are arbitrary

constants. There are many other cases which are omitted her

for convenience to the reader. In this case the traveling wave

solution of Eq.(20) takes the following form:

)(/)(

)](/)(2[

)](/)(2[

)(/)())(4(2)(

21

21

2

21

2

21

GGkA

GGBE

GGBEAk

GGACEB

k

kk

(23)

There are many families to discuss the types of the traveling

wave solutions of Eq.(20) as follows:

Family 1. When 0B , 0)(42 CAEB , we

obtain the hyperbolic exact solution of Eq.(20) takes the

following:

.

)]2

sinh()()2

cosh()[(

)]2

sinh(})4{()2

cosh(})4[{(

)]2

sinh(})4{()2

cosh(})4[{(

)]2

sinh()()2

cosh()[(

2)(

122121

122

212

122

2122

1

1221

21

2

21

CBCCBCkA

BCCBEBCCBE

BCCBEBCCBEAk

CBCCBC

k

kk



5336

Consequently the hyperbolic traveling wave solution of Eq.(14) has the following form:

.

)]2

sinh()()2

cosh()[(

)]2

sinh(})4{()2

cosh(})4[{(

)]2

sinh(})4{()2

cosh(})4[{(

)]2

sinh()()2

cosh()[(

2),,,(

12211

122

212

122

212

1

1221

12

211

CBCCBCkA

BCCBEBCCBE

BCCBEBCCBEAk

CBCCBC

k

kktzyxu

(24)

where

)](63

1212) )(4(4[2

23

22

22222

222

21222

1222

21

2321

kkAkA

kkAkAACEBAk

tzkykxk

Family 2. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solution of Eq.(20) takes the

following form

.

)]2

sin()()2

cos()[(

)]2

sin(})4{()2

cos(})4[{(1

)]2

sin(})4{()2

cos(})4[{(

)]2

sin()()2

cos()[(

2)(

1221

122

212

21

122

2122

1

1221

21

2

21

CBCCBC

BCCBEBCCBE

kA

BCCBEBCCBEAk

CBCCBC

k

kk

(25)

Consequently the periodic trigonometric traveling wave solution of Eq.(14) has the following form:

,

)]2

sin()()2

cos()[(

)]2

sin(})4{()2

cos(})4[{(1

)]2

sin(})4{()2

cos(})4[{(

)]2

sin()()2

cos()[(

2),,,(

1221

122

212

1

122

212

1

1221

12

212

CBCCBC

BCCBEBCCBE

kA

BCCBEBCCBEAk

CBCCBC

k

kktzyxu

(26)



5337

Family 3. When 0B , 0 , we obtain the rational exact


.2)(

]2))(4[{(1

2)(

221

2212

21

21

2

21

CCCB

BCCCBE

kA

k

kk

(27)

Consequently the rational traveling wave solution of Eq.(14)

has the following form:

,2)(

]2))(4[{(1

2),,,(

221

2212

1

12

213

CCCB

BCCCBE

kA

k

kktzyxu

(28)

Family 4. When 0B , 0 E , we obtain the

hyperbolic exact solution of Eq.(15) takes the following form:

.

)]cosh()sinh([

)]sinh()cosh([2

)]sinh()cosh([

)]cosh()sinh([2

2)(

21

21

21

21

21

21

21

2

21

CC

CC

kA

E

CC

CC

kA

k

kk

(29)



.

)]cosh()sinh([

)]sinh()cosh([2

)]sinh()cosh([

)]cosh()sinh([2

2),,,(

21

21

1

21

21

1

12

214

CC

CC

kA

E

CC

CC

kA

k

kktzyxu

(30)

where

)].(6312

12) )(16[2

23

22

22222

22221

2

221

22

21

2

321

kkAkAkkA

kAACEAk

t

zkykxk

(31)



.

)]cos()sin([

)]sin()cos([2

)]sin()cos([

)]cos()sin([2

2)(

21

21

21

21

21

21

21

2

21

CC

CC

kA

E

CC

CC

kA

k

kk

(32)



,

)]cos()sin([

)]sin()cos([2

)]sin()cos([

)]cos()sin([2

2),,,(

21

21

1

21

21

1

12

215

CC

CC

kA

E

CC

CC

kA

k

kktzyxu

(33)

where is defined as Eq.(31).

Generalized Kudryashov method to the (3+1) dimensional

Gardner type integro- differential equation :

In this subsection we discuss the solution of Eq.(20) by using

generalized Kudryashov method. Balancing the highest order

derivative 2 with the nonlinear term

4 , we have

1MN . (34)

Equation (34) has infinitely solutions, in the special if

1M then 2N . Consequently the solution formula of



5338

Eq.(20) has the following form:

)(

)()()(

10

2110

Qbb

QaQaa

(35)

where 10210 ,,,, bbaaa are constants to be determined later

and

)()()( 2 BQAQQ . (36)

Substituting Eqs. (35) and (36) into Eq. (20), we obtain a

polynomial in ,...)2,1,0,(, jiQ ji, Setting all

coefficients of this polynomial to be zero, we obtain a system

of algebraic equations which can be solved by the Maple or

Mathematica software package to get the unknown

parameters 12110210 ,,,,,,, ckkbbaaa and w .

,2

,4

,)22(2

,)22(

01

20

2

21

2

22110

1

21

2

22110

0

B

Abb

B

Aba

kB

kkBkAba

k

kkBkba

}64

91212{2

1

}32

1632

1688

2432{8

1

},46

1288{2

1

23

222241

22

2221

21

2

12

261

22

481

4442

42

251

3

41

4241

422

23321

3

222

21

222

31

3

41

62

2412

32221

2

221

2332

331

322512

141

kBk

kkkkk

w

Bk

BkkkBk

kBkkkk

kkkkk

c

Bkkkk

kkkkBkk

c

(37)

where 0b , ,,,,, 321 kkk , and BA, are arbitrary

constants . In this case the traveling wave solution takes the

form:

)(2

)(4)()22(2)22()(

21

200

21

2

221

22

2112

211

QkAbBbk

QkAQkkBkAkkBkB

(38)

Substituting by the general solutions of Eq.(13) into (38) we have the rational traveling wave solution:

)}1()22(2)1)(22(

4{)]1(2)1(

1)(

2211

22

211

2221

2

21

20

20

21

2

nBnBnB

B

BBB

CeAekkBkACCeAkkBk

eCBkACeACekAbCeAbk

(39)

where

}6491212{2

23

222241

22

2221

21

2

12321 kBkkkkkk

tzkykxk

There are many other cases which omit for convenience to the reader.

Remark 1: The general GG / expansion method is more

effective than the generalized Kuderyshov method. The

general GG / expansion is complicated than the generalized

Kuderyshov method but determine many types of exact

solutions such as the hyperbolic functions, trigonometric

functions and rational function but the generalized

Kuderyshov method determine only one type of solution.

TRAVELING WAVE SOLUTIONS FOR (2+1)

DIMENSIONAL SAWADA- KOTERA NONLINEAR

INTEGRO PARTIAL DIFFERENTIAL EQUATION

In this section, we use genealized ( GG / ) expansion

method to discuss the exact solutions for the nonlinear

evolution equations in mathematical physics via (2+1)

dimensional Sawada- Kotera nonlinear integro partial

differential equation which are very important in the

mathematical science and have been paid attention by many

researchers in physics and engineering. The (2+1)

dimensional Sawada- Kotera nonlinear integro partial



5339

differential equation takes the following form:

x

yxy

x

yy

xxyxxxxxxt

dxuuuudxu

uuuuuu

'55'5

)3

55( 3

(40)

The transformation (15) convert the (2+1) dimensional

Sawada- Kotera nonlinear integro partial differential equation

(40) to the following partial differential equation

yxxxyxyy

xxxyxxxxxxxt

vvvvv

vvvvvv

355

)3

55( 3

5

(41)

Traveling wave transformation (17) permits us to convert the

nonlinear partial differential equation (41) to the following

ordinary differential equation

.0105)

3

55(

221

222

21

331

41

)5(5111

kkkkk

kkkkwk (42)

By using the integration equation (42) can be written in the

following form:

,05)3

5

5()5(

12

2212

21

331

41

)5(511

221

ckkkkk

kkkkwk

(43)

where 1c is the integration constant. If , we take

)()( equation (43) can be reduced to the following

ODE’s:

,05

3

55)5(

12

2212

31

341

51

)4(61

221

ckkkk

kkkkwk

(44)

We discuss the solution of Eq.(44) by using generalized

rational ( GG / ) expansion method. Balancing the highest

order derivative )4( with the nonlinear term

3 , we get

the solution formula of Eq.(44) has the following form:

2

22

2

22

110

)](/)([

)](/)(1[

)](/)(1[

)](/)([

)(/)(

)](/)(1[

)](/)(1[

)(/)()(

GG

GGb

GG

GGa

GG

GGb

GG

GGaa

(45)

where 22110 ,,,, babaa and are constants to be

determined later. Substituting Eq. (45) along with (5) into

Eq. (44) and cleaning the denominator and collecting all

terms with the same order of ( )(/)( GG ) together, the

left hand side of Eq. (44) are converted into polynomial in

( )(/)( GG ). Setting each coefficient of these

polynomials to be zero , we derive a set of algebraic equations

for 132122110 ,,,,,,,,, cwkkkbabaa and . Solving the

set of algebraic equations by using Maple or Mathematica ,

software package to get the following results:

Case 1.

,12

))],(4(109[5

1

2

12

2

231

222

120

A

kEb

ACEBkAkkA

a

}84480

2112042240

26406144061440

21120204800204800

2433200307200

38400153600153600

3840042240{75

1

}17))(4(40{5

2

,0

,2

,4

)](4[3

2326

1

2326

12426

1

22

461

2391

2391

2226

1339

1339

1

632

691

2291

491

22291

22291

491

2222

612

161

422

22614

1

11

22

221

2

kCAEk

kEABkkAEk

AkBkCAEkACEk

ECkABkCEkAEk

AkBkCAEBk

EABkAEBkCEBk

ECBkACEkkkA

c

AkACEBkAk

w

ba

E

B

EA

ACEBka

(46)

where ,,,,, AEBC 21, kk are arbitrary constants. There

are many other cases which are omitted her for convenience to

the reader. In this case the traveling wave solutions of

Eq.(44) take the following form:

2

22

12

222

2221

231

222

12

)](/)([

)](/)(1[12

)](/)(1[4

)](/)([)](4[3

))](4(109[5

1)(

GGA

GGkE

GGEA

GGACEBk

ACEBkAkkA

(47)

There are many families to discuss the types of the traveling

wave solutions of Eq.(44) as follows:



5340

Family 1. When 0B , 0)(42 CAEB , CA we obtain the hyperbolic exact solution of Eq.(44) takes

the following:

212

221

22

21221

21

231

222

12

)]2

sinh(})4{()2

cosh(})4[{(

)]2

sinh()()2

cosh()[(3

))](4(109[5

1)(

BCCBEBCCBEA

CBCCBCk

ACEBkAkkA

21221

2

212

221

21

)]2

sinh()()2

cosh()[(

)]2

sinh(})4{()2

cosh(})4[{(3

CBCCBCA

BCCBEBCCBEk

(48)

Consequently the hyperbolic traveling wave solution of Eq.(40) has the following form:

21221

2

212

221

221

212

221

22

21221

221

231

2221

)]2

sinh()()2

cosh()[(

)]2

sinh(})4{()2

cosh(})4[{(3

)]2

sinh(})4{()2

cosh(})4[{(

)]2

sinh()()2

cosh()[(3

))](4(109[5

1),,(

CBCCBCA

BCCBEBCCBEk

BCCBEBCCBEA

CBCCBCk

ACEBkAkA

tyxu

(49)

where

}17))(4(40{5

2 422

22614

1

21 AkACEBkAk

tykxk (50)

Family 2. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solution of Eq.(44) takes the

following form

.

)]2

sin()()2

cos()[(

)]2

sin(})4{()2

cos(})4[{(3

)]2

sin(})4{()2

cos(})4[{(

)]2

sin()()2

cos()[(3

))](4(109[5

1)(

21221

212

221

2

2

1

212

221

22

21221

21

231

22

12

CBCCBC

BCCBEBCCBE

A

k

BCCBEBCCBEA

CBCCBCk

ACEBkAkkA

(51)



5341

Consequently the periodic trigonometric traveling wave solution of Eq.(40) has the following form:

.

)]2

sin()()2

cos()[(

)]2

sin(})4{()2

cos(})4[{(3

)]2

sin(})4{()2

cos(})4[{(

)]2

sin()()2

cos()[(3

))](4(109[5

1),,(

21221

212

221

2

2

21

212

221

22

21221

221

231

2222

CBCCBC

BCCBEBCCBE

A

k

BCCBEBCCBEA

CBCCBCk

ACEBkAkA

tyxu

(52)


Family 3. When 0B , 0 , we obtain the rational exact


2221

2221

2

2

1

21

2

]2)([

]2))(4[{(3

5

9)(

CCCB

BCCCBE

A

k

k

k

(53)



,]2)([

]2))(4[{(3

5

9),,,(

2221

2221

2

2

21

1

23

CCCB

BCCCBE

A

k

k

ktzyxu

(54)




221

2

2211

221

221

2

1

31

222

12

)]cosh()sinh([

)]sinh()cosh([12

)]sinh()cosh([

)]cosh()sinh([

4

48

]409[5

1)(

CCA

CCk

CC

CC

A

k

kAkkA

(55)



221

2

221

21

221

221

2

21

31

22

124

)]cosh()sinh([

)]sinh()cosh([12

)]sinh()cosh([

)]cosh()sinh([

4

48

]409[5

1),,(

CCA

CCk

CC

CC

A

k

kAkkA

tyxu

(56)

where

}17640{5

2 422

2614

1

21 AkkAk

tykxk (57)





5342

.

)]cos()sin([

)]sin()cos([12

)]sin()cos([

)]cos()sin([

4

48

]409[5

1)(

221

2

2211

221

221

2

1

31

222

12

CCA

CCk

CC

CC

A

k

kAkkA

(58)



.

)]cos()sin([

)]sin()cos([12

)]sin()cos([

)]cos()sin([

4

48]409[

5

1),,(

221

2

221

21

221

221

2

213

12

2

125

CCA

CCk

CC

CC

A

kkAk

kAtyxu

(59)


3.2. Numerical solutions for KdV equation

In this section we give some figures to illustrate some of our results which obtained in this section. To this end , we select some

special values of the parameters to show the behavior of the extended rational ( GG / ) expansion method for the KdV

equation.

5 10 15 20x

0.705

0.710

0.715

0.720

0.725

0.730

u1

Figure 1. The exact extended ( GG / ) expansion solution 1U in Eq. (24) and its projection at 0t when the parameters take

special values ,1E ,21C ,2A B=5, ,1C 5,3,1,3,11,7,3,5 321 yzkkk



5343

5 10 15 20x

40

60

80

100

u2


special values ,1E ,21C ,32 C ,1A B=1, ,10C 5,3,1,3,11,7,3,5 321 yzkkk

.

5 10 15 20x

5

10

15

20

25

30

35

u3



5 10 15 20x

2.38

2.40

2.42

2.44

u4



.



5344

5 10 15 20x

40

30

20

10

u5



CONCLUSION

In this paper we use the generalized ( GG / ) expansion

method and generalized Kudryashov method to construct a

series of some new traveling wave solutions for some

nonlinear integro- partial differential equations in the

mathematical physics. We constructed the rational exact

solutions in many different functions such as hyperbolic

function solutions, trigonometric function solutions and

rational exact solution. The performance of this method

reliable, effective and powerful for solving the nonlinear

partial differential equations.

Conflict of Interests The authors declare that there is no

conflict of interests regarding the publication of this paper.

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