extended rational expansion method for nonlinear partial ... · expansion method for nonlinear...

Published by World Academic Press, World Academic Union

ISSN 1746-7659, England, UK Journal of Information and Computing Science

Vol. 11, No. 1, 2016, pp. 030-057

Extended rational GG / expansion method for nonlinear partial differential equations

Khaled A. Gpreel 2,1 , Taher A. Nofal2,3 and Khulood O. Alweail2 1 Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt.

2 Mathematics Department, Faculty of Science, Taif University, Kingdom Saudi Arabia. 3Mathematics Department, Faculty of Science, El-Minia University, Egypt.

(Received August 22, 2015, accepted October 17, 2015)

Abstract. In this article, we use the extended rational ( GG / )- expansion function method to construct exact solutions for some nonlinear partial differential equations in mathematical physics via the (2+1)-dimensional Wu-Zhang equations, the KdV equation, and generalized Hirota–Satsuma coupled KdV equation in terms of the hyperbolic functions, trigonometric functions and rational function, when G satisfies a nonlinear second order ordinary 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: The extended rational( GG / )- expansion function method, Traveling wave solutions, The (2+1)-dimensional Wu-Zhang equations, The KdV equation, The generalized Hirota–Satsuma coupled KdV equation.

1. Introduction Nonlinear partial differential equations play an important role in describing the various phenomena not

only in physics, but also in biology and chemistry, and several other fields of science and engineering. It is one of the important jobs in the study of the nonlinear partial differential equations are searching for finding the traveling wave solutions. 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 . In this paper we use the extended rational ( GG / )- expansion function method when G satisfies a nonlinear differential equations

,0)( 22 GCEGGBGGAG where ECBA ,,, are real arbitrary constants to find the traveling wave solutions for some nonlinear partial differential equations in mathematical physics namely the (2+1)-dimensional Wu-Zhang equations, the KdV equation and the generalized Hirota–Satsuma coupled KdV equation. We obtain some new kind of traveling wave solutions when the parameter are taking some special values.

2. Description of the extended rational ( GG / ) expansion function method for NPDEs In this part of the manuscript, the extended rational ( GG / ) expansion method will be given. In order

to apply this method to nonlinear partial differential equations we consider the following steps. Step 1. We consider the nonlinear partial differential equation, say in two independent variables x and t is given by

Journal of Information and Computing Science, Vol. 11(2016) No. 1, pp 030-057

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31

,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:

,),( ktxUu (2) where k 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 , we obtain the hyperbolic exact solution of Eq.(5) takes the following form:

)(2

2

2

1

)(22

)(2)

2)(4

sinh()2

)(4cosh(

)](4[

)(CA

A

CA

A

CA

B

CA

CAEBC

A

CAEBC

CAEB

eG

(6)

where 1C and 2C are arbitrary constants. In this case the ratio between Gand G takes the form

)2

)(4sinh()

2)(4

cosh(

)2

)(4cosh()

2)(4

sinh(

)(2)(4

)(2 2

2

2

1

2

2

2

12

A

CAEBC

A

CAEBC

A

CAEBC

A

CAEBC

CA

CAEB

CA

B

G

G (7)

(ii) When 0B , 0)(42 CAEB , we obtain the trigonometric exact solution of Eq.(5) takes the form

)2

)(4sin()

2)(4

cos(

)2

)(4cos()

2)(4

sin(

)(2)](4

)(2 2

2

2

1

2

2

2

12

A

CAEBC

A

CAEBC

A

CAEBC

A

CAEBC

CA

CAEB

CA

B

G

G

(8)

Khaled A. Gpreel et.al : Extended rational GG / expansion method for nonlinear partial differential equations

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32

(iii) When 0B , 0)(42 CAEB , we obtain the rational exact solution of Eq.(5) takes the form

21

2)(2 CC

C

CA

B

G

G

(9)

(iv) When 0B , 0)(4 CAE , we obtain the hyperbolic exact solution of Eq.(5) takes the following form:

)2

)(4sinh()

2)(4

cosh(

)2

)(4cosh()

2)(4

sinh(

)(2)(4

21

21

A

CAEC

A

CAEC

A

CAEC

A

CAEC

CA

CAE

G

G (10)

(v) When 0B , 0)(4 CAE , we obtain the hyperbolic exact solution of Eq.(5) takes the following form:

)2

)(4sin()

2)(4

cos(

)2

)(4cos()

2)(4

sin(

)(2)](4

21

21

A

CAEC

A

CAEC

A

CAEC

A

CAEC

CA

CAE

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

3. Applications of extended rational ( GG / ) expansion method for NPDEs In this section, we use the extended rational ( GG / ) expansion method to find the exact solutions for

nonlinear evolution equations in mathematical physics namely the (2+1)-dimensional Wu-Zhang equations, the KdV equation and the generalized Hirota–Satsuma coupled KdV equation which are very important in the mathematical science and have been paid attention by many researchers in physics and engineering. 3.1. Example 1 . Extended rational ( GG / )- expansion method for KdV equation In this section , we study the exact solution of the following KdV equation:-

0u t xxxxx uuuu (12)

where and are arbitrary constant. The Korteweg–de Vries equation is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, an harmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves [25]. Let us assume the traveling wave solutions of Eq. (12) in the following form:

)(),( Utxu , ktx (13) where k is an arbitrary constant. The transformation (13) permits us to convert PDE (12) to the following ODE :-

0)( UUUUk (14) By balancing the highest order derivative term and nonlinear term in Eq. (14), we suppose the solution of Eq. (14) has the following form:

2

2

4

2

2

321

0

)()(

)()(1

)()(1

)()(

)()(

)()(1

)()(1

)()(

G

G

G

Ga

G

G

G

Ga

G

G

G

Ga

G

G

G

Ga

aU . (15)

where 3210 ,,, aaaa and 4a are constants to be determined later. Substituting Eq. (15) along with (5) into Eq. (14) and cleaning the denominator and collecting all terms with the same order of ( )(/)( GG ) together, the left hand side of Eq. (14) are converted into polynomial in ( )(/)( GG ). Setting each coefficient of these polynomials to be zero , we derive a set of algebraic equations for



33

kaaaaa ,,,,,,,, 43210 and . Solving the set of algebraic equations by using Maple or Mathematica , software package to get the following results: Case 1.

,12,)2(12,)881212(2

2

4232

22222

0A

Ea

A

BEEa

A

AkAEACEBBEEa

021 aa , where ,,,,,, AEBC and are arbitrary constants. In this case the traveling wave solutions of the KdV equation take the following form:

Family 1. When 0B , 0)(42 CAEB , we obtain the hyperbolic exact solution of Eq.(15) takes the following:

.

2)

sinh(][)2

)cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[(12

)2

sinh(]))(2[()2

cosh(]))(2[(

2)

sinh(][)2

)cosh(][)2(12

)881212(

2

12212

2

12212

2

12212

2

1221

2

22222

1

ACBC

ACBCA

ACCBCA

ACCBCAE

ACCBCA

ACCBCAA

ACBC

ACBCBEE

A

AkAEACEBBEEU

(17)

Family 2. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solution of Eq.(15) takes the following form

2

12212

2

1221

2

22222

2

)2

sin(]))(2[()2

cos(]))(2[(

2sin(][)

2cos(][)2(12

)881212(

ACCBCA

ACCBCAA

ACBC

ACBCBEE

A

AkAEACEBBEEU

2

12212

2

12212

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[(12

ACBC

ACBCA

ACCBCA

ACCBCAE

(18)

Family 3. When 0B , 0)(42 CAEB , we obtain the rational exact solution of Eq.(15) takes the following form:



34

.)()(2

)(2))(2)((12

)(2))(2)(()()(2)2(12

)881212(

2212

2

2221

2

2221

2

2212

2

22222

3

CCBCACA

CACBCACCE

CACBCACCA

CCBCACBEE

A

AkAEACEBBEEU

(19)

There are other families of exact solutions which omitted here for convenience . Case 2.

2

22

22

22222

0)(12,)881212(

A

ACBEa

A

AkAEACEBBEEa

2

3222

1)2223(12

A

EAECEEBCBABBa

, (20)

043 aa where ,,,,,, AEBC and are arbitrary constants. In this case the following traveling wave solutions of nonlinear KdV equation takes the following form:

Family 4. When 0B , 0)(42 CAEB , we obtain the hyperbolic exact solution of Eq.(15) takes the following form:

)2

sinh(][)2

cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[()(12

)2

sinh(]))(2[()2

cosh(]))(2[(

)2

sinh(][)2

cosh(][)2223(12

)881212(

12212

122122

12212

12213222

2

22222

4

ACBC

ACBCA

ACCBCA

ACCBCAACBE

ACCBCA

ACCBCAA

ACBC

ACBCEAECEEBCBABB

A

AkAEACEBBEEU

(21)

Family 5. When 0B , 0)(42 CAEB , we obtain the trigonometric traveling wave solution of Eq.(15) takes the following form:

.

2)

sin(][)2

)cos(][

)2

sin(]))(2[()2

cos(]))(2[()(12

)2

sin(]))(2[()2

cos(]))(2[(

)2

sin(][)2

cos(][)2223(12

)881212(

12212

122122

12212

12213222

2

22222

5

ACBC

ACBCA

ACCBCA

ACCBCAACBE

ACCBCA

ACCBCAA

ACBC

ACBCEAECEEBCBABB

A

AkAEACEBBEEU

(22)



35

Family 6. When 0B , 0)(42 CAEB , we obtain the rational exact solution of Eq.(15) takes the form:

.)()(2

)(2))(2)(()(12)(2))(2)((

)()(2)2223(12

)881212(

2122

22122

2212

2123222

2

22222

6

CCBCACA

CACBCACCACBE

CACBCACCA

CCBCACEAECEEBCBABB

A

AkAEACEBBEEU

(23)

There are other families of exact solutions which omitted here for convenience . 3.2. Numerical solutions for the exact solutions of the 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. (17) and its projection at 0t when the parameters take special values ,2E ,5.01C ,3A ,1C ,5.2k ,75.02 C ,75.1

,1B ,2.2 25.1 and 25.0 .

5 10 15 20x

40

60

80

100u2

Figure 2. The exact extended ( GG / ) expansion solution 2U in Eq. (18) and its projection at 0t when the parameters take special values ,2E ,5.01C ,1A ,3C ,5.2k ,75.02 C ,75.1

,1B ,2.2 25.1 and 25.0 .



36

5 10 15 20x

5

10

15

20

25

30

35

u3

Figure 3.The exact extended ( GG / ) expansion solution 3U in Eq. (19) and it s projection at 0t when t he

parameters take special values ,2E ,5.01C ,3A ,1C ,5.2k ,75.02 C ,75.1 ,1B ,2.2 25.1 and 25.0 .

5 10 15 20x

2.38

2.40

2.42

2.44u4

Figure 4. The exact extended ( GG / ) ex pansion solution 4U in Eq. (21) and it s pro jection at 0t when t he parameters take special values ,2E ,5.01C ,3A ,1C ,5.2k ,75.02 C ,75.1

,1B ,2.2 25.1 and 25.0 .

5 10 15 20x

40

30

20

10

u5

Figure 5. The e xact extended ( GG / ) expansion so lution 5U in E q. (22) a nd its pro jection at 0t when th e




37

5 10 15 20x

2

4

6

8

10u6

Figure 6. The exact extended ( GG / ) expansion solution 6U in Eq. ( 23) and its pro jection at 0t when t he


3.3. Exam ple 2. Ex tended rational ( GG / ) expansion funct ion method for the (2+1)-dimensional Wu-Zhang equations In this section, we study the (2+1)-dimensional Wu-Zhang equations [26,27].

.0)(31)()(

,0

,0

yyyxxyxyyxxxyxt

yyxt

xyxt

vvuuuwvuw

wvvvuv

wuvuuu

(24)

where w is the elevation of the water, is the surface velocity of water along -direction, and is the surface velocity of water along -direction. The explicit solutions of Eqs. (24) are very helpful for costal and civil engineers to apply the nonlinear water wave model in harbor and coastal design [26,27]. Let us assume the traveling wave solutions of Eqs. (24) in the following forms:

,),(),,(),(),,(),(),,( tkyxWtyxwVtyxvUtyxu (25) where k is an arbitrary constant. The transformations (25) permit us to convert NPDE’s (24) to the following NODE’s :

,0)(32

,0,0

LVUUWUVWk

WVVVUVkWUVUUUk

(26)

where L is the integration constant. By balancing the highest order derivative terms and nonlinear terms in Eqs. (26), we suppose that Eqs. (26) have the following solutions :

,

)()(

)()(1

)()(

)()(1

)()(1

)()(

)()(1

)()(

,

)()(

)()(1

)()(1

)()(

,

)()(

)()(1

)()(1

)()(

2

2

43

2

2

21

0

21

0

21

0

GG

GGc

GG

GGc

GG

GGc

GG

GGc

cW

GG

GGb

GG

GGb

bV

GG

GGa

GG

GGa

aU

(27)

where 3210210210 ,,,,,,,,, ccccbbbaaa and 4c are arbitrary c onstants to be deter mined later. Su bstituting Eqs. (27) along with (5) into Eqs. (26) and cleaning the denominator and collecting all terms with the same



38

order of ( )(/)( GG ) together, the left hand side of Eqs. (26) are converted into polynomials in ( )(/)( GG ). Setting each coefficient of these polynomials to be zero , we get a set of algebraic equations for ,, 10 aa Lkcccccbbba ,,,,,,,,,, 432102102 and . Solving these set of algebraic equations by using the software backage such as Maple or Mathematica , we get the following results: Case1.

,422236

63,

362,

3)(62

002

2

1

EBAAaAk

Ab

A

Ea

A

BACEa

,)(3

8,3

62,3

)(62 22222

2

1 BACEA

cA

Eb

A

BACEb

],612126126

9331661646336[

61

00

222220

20

22220

220

BAaAEkAEaABAEkBA

kAAkAaBEAaEBkAaAA

c

,42)(36)(2

02

21

EBAAaAkBACE

Ac

,42)(362

023

EBAAaAk

A

Ec ,

38

2

2

4A

Ec

)],122748272748129279948

48(36181836363672128

1281283618192646464[631

20

30

220

320

3220

3233330

30

20

2220

220

2220

233

22222223

AkBkaA

EaAkaAkaAkEABAakAkAkAaAECAa

AkECBaABkABkaAEaAEkAEkaAE

AECEEkABkABECEBEABEBA

L

(28) where ,,,,,, 0 ECBAak are arbitrary constants. In this case the following traveling wave solutions of the

(2+1)-dimensional Wu-Zhang equations take the following form: Family 1. When 0B , 0)(42 CAEB , we obtain the hyperbolic exact solution of Eq.(27) takes the following form:

.

2sinh(][)

2cosh(][3

)2

sinh(]))(2[()2

cosh(]))(2[(62

)2

sinh(]))(2[()2

cosh(]))(2[(3

2sinh(][)

2cosh(][)(62

1221

1221

1221

12212

01

ACBC

ACBCA

ACCBCA

ACCBCAE

ACCBCA

ACCBCAA

ACBC

ACBCBACE

aU

)2

sinh(]12))(2[()2

cosh(]21))(2[(3

2sinh(]12[)

2cosh(]21[)2(62

42102236

63

1

ACCBCA

ACCBCAA

ACBC

ACBCBACE

EBakA

AV



39

.

2sinh(]12[)

2cosh(]21[3

)2

sinh(]12))(2[()2

cosh(]21))(2[(62

ACBC

ACBCA

ACCBCA

ACCBCAE

.2

2sinh(]12[)

2cosh(]21[23

2)

2sinh(]12))(2[()

2cosh(]21))(2[(28

2sinh(]12[)

2cosh(]21[23

)2

sinh(]12))(2[()2

cosh(]21))(2[()42(3)0(62

2)

2sinh(]12))(2[()

2cosh(]21))(2[(23

2

2sinh(]12[)

2cosh(]21[2)2(8

)2

sinh(]12))(2[()2

cosh(]21))(2[(23

2sinh(]12[)

2cosh(]21[)42(3)0(6)2(2

]06120126126

292232203162

0622162420623

36[

26

11

ACBC

ACBCA

ACCBCA

ACCBCAE

ACBC

ACBCA

ACCBCA

ACCBCAEBAAaAkE

ACCBCA

ACCBCAA

ACBC

ACBCBACE

ACCBCA

ACCBCAA

ACBC

ACBCEBAAaAkBACE

BAaAEkAEaABAEkBA

kAAkAaBEAaEBkAaA

A

W

(29) Family 2. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solution of Eq.(27) takes the following form:

.

2sin(][)

2cos(][3

)2

sin(]))(2[()2

cos(]))(2[(62

)2

sin(]))(2[()2

cos(]))(2[(3

2sin(][)

2cos(][)(62

1221

1221

1221

12212

02

ACBC

ACBCA

ACCBCA

ACCBCAE

ACCBCA

ACCBCAA

ACBC

ACBCBACE

aU



40

.

2sin(][)

2cos(][3

)2

sin(]))(2[()2

cos(]))(2[(62

)2

sin(]))(2[()2

cos(]))(2[(3

2sin(][)

2cos(][)(62

422236

63

1221

1221

1221

12212

02

ACBC

ACBCA

ACCBCA

ACCBCAE

ACCBCA

ACCBCAA

ACBC

ACBCBACE

EBAAaAkA

V

.

2sin(][)

2cos(][3

)2

sin(]))(2[()2

cos(]))(2[(8

2sin(][)

2cos(][3

)2

sin(]))(2[()2

cos(]))(2[()42(3)(62

)2

sin(]))(2[()2

cos(]))(2[(3

2sin(][)

2cos(][)(8

)2

sin(]))(2[()2

cos(]))(2[(3

2sin(][)

2cos(][)42(3)(6)(2

]612126126

9331661646336[

61

2

12212

2

12212

12212

12210

2

12212

2

122122

12212

122102

00

222220

20

22220

222

ACBC

ACBCA

ACCBCA

ACCBCAE

ACBC

ACBCA

ACCBCA

ACCBCAEBAAaAkE

ACCBCA

ACCBCAA

ACBC

ACBCBACE

ACCBCA

ACCBCAA

ACBC

ACBCEBAAaAkBACE

BAaAEkAEaABAEkBA

kAAkAaBEAaEBkAaAA

W

(30)

Family 3. When 0B , 0)(42 CAEB , we obtain the rational exact solution of Eq.(27) takes the following form:

.)()(23

)(2))(2)((62

)(2))(2)((3)()(2)(62

212

221

221

2122

03

CCBCACA

CACBCACCE

CACBCACCA

CCBCACBACEaU

.)()(23

)(2))(2)((62

)(2))(2)((3)()(2)(62

422236

63

212

221

221

2122

03

CCBCACA

CACBCACCE

CACBCACCA

CCBCACBACE

EBAAaAkA

V



41

.

)()(23

)(2))(2)((8

)()(23

)(2))(2)(()42(3)(62

)(2))(2)((3

)()(2)(8

)(2))(2)((3

)()(2)42(3)(6)(2

]612126126

9331661646336[

61

2212

2

2221

2212

22210

2221

2

2212

22221

22120

200

222220

20

22220

223

CCBCACA

CACBCACCE

CCBCACA

CACBCACCEBAAaAkE

CACBCACCA

CCBCACBACE

CACBCACCA

CCBCACEBAAaAkBACE

BAaAEkAEaABAEkBA

kAAkAaBEAaEBkAaAA

W

(31)

There are other families of exact are omitted here for convenience . Case 2.

,24)(36

63

0

BEAkA

Aa ),(

362 2

1 BACEA

a

,2436

63

0

BEA

Ab ,

362 2

1 BACEA

b

),84824324(3

1 222220 CEBEAEBkAE

Ac

),222222(3

8 22222232422 CACABABCBEAECBEE

Ac

].89696228)128121632

88192163236181836(3663[1

22223332

2222222333

CAEAEEABABEAEBBCECE

EBBEBEAEAEABABkAEkAkAAA

L

.043122 cccba

(32) where ,,,,, kECBA are arbitrary constants . In this case the following traveling wave solutions of the (2+1)-dimensional Wu-Zhang equations take the following form: Family 4. When 0B , 0)(42 CAEB , we obtain the hyperbolic exact solution of Eq.(15) takes the following form:

.)

2sinh(]))(2[()

2cosh(]))(2[(3

2sinh(][)

2cosh(][)(62

24)(36

63

1221

12212

4

ACCBCA

ACCBCAA

ACBC

ACBCBACE

BEAkAA

U

.)

2sinh(]))(2[()

2cosh(]))(2[(3

2sinh(][)

2cosh(][)(62

2436

63

1221

12212

4

ACCBCA

ACCBCAA

ACBC

ACBCBACE

BEAA

V



42

2

12212

2

1221

222222324

2222

24

)2

sinh(]))(2[()2

cosh(]))(2[(3

2sinh(][)

2cosh(][

)222222(8

)84824324(3

1

ACCBCA

ACCBCAA

ACBC

ACBC

CACABABCBEAECBEE

CEBEAEBkAEA

W

(33)

Family 5. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solution of Eq.(27) takes the following form

.)

2sin(]))(2[()

2cos(]))(2[(3

2sin(][)

2cos(][)(62

24)(36

63

1221

12212

5

ACCBCA

ACCBCAA

ACBC

ACBCBACE

BEAkAA

U

.)

2sin(]))(2[()

2cos(]))(2[(3

2sin(][)

2cos(][)(62

2436

63

1221

12212

5

ACCBCA

ACCBCAA

ACBC

ACBCBACE

BEAA

V

.

)2

sin(]))(2[()2

cos(]))(2[(3

2sin(][)

2cos(][

)222222(8

)84824324(3

1

2

12212

2

1221

222222324

2222

25

ACCBCA

ACCBCAA

ACBC

ACBC

CACABABCBEAECBEE

CEBEAEBkAEA

W

(34) Family 6. When 0B , 0)(42 CAEB , we obtain the rational exact solution of Eq.(27) takes the following form:

.)(2))(2)((3

)()(2)(6224)(36

63

221

2122

6CACBCACCA

CCBCACBACEBEAkA

AU

.)(2))(2)((3

)()(2)(622436

63

221

2122

6CACBCACCA

CCBCACBACEBEA

AV



43

.)(2))(2)((3

)()(2)222222(8

)84824324(3

1

2221

2

2212

222222324

222226

CACBCACCACCBCACCACABABCBEAECBEE

CEBEAEBkAEA

W

(35) There are other cases of exact solutions are omitted here for convenience . 3.4. Numerical solutions for the exact solutions for (2+1)-dimensional Wu-Zhang equations In this section we give some figures to illustrate the behavior of the exact solutions which obtained in above section To this end , we select so me special values of th e parameters to show the behavior of extended rational ( GG / ) - expansion method for (2+1)-dimensional Wu-Zhang equations

5 10 15 20x

7.3668

7.3666

7.3664

u1

5 10 15 20x

1.00010

1.00005

1.00000

v1

5 10 15 20

x

6.50003

6.50002

6.50001

6.50000

w1

Figure 7. The exact extended ( GG / ) expansion solutions 11,VU and 1W in E q. (29) and it s projection at

0t when the parameters take special values ,3A ,1C ,2E ,1B ,5.01 C ,3y ,5.10 a

,75.02 C 5.2k and .25.0

5 10 15 20x

50

40

30

20

10

10

u2

5 10 15 20x

10

5

v2

5 10 15 20x

200

100

100

w2



44

Figure 8. The exact extended ( GG / ) expansion s olutions 22 ,VU and 2W in Eq. (30) a nd it s projection a t

0t when the parameters take special values ,1B ,1A ,3C ,2E ,5.01C ,3y ,5.10 a

,75.02 C 5.2k and .25.0

5 10 15 20x

0.5

1.0

1.5

2.0

2.5

3.0

u3

0 5 10 15 20x

2.4

2.6

2.8

3.0

3.2

v3

5 10 15 20

x

6

8

10

12

w3

Figure 9. The exact extended ( GG / ) expansion s olutions 33 ,VU and 3W in Eq. (31) a nd it s projection a t

0t when the parameters take special values ,3A ,1C ,2E ,1B ,5.01C ,3y ,5.10 a

,75.02 C 5.2k and .25.0

5 10 15 20x

4.62220

4.62222

4.62224

4.62226u4

5 10 15 20x

3.24440

3.24445

3.24450

v4

5 10 15 20x

2.44785

2.44780

2.44775

w4

Figure 10. The exact extended ( GG / ) expansion s olutions 44 ,VU and 4W in Eq. ( 33) and it s pr ojection at


,75.02 C 5.2k and .25.0



45

5 10 15 20x

5

5

10

15

20

u5

5 10 15 20x

40

30

20

10

10

20

v5

5 10 15 20x

200

150

100

50

w5

Figure 11. The exact extended ( GG / ) expansion s olutions 55 ,VU and 5W in Eq. ( 34) and it s pro jection a t


,75.02 C 5.2k and .25.0

5 10 15 20x

2.90

2.95

3.00

u6

5 10 15 20

x

1.8

1.9

2.0

v6

5 10 15 20x

0.45

0.40

0.35

w6

Figure 12. The exact extended ( GG / ) expansion s olutions 66 ,VU and 6W in Eq. ( 35) and it s pr ojection at


,75.02 C 5.2k and .25.0 3.5. Example 3. Extended ( GG / ) expansion method for generalized Hirota–Satsuma coupled KdV equations In th is sec tion we study t he following gen eralized Hirota–Satsuma coupled KdV equ ations by use the extended rational ( GG / ) expansion method [28].

0303

0)(3321

xxxxt

xxxxt

xxxxxt

uwwwuvvv

vwuuuu

(36)



46

The Hirota-Satsuma equations are widely used as models to describe complex physical phenomena in various fields of science, especially in fluid mechanics, solid stat physics, plasma physics. Various methods have been used to explore different kinds of solutions of physical models described by nonlinear PDEs [29]. Let us assume the traveling wave solutions of Eqs (36) in the following forms:

tkxWtxwVtxvUtxu ),(),(),(),(),(),( (37) where k is an arbitrary constant. Substituting (37) into Eqs. (36), we have:

0303

0)(3321

WUWWk

VUVVk

WVWVUUUUk

(38)

By balancing the highest order derivative terms and nonlinear terms in Eqs. (38), we suppose that Eqs. (38) own the solutions in the following:

,

)()(

)()(1

)()(

)()(1

)()(1

)()(

)()(1

)()(

,

)()(

)()(1

)()(

)()(1

)()(1

)()(

)()(1

)()(

,

)()(

)()(1

)()(

)()(1

)()(1

)()(

)()(1

)()(

2

2

43

2

2

21

0

2

2

43

2

2

21

0

2

2

43

2

2

21

0

G

G

G

Gc

G

G

G

Gc

G

G

G

Gc

G

G

G

Gc

cW

G

G

G

Gb

G

G

G

Gb

G

G

G

Gb

G

G

G

Gb

bV

G

G

G

Ga

G

G

G

Ga

G

G

G

Ga

G

G

G

Ga

aU

(39)

where 32104321043210 ,,,,,,,,,,,,, ccccbbbbbaaaaa and 4c are constants to be determined later. Substituting Eqs. (39) along with (5) into Eqs. (38) and cleaning the denominator and collecting all terms with the same order of ( )(/)( GG ) together, the left hand side of Eqs. (38) are converted into polynomials in ( )(/)( GG ). Setting each coefficient of these polynomials to be zero , we derive a set of algebraic equations for kcccccbbbbbaaaaa ,,,,,,,,,,,,,,, 432104321043210 and . Solving the set of algebraic equations by using Maple or Mathematica , software backage to get the following results: Case 1:

,3

881212),2232(22

2222

02232

21A

BECEAEBEkAaABCBBAEECEBE

Aa

),222222(2 422222322

22 EACEACEACABBCEBBA

a

),( 231 CEAB

E

bb

23)2(2

A

BEEa

, 2

2

42A

Ea , (40)

),448267220408

12812440208(3

20

333

23

2233

23

33

20

20

022

00232

32

332

3423

0

kEAbEbEBbBEbCBEbCEbBbEAbEBb

BbECEbbEBAkbAEbBEAbAEkbAb

Ec

),)(1288412(3

22222

341 CEABBEECEAkAEBbA

Ec



47

).1288412(3

2222

34

2

3 BEECEAkABEbA

Ec

04242 ccbb

where 30 ,,,,,, bbAEBC and k are arbitrary constants. In this case the following traveling wave solutions of the generalized Hirota–Satsuma coupled KdV equations take the following forms:

Family 1. When 0B , 0)(42 CAEB , we obtain the hyperbolic exact solutions of Eqs.(39) take the following forms:



48

.

2sinh(][)

2cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[(2

2sinh(][)

2cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[()2(2

)2

sinh(]))(2[()2

cosh(]))(2[(

2sinh(][)

2cosh(][)222222(2

)2

sinh(]))(2[()2

cosh(]))(2[(

2sinh(][)

2cosh(][)2232(2

3881212

2

12212

2

12212

12212

1221

2

12212

2

1221422222322

12212

12212232

2

2222

1

ACBC

ACBCA

ACCBCA

ACCBCAE

ACBC

ACBCA

ACCBCA

ACCBCABEE

ACCBCA

ACCBCAA

ACBC

ACBCEACEACEACABBCEBB

ACCBCA

ACCBCAA

ACBC

ACBCABCBBAEECEBE

A

BECEAEBEkAU

.

2sinh(][)

2cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[(

)2

sinh(]))(2[()2

cosh(]))(2[(

2sinh(][)

2cosh(][)(

1221

12213

1221

12212

3

01

ACBC

ACBC

ACCBCA

ACCBCAb

ACCBCA

ACCBCAE

ACBC

ACBCCEABb

bV

.

2)

sinh(][)2

)cosh(][3

)2

sinh(]))(2[()2

cosh(]))(2[()1288412(

)2

sinh(]))(2[()2

cosh(]))(2[(3

2)

sinh(][)2

)cosh(][))(1288412(

)448267220408

12812440208(3

122134

122122222

122134

122122222

20

333

23

2233

23

33

20

20

022

00232

32

332

3423

1

ACBC

ACBCbA

ACCBCA

ACCBCABEECEAkABEE

ACCBCA

ACCBCAbA

ACBC

ACBCCEABBEECEAkAEBE



EW

(41)

Family 2. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solutions of Eqs.(39) take the following forms:



49

.

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[(2

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[()2(2

)2

sin(]))(2[()2

cos(]))(2[(

2sin(][)

2cos(][)222222(2

)2

sin(]))(2[()2

cos(]))(2[(

2sin(][)

2cos(][)2232(2

3881212

2

12212

2

12212

12212

1221

2

12212

2

1221422222322

12212

12212232

2

2222

2

ACBC

ACBCA

ACCBCA

ACCBCAE

ACBC

ACBCA

ACCBCA

ACCBCABEE

ACCBCA

ACCBCAA

ACBC

ACBCEACEACEACABBCEBB

ACCBCA

ACCBCAA

ACBC

ACBCABCBBAEECEBE

A

BECEAEBEkAU

.

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[(

)2

sin(]))(2[()2

cos(]))(2[(

2sin(][)

2cos(][)(

1221

12213

1221

12212

3

02

ACBC

ACBC

ACCBCA

ACCBCAb

ACCBCA

ACCBCAE

ACBC

ACBCCEABb

bV

.

2sin(][)

2cos(][3

)2

sin(]))(2[()2

cos(]))(2[()1288412(

)2

sin(]))(2[()2

cos(]))(2[(3

2sin(][)

2cos(][))(1288412(

)448267220408

12812440208(3

122134

122122222

122134

122122222

20

333

23

2233

23

33

20

20

022

00232

32

332

3423

2

ACBC

ACBCbA

ACCBCA

ACCBCABEECEAkABEE

ACCBCA

ACCBCAbA

ACBC

ACBCCEABBEECEAkAEBE



EW

(42) Family 3. When 0B , 0)(42 CAEB , we obtain the rational exact solutions of Eqs.(39) take the forms:



50

.)()(2

)(2))(2)((2

)()(2)(2))(2)(()2(2

)(2))(2)(()()(2)222222(2

)(2))(2)(()()(2)2232(2

3881212

2212

2

2221

2

2122

221

2221

2

2212

422222322

2212

2122232

2

2222

3

CCBCACA

CACBCACCE

CCBCACA

CACBCACCBEE

CACBCACCA

CCBCACEACEACEACABBCEBB

CACBCACCA

CCBCACABCBBAEECEBE

A

BECEAEBEkAU

.)()(2

)(2))(2)(()(2))(2)((

)()(2)(

212

2213

221

2122

303

CCBCAC

CACBCACCb

CACBCACCE

CCBCACCEABbbV

.)()(23

)(2))(2)(()1288412()(2))(2)((3

)()(2))(1288412()448267220408

12812440208(3

21234

22122222

22134

21222222

20

333

23

2233

23

33

20

20

022

00232

32

332

3423

3

CCBCACbA

CACBCACCBEECEAkABEE

CACBCACCbA

CCBCACCEABBEECEAkAEBE



EW

(43) There are other cases of exact solutions are omitted here for convenience . Case 2:

,3

8812122

2222

0A

BECEAEBEkAa

,)2(423

A

BEEa

,42

2

4A

Ea

,)2(4

3E

BEbb

,)2(4

44

3

3bA

BEEc

,4

44

4

4bA

Ec

),64121288(3

2 204

224

22444442

40 EbbkABbEbEBbCEbAEb

Ab

Ec

.0212121 ccbbaa (44)

where 40 ,,,,,, bbAEBC and k are arbitrary constants. In this case the following traveling wave solutions of the generalized Hirota–Satsuma coupled KdV equations take the following forms:

Family 4. When 0B , 0)(42 CAEB , we obtain the hyperbolic exact solutions of Eqs.(39) take the following forms:



51

.

2sinh(][)

2cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[(4

2sinh(][)

2cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[()2(4

3881212

2

12212

2

12212

12212

1221

2

2222

4

ACBC

ACBCA

ACCBCA

ACCBCAE

ACBC

ACBCA

ACCBCA

ACCBCABEE

A

BECEAEBEkAU

.

2sinh(][)

2cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[(

2sinh(][)

2cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[()2(

2

1221

2

12214

1221

12214

04

ACBC

ACBC

ACCBCA

ACCBCAb

ACBC

ACBCE

ACCBCA

ACCBCABEb

bV

.

2)

sinh(][)2

)cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[(4

2)

sinh(][)2

)cosh(][

)2

sinh(]))(2[()2

cosh(]))(2[()2(4

)64121288(3

2

2

122144

2

12214

122144

12213

204

224

22444442

4

2

4

ACBC

ACBCbA

ACCBCA

ACCBCAE

ACBC

ACBCbA

ACCBCA

ACCBCABEE

EbbkABbEbEBbCEbAEbAb

EW

(45)

Family 5. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solutions of Eqs.(39) take the following forms:

.

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[(4

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[()2(4

3881212

2

12212

2

12212

12212

1221

2

2222

5

ACBC

ACBCA

ACCBCA

ACCBCAE

ACBC

ACBCA

ACCBCA

ACCBCABEE

A

BECEAEBEkAU



52

.

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[(

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[()2(

2

1221

2

12214

1221

12214

05

ACBC

ACBC

ACCBCA

ACCBCAb

ACBC

ACBCE

ACCBCA

ACCBCABEb

bV

.

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[(4

2sin(][)

2cos(][

)2

sin(]))(2[()2

cos(]))(2[()2(4

)64121288(3

2

2

122144

2

12214

122144

12213

204

224

22444442

4

2

5

ACBC

ACBCbA

ACCBCA

ACCBCAE

ACBC

ACBCbA

ACCBCA

ACCBCABEE


EW

(46) Family 6. When 0B , 0)(42 CAEB , we obtain the rational exact solutions of Eqs.(39) take the forms:

.)()(2

)(2))(2)((4

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

3881212

2212

2

2221

2

2122

221

2

2222

6

CCBCACA

CACBCACCE

CCBCACA

CACBCACCBEE

A

BECEAEBEkAU

.)()(2

)(2))(2)((

)()(2)(2))(2)(()2(

2212

22214

212

221406

CCBCAC

CACBCACCb

CCBCACE

CACBCACCBEbbV

.)()(2

)(2))(2)((4.

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

)64121288(3

2

22124

4

2221

4

21244

2213

204

224

22444442

4

2

6

CCBCACbA

CACBCACCE

CCBCACbA

CACBCACCBEE


EW

(47) There are other families of exact are omitted here for convenience . 3.6. Numerical solutions of the generalized Hirota–Satsuma coupled KdV equations 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 extended ( GG / ) expansion method for the generalized Hirota–Satsuma coupled KdV equations.



53

5 10 15 20x

0.203695

0.203690

0.203685

0.203680

0.203675

0.203670

0.203665u1

5 10 15 20x

1.230

1.235

1.240

v1

5 10 15 20

x

508.5

508.0

507.5

507.0

w1

Figure 13.The exact extended ( GG / ) expansion solutions 11,VU and 1W in Eqs.(41) and its projection at 0t when the parameters take special values ,3A ,1C ,2E ,5.01C ,5.10 b ,75.02 C

,1B ,5.2k 75.13 b and .25.0

5 10 15 20x

20

30

40

50

60

70

80

u2

5 10 15 20x

6

4

2

v2

5 10 15 20x

140

120

100

80

60

40

w2

Figure 14.The exact extended ( GG / ) expansion solutions 22 ,VU and 2W in Eqs. (42) and its projection at 0t when the parameters take special values ,1A ,3C ,2E ,5.01C ,1B ,5.10 b

,75.02 C ,5.2k 75.13 b and .25.0



54

5 10 15 20x

2

4

6

8

u3

5 10 15 20x

2

4

6

8

u3

5 10 15 20x

1000

800

600

400

200

w3

Figure 15.The exact extended ( GG / ) expansion solutions 33,VU and 3W in Eqs. (43) and its projection

at 0t when the parameters take special values ,3A ,1C ,2E ,5.01C ,5.10 b ,75.02 C,5.2k ,1B 75.13 b and .25.0

5 10 15 20x

0.215

0.210

0.205

u4

0 5 10 15 20x

3.345

3.350

3.355

v4

5 10 15 20x

2.244

2.243

2.242

2.241

2.240w4

Figure 16.The exact extended ( GG / ) expansion solutions 44 ,VU and 4W in Eqs. (45) and its projection at 0t when th e parameters take special values ,3A ,1C ,2E ,1B ,5.01C ,5.10 b

,75.02 C ,5.2k 75.14 b and .25.0



55

5 10 15 20x

5

10

15

20u5

5 10 15 20

x

3

4

5

6

v5

5 10 15 20x

20

40

60

80

100

w5

Figure 17.The exact extended ( GG / ) expansion solutions 55 ,VU and 5W in Eqs. (46) and its projection

at 0t when the parameters take special values ,1A ,3C ,2E ,1B ,5.01C ,5.10 b ,75.02 C ,5.2k 75.14 b and .25.0

5 10 15 20x

5

10

15

20

u6

5 10 15 20x

100

200

300

400

v6

5 10 15 20x

2000

4000

6000

8000

w6

Figure 18. The exact extended ( GG / ) expansion solutions 66 ,VU and 6W in Eqs. (47) and its projection

at 0t when the parameters take special values ,1A ,3C ,2E ,1B ,5.01C ,5.10 b ,75.02 C ,5.2k 75.14 b and .25.0

4. Conclusion

In this paper we use the extended ( GG / ) expansion method to construct a series of some new traveling wave solutions for some nonlinear 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.

5. References



56

[1] M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge Univ. Press, Cambridge, 1991.

[2] R.Hirota, Exact solution of the KdV equation for multiple collisions of solutions, Phys. Rev. Letters 27 (1971) 1192-1194.

[3] M.R.Miura, Backlund Transformation,Springer-Verlag, Berlin,1978. [4] A. Bekir , F. Tascan and O. Unsal, Exact solutions of the Zoomeron and Klein Gordon Zahkharov equations, J.

Associat. Arab Univ. Basic Appl. Sci. 17 (2015) 1-5. [5] J.Weiss, M.Tabor and G.Garnevalle, The Paineleve property for partial differential equations, J.Math.Phys. 24

(1983) 522-526 [6] D.S.Wang, Y.J.Ren and H.Q.Zhang, Further extended sinh-cosh and sin-cos methods and new non traveling

wave solutions of the (2+1)-dimensional dispersive long wave equations, Appl. Math.E-Notes, 5 (2005) 157-163. [7] M.L. Wang, Exact solutions for a compound KdV-Burgers equation , Phys. Lett. A 213 (1996) 279-287. [8] K.A. Gepreel and T.A. Nofal, Extended trial equation method for nonlinear partial differential equations, Z.

Naturforsch A70(2015) 269-279. [9] F. Belgacem, H Bulut, H. Baskonus and T. Aktuk, Mathematical analysis of generalized Benjamin and Burger-

KdV equation via extended trial equation method, J. Associat. Arab Univ. Basic Appl. Sci. 16 (2014) 91-100. [10] J.H. He, Homotopy perturbation method for bifurcation of nonlinear wave equations, Int. J. Nonlinear Sci. Numer.

Simul., 6 (2005) 207-208 . [11] E.M.E.Zayed, T.A. Nofal and K.A.Gepreel, The homotopy perturbation method for solving nonlinear Burgers and

new coupled MKdV equations, Zeitschrift fur Naturforschung Vol. 63a (2008) 627 -633. [12] H.M. Liu, Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Chaos,

Solitons & Fractals, 23 (2005) 573 -576. [13] H.A. Abdusalam, On an improved complex tanh -function method, Int. J. Nonlinear Sci.Numer. Simul., 6 (2005)

99-106. [14] E.M.E.Zayed ,Hassan A.Zedan and Khaled A. Gepreel, Group analysis and modified extended Tanh- function to

find the invariant solutions and soliton solutions for nonlinear Euler equations , Int. J. Nonlinear Sci. Numer. Simul., 5 (2004) 221-234.

[15] Y. Chen and Q. Wang, Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1) dimensional dispersive long wave equation, Chaos, Solitons and Fractals, 24 (2005) 745-757 .

[16] S.Liu, Z. Fu, S.D. Liu and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Letters A, 289 (2001) 69-74.

[17] S. Zhang and T.C. Xia, A generalized F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations, Appl. Math. Comput., 183 (2006) 1190-1200 .

[18] J.H. He and X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006) 700-708.

[19] M.A. Abdou, The extended F-expansion method and its applications for a class of nonlinear evolution equation, Chaos, Solitons and Fractals 31(2007) 95 -104 .

[20] M .Wang and X. Li, Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation, Chaos, Solitons and Fractals 24 (2005) 1257- 1268.

[21] M.Wang, X.Li and J.Zhang, The ( GG / )- expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys.Letters A, 372 (2008) 417-423.

[22] E.M.E.Zayed and K.A.Gepreel, The ( )- expansion method for finding traveling wave solutions of nonlinear PDEs in mathematical physics, J. Math. Phys., 50 (2009) 013502-013514.

[23] E.M.E.Zayed and Khaled A.Gepreel, Some applications of the ( GG / ) expansion method to non-linear partial differential equations, Appl. Math. and Comput., 212 (2009) 1-13.

[24] A. R. Shehata, E.M.E.Zayed and K.A Gepreel, Eaxct solutions for some nonlinear partial differential equations in mathematical physics , J. Information and computing Science 6 (2011) 129-142.

[25] R. M. Miura, The Korteweg-de Vries equations: A Survey of results, SIAM Rev. 18 (2006) 412-549. [26] Z.Y.Ma, Homotopy perturbation method for the Wu-Zhang equation in fluid dynamics, J. Phys.: Conf. Ser. 96

(2008) 012182-012188. [27] X. Ji, C. Chen, J.E. Zhang and Y. Li, Lie symmetry analysis and some new exact solutions of the Wu-Zhang

equation, Journal Math. Phys., (2004) 448-460.



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[28] M Alquran and R. Al-omary, Soliton and periodic solutions to the generalized Hirota-Satsuma coupled system using trigonometric and hyperbolic function methods, Int.J. Nonlinear Sci. 14(2012) 150-159.

[29] E.M.E.Zayed, Khaled A. Gepreel and M. M. El-Horbaty, Modified simple equation method to the nonlinear Hirota Satsuma KdV system, Journal of Information and computing Science, 10( 2015) 054-062.

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