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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,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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(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
Journal of Information and Computing Science, Vol. 11(2016) No. 1, pp 030-057
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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:
Khaled A. Gpreel et.al : Extended rational GG / expansion method for nonlinear partial differential equations
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.)()(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)
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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 .
Khaled A. Gpreel et.al : Extended rational GG / expansion method for nonlinear partial differential equations
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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
parameters take special values ,2E ,5.01C ,1A ,3C ,5.2k ,75.02 C ,75.1 ,1B ,2.2 25.1 and 25.0 .
Journal of Information and Computing Science, Vol. 11(2016) No. 1, pp 030-057
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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
parameters take special values ,2E ,5.01C ,3A ,1C ,5.2k ,75.02 C ,75.1 ,1B ,2.2 25.1 and 25.0 .
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
Khaled A. Gpreel et.al : Extended rational GG / expansion method for nonlinear partial differential equations
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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
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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
Khaled A. Gpreel et.al : Extended rational GG / expansion method for nonlinear partial differential equations
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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
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.
)()(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
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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
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.)(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
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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
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
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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
0t when the parameters take special values ,1A ,3C ,2E ,1B ,5.01C ,3y ,5.10 a
,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
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 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)
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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
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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:
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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
kEAbEbEBbBEbCBEbCEbBbEAbEBb
BbECEbbEBAkbAEbBEAbAEkbAb
EW
(41)
Family 2. When 0B , 0)(42 CAEB , we obtain the trigonometric exact solutions of Eqs.(39) take the following forms:
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.
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
kEAbEbEBbBEbCBEbCEbBbEAbEBb
BbECEbbEBAkbAEbBEAbAEkbAb
EW
(42) Family 3. When 0B , 0)(42 CAEB , we obtain the rational exact solutions of Eqs.(39) take the forms:
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.)()(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
kEAbEbEBbBEbCBEbCEbBbEAbEBb
BbECEbbEBAkbAEbBEAbAEkbAb
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:
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.
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
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.
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
EbbkABbEbEBbCEbAEbAb
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
EbbkABbEbEBbCEbAEbAb
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.
Journal of Information and Computing Science, Vol. 11(2016) No. 1, pp 030-057
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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
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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
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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
Khaled A. Gpreel et.al : Extended rational GG / expansion method for nonlinear partial differential equations
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