abundant solutions with distinct physical structure for ... · partial differential equations ......
TRANSCRIPT
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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)
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
5337
Family 3. When 0B , 0 , we obtain the rational exact
solution of Eq.(20) takes the following form:
.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)
Consequently the rational traveling wave solution of Eq.(14)
has the following form:
.
)]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)
Family 5. When 0B , 0 E , we obtain the
hyperbolic exact solution of Eq.(20) takes the following form:
.
)]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)
Consequently the rational traveling wave solution of Eq.(14)
has the following form:
,
)]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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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:
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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)
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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)
where is defined as Eq.(50).
Family 3. When 0B , 0 , we obtain the rational exact
solution of Eq.(44) takes the following form:
2221
2221
2
2
1
21
2
]2)([
]2))(4[{(3
5
9)(
CCCB
BCCCBE
A
k
k
k
(53)
Consequently the rational traveling wave solution of Eq.(40)
has the following form:
,]2)([
]2))(4[{(3
5
9),,,(
2221
2221
2
2
21
1
23
CCCB
BCCCBE
A
k
k
ktzyxu
(54)
where is defined as Eq.(50).
Family 4. When 0B , 0 E , we obtain the
hyperbolic exact solution of Eq.(44) takes the following form:
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)
Consequently the rational traveling wave solution of Eq.(40)
has the following form:
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)
Family 5. When 0B , 0 E , we obtain the
hyperbolic exact solution of Eq.(44) takes the following form:
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
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)
Consequently the rational traveling wave solution of Eq.(40)
has the following form:
.
)]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)
where is defined as Eq.(51).
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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
5343
5 10 15 20x
40
60
80
100
u2
Figure 2. The exact extended ( GG / ) expansion solution 2U in Eq. (26) and its projection at 0t when the parameters take
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
Figure 3. The exact extended ( GG / ) expansion solution 3U in Eq. (29) and its projection at 0t when the parameters take
special values ,1E ,21C ,32 C ,1A B=2, ,2C 5,3,1,3,11,7,3,5 321 yzkkk
5 10 15 20x
2.38
2.40
2.42
2.44
u4
Figure 4. The exact extended ( GG / ) expansion solution 4U in Eq. (30) and its projection at 0t when the parameters take
special values ,1E ,21C ,32 C ,1A B=0, ,2C 5,3,1,3,11,7,3,5 321 yzkkk
.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
5344
5 10 15 20x
40
30
20
10
u5
Figure 5. The exact extended ( GG / ) expansion solution 5U in Eq. (33) and its projection at 0t when the parameters take
special values ,1E ,21C ,32 C ,2A B=0, ,1C 5,3,1,3,11,7,3,5 321 yzkkk
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.
REFERENCES
[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
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5332-5345
© Research India Publications. http://www.ripublication.com
5345
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 ( GG / )-
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] MA Akbar, N. Ali, E. Zayed , A generalized and
improved (G′/G)-expansion method for nonlinear
evolution equations. Mathematical Problems in
Engineering (2012) Article ID: 459879, 22 pages,
doi:10.1155/2012/459879.
[26] Khaled A. Gepreel, Taher A. Nofaland Khulood O.
Alweail, Extended rational (G′/G) - expansion method
for nonlinear partial differential equations, Journal of
Information and Computing Science, 11 ( 2016) 030-
057.
[27] H. Naher and F. Abdullah, New generalized (G′/G )-
expansion method to the Zhiber- Shabat Equation and
Liouville Equations , IOP Conf. Series: Journal of
Physics: Conf. Series 890 (2017) 012018.
[28] A. A. Al-Shawba, K. A. Gepreel, F. A. Abdullah, A.
Azmi,Abundant Closed Form Solutions of the
Conformable Time Fractional Sawada-Kotera-Ito
Equation Using (G'/G) - Expansion Method, Accepted
for publication in J. Results in Physics (2018) 9
(2018) 337–343.
[29] Khaled A Gepreel, Taher A Nofal and AMERA
O.ALASMARI, Exact Solutions for Nonlinear Integro-
Partial Differential Equations Using the Generalized
Kudryashov Method, Journal of the Egyptian
Mathematical Society 25 (2017), 438-444.
[30] Khaled A Gepreel, Taher A Nofal , Explicit traveling
wave solutions for nonlinear differential difference
equations in mathematical physics, International
Journal of Applied Engineering Research 13(5) (2018)
3034-3042.