The Riccati equation method with variable expansion coefficients. II. Solving the KdV equation
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Transcript of The Riccati equation method with variable expansion coefficients. II. Solving the KdV equation
The Riccati equation method
with variable expansion coefficients.
II. Solving the KdV equation.
Solomon M. Antoniou
SKEMSYS Scientific Knowledge Engineering
and Management Systems
37 Κoliatsou Street, Corinthos 20100, Greece [email protected]
Abstract
The Riccati equation method with variable expansion coefficients, introduced in a
previous paper, is used to find travelling wave solutions to the KdV equation
0uauuu xxxxt =++ . The −ξ dependent coefficients A and B of the Riccati
equation 2BYAY +=′ satisfy their own nonlinear ODEs, which can be further
solved by one of the known methods, like Jacobi's elliptic equation method. The
KdV equation is also solved by the −′ )G/G( expansion method with variable
expansion coefficients.
Keywords: Riccati method, nonlinear evolution equations, traveling wave
solutions, KdV equation, exact solutions.
2
1. Introduction .
Nonlinear partial differential equations arise in a number of areas of Mathematics
and Physics in an attempt to model physical processes, like Chemical Kinetics
(Gray and Scott [37]), Fluid Mechanics (Whitham [94]), or biological processes
like Population Dynamics (Murray [65]). In the recent past there are a number of
new methods which have been invented in solving these equations. Among the
new methods are the inverse scattering method (Ablowitz and Segur [13],
Ablowitz and Clarkson [12], Novikov, Manakov, Pitaevskii and Zakharov [69]),
Hirota’s bilinear method (Hirota [43] and [44), the Bäcklund transformation
method (Rogers and Shadwick [77]), the Painleve truncated method (Weiss,
Tabor and Carnevale [92] and [93]), the Cole-Hopf transformation method (Salas
and Gomez [80]), the algebro-geometric approach (Belokolos et al [19]), the Lie
symmetry method (Lie point symmetries, potential symmetries, nonclassical
symmetries, the direct method) (Bluman and Kumei [20], Hydon [45], Olver [70],
Ovsiannikov [71], Stephani [82]), the tanh-coth method (Malfliet [58] and [59],
Malfliet and Hereman [60] and [61], El-Wakil and Abdou [26], Fan [30], Griffiths
and Sciesser [38], Fan and Hon [31], Parkes and Duffy [73], Parkes, Zhou, Duffy
and Huang [75], Wazwaz [90] ), the sn-cn method (Baldwin et al [17]), the F-
expansion method (Abdou [4] and [7], Wang and Li [87]), the Jacobi elliptic
function method (Abbott, Parkes and Duffy [1], Abdou and Elhanbaly [10], Chen
and Zhang [22], Chen and Wang [23], Fan and Zhang [32], Inc and Ergüt [46],
Liu, Fu, Liu and Zhao [55], [56], Lu and Shi [57], Parkes, Duffy and Abbott [74]),
the Riccati equation method (Zhang and Zhang [101], Abdou [3], Antoniou [15]),
the Weierstrass elliptic function method (Kudryashov [51], [53]), the exp-
function method (He and Wu [41], Abdou [8], Aslan [16], Bekir and Boz [18],
Ebaid [24], El-Wakil, Abdou and Hendi [27], He and Abdou [40], Naher,
Abdullah and Akbar [66] and [67]), the Adomian decomposition method
(Adomian [14], Abdou [2], Wazwaz [89] and [91]), the −′ G/G expansion
3
method (Borhanibar and Moghanlu [21], Feng, Li and Wan [33], Jabbari, Kheiri
and Bekir [47], Naher, Abdullah and Akbar [68], Ozis and Aslan [72], Wang, Li
and Zhang [88], Zayed [98], Zayed and Gepreel [100]), the homogeneous
balance method (Fan [29], Wang, Zhou and Li [86], El-Wakil, Abulwafa,
Elhanbaly and Abdou [25], Wang, Zhou and Li [86]), the direct algebraic method
(Soliman and Abdou [81]), the basic equation method (Kudryashov [52]) and its
variants, like the simplest equation method (Abdou [6], Jawad, Petkovich and
Biswas [48], Vitanov [85], Yefimova [96], Zayed [99]), the first integral method
(Feng [34], Raslan [76]), the integral bifurcation method (Rui, Xie, Long and He
[78]), the reduced differential transform method (Keskin and Oturanc [49]), the
homotopy perturbation method (Taghizadeh, Akbari and Ghelichzadeh [83],
Yahya et al [95], Liao [54], El-Wakil and Abdou [28]), the variational iteration
method (He [39], Abdou [5], Abdou and Soliman [9], Wazwaz [91]). A more
detailed (but not complete) set of references of the above methods appears in
Antoniou [15]. The implementation of most of these methods was made possible
only using Symbolic Languages like Mathematica, Macsyma, Maple, etc.
In this paper we introduce the Riccati equation method with variable expansion
coefficients and we find traveling wave solutions of the KdV equation. The paper
is organized as follows: In Section 2 we introduce the basic ingredients of the
method used. In Section 3 we consider KdV equation and Riccati’s equation
method of solution where the expansion coefficients depend on the variable ξ . In
Section 4 the KdV equation is solved using the −′ )G/G( expansion method with
variable expansion coefficients.
2. The Method.
We consider an evolution equation of the general form
),u,u,u(Gu xxxt L= or ),u,u,u(Gu xxxtt L= (2.1)
where u is a sufficiently smooth function.
We introduce a new variable ξ given by
4
)tx(k ω−=ξ (2.2)
where k and ω are constants. Changing variables, since
)(u)k(ut ξ′ω−= , )(ukux ξ′= , )(uku 2xx ξ′′= , … (2.3)
equation (2.1) becomes an ordinary nonlinear differential equation
ξξ=
ξω− L,
d
udk,
ddu
k,uGddu
)k(2
22 (2.4)
or
ξξ=
ξL,
d
udk,
ddu
k,uGd
udk
2
22
2
22 (2.5)
Equations (2.4) or (2.5) will be solved considering expansions of the form
∑=
=ξn
0k
kkYa)(u (2.6)
or
∑∑==
+=ξn
1kkk
n
0k
kk
Y
bYa)(u (2.7)
where all the expansion coefficients depend on the variable ξ ,
)(aa kk ξ≡ , )(bb kk ξ≡ for every n,,2,1,0k L=
contrary to the previously considered cases where the expansion coefficients were
considered as constants. The function )(YY ξ≡ satisfy Riccati’s equations
2YBA)(Y ⋅+=ξ′ (2.8)
where again the coefficients A and B depend on the variable ξ .
In solving equations (2.4) or (2.5), we consider the expansions (2.6) or (2.7) and
then we balance the nonlinear term with the highest derivative of the function
)(u ξ which determines n (the number of the expansion terms). Equating similar
powers of the function )(Yξ we can determine the various coefficients and thus
find the solution of the equation considered.
5
3. The KdV equation and its solutions.
The KdV equation was introduced by D. J. Korteweg and G. de Vries (Korteweg
and de Vries [50]) in modeling the motion of surface waves in shallow, narrow
canals, according to an observation made by J. S. Russell (Russell [79]). The
interest to Russell's observation was triggered by a computer experiment at the Los
Alamos Laboratory by Fermi, Pasta and Ulam (Fermi, Pasta and Ulam [35]). In
the above list it is fair to include the name of Mary Tsingou (see also Tuck and
Menzel [84], where Mary Tsingou appears under the name M. T. Menzel). Ten
years later, N. J. Zabusky and M. D. Kruskal (Zabusky and Kruskal [97]),
published a paper in which they provided a satisfactory interpretation of Fermi-
Pasta-Ulam experiment and introduced the term "soliton" to name the solitary-
wave solutions of the KdV equation. The KdV equation can be solved through the
famous Inverse Scattering Transform (AKNS [11], Gardner, Greene, Kruskal and
Miura [36], Miura [62] and [64], Miura, Gardner and Kruskal [63]), and the
algebro-geometric approach (Belokolos et al [19]).
Some solutions have also been found using the tanh-method (Hereman and
Malfliet [42]):
))tx(k(tanhka12ka8)t,x(u 222 ω−−+ω=
We consider the KdV equation in the form
0uauuu xxxxt =++ (3.1)
and try to find traveling wave solutions of this equation. We introduce a new
variable ξ given by
)tx(k ω−=ξ (3.2)
where k and ω are constants. Changing variables, since
)(u)k(ut ξ′ω−= , )(ukux ξ′= and )(uku 3xxx ξ′′=
equation (3.1) becomes an ordinary nonlinear differential equation
0)(uka)(u)(uk)(u)k( 3 =ξ′′′+ξ′ξ+ξ′ω− (3.3)
6
Integrating the above equation once, we obtain
022 c)(uka)(u
21
)(u)( =ξ′′+ξ+ξω− (3.4)
where 0c is a constant.
We consider the following cases in solving equation (3.4).
3.1. First Case. The Riccati Method.
We consider the solution of equation (3.4) to be of the form
∑=
=ξn
0k
kkYa)(u (3.5)
where )(aa kk ξ≡ , for every n,,2,1,0k L= .
3.1.1. Method I. We first consider that )(YY ξ≡ satisfies Riccati’s equation
2BYA)(Y +=ξ′ (3.6)
where A and B depend on the variable ξ : )(AA ξ≡ and )(BB ξ≡ .
We substitute (3.5) into (3.4) and take into account Riccati’s equation (3.6). We
then balance the second order derivative term with that of the nonlinear term. The
order of the nonlinear term )(u2 ξ is n2 and that of the second order derivative
term is 2n + . We thus get the equation 2nn2 += from which we obtain 2n = .
Therefore
2210 YaYaa)(u ++=ξ (3.7)
We calculate )(u ξ′′ from the above equation, taking into account Riccati’s
equation (3.6) and that the various coefficients 0a , 1a , 2a , A and B depend on
ξ . We find
++′+′+′=ξ′′ }A)Aa2a()Aaa{()(u 2110
+′++′+′+ Y}A)aBa(2)Aa2a{( 2121
++′′+++′+ 222121 Y}BAa6)aBa(B)Aa2a{(
422
3221 Y)Ba(6Y})Ba(B)aBa({2 +′+′++ (3.8)
7
Because of (3.7) and (3.8), equation (3.4) becomes
++++++ω− 22210
2210 )YaYaa(
21
)YaYaa()(
++′+′+′+ }A)Aa2a()Aaa[{(ka 21102
+′++′+′+ Y}A)aBa(2)Aa2a{( 2121
++′′+++′+ 222121 Y}BAa6)aBa(B)Aa2a{(
042
23
221 c]Y)Ba(6Y})Ba(B)aBa({2 =+′+′++
We first arrange the above equation in powers of Y, and then equate the
coefficients of each one power to zero. We obtain
coefficient of 0Y :
0211022
00 c]A)Aa2a()Aaa([kaa21
a)( =+′+′+′++ω− (3.9)
coefficient of Y :
0]A)aBa(2)Aa2a([kaaaa)( 21212
101 =′++′+′++ω− (3.10)
coefficient of 2Y :
++′+++ω− B)Aa2a([ka)aaa2(21
a)( 2122
1202
0]BAa6)aBa( 221 =+′′++ (3.11)
coefficient of 3Y :
0])Ba(B)aBa([ka2aa 2212
21 =′+′++ (3.12)
coefficient of 4Y :
0Baka6a21 2
222
2 =+ (3.13)
We now have to solve the system of simultaneous equations (3.9)-(3.13).
From equation (3.13), ignoring the trivial case, we determine the coefficient 2a :
222 Bka12a −= (3.14)
From equation (3.12), because of (3.14), we obtain
8
Bka12a 21 ′−= (3.15)
From equation (3.11), using the values of 1a and 2a , given by (3.15) and (3.14)
respectively, we obtain the equation
BAka8B
Bka4
B
Bka3a 22
22
0 −′′
−
′+ω= (3.16)
From equation (3.10), using (3.14) and (3.15), we derive another expression for
the coefficient 0a :
BAka10BBA
ka2BB
kaa 22
220 −
′′
−′′′′
−ω= (3.17)
Finally from equation (3.9), using again (3.14) and (3.15), we obtain the equation
0]BA2)BA(BA[ka12aa21
aka 22420
200
2 =+′′+′′−ω−+′′ (3.18)
Equating the two different expressions of 0a , given by (3.16) and (3.17), we
obtain the equation
0B
BA2
B
BBA2
B
B4
B
B3
22
=′
′+
′′′′
++′′
−
′
The above equation can be written as
0)AB(2BB
BB
BB
BB
2B
BB3
BB 23
2=′+
′−
′′
′−
′+
′′′−
′′′ (3.19)
and can easily be integrated, considering the substitution
BB
F′
= (3.20)
Equation (3.19) thus becomes 0)AB(2FFF =′+′−′′ which can be integrated
12 cAB2F
21
F =+−′ (3.21)
From the above equation we get
MBB
41
BB
21
AB2
+
′+
′
′−= (3.22)
9
where M is an arbitrary constant.
Equations (3.16) or (3.17), give because of (3.22):
Mka8B
Bka3a 2
22
0 −
′−ω= (3.23)
Finally, from equation (3.18), because of (3.23) and (3.22), we derive (after some
lengthy calculations) the equation
02422 cMka8
21 =+ω− (3.24)
which is a consistency condition between the different parameters and constants. It
can also be considered as an equation which determines ω .
Riccati’s equation 2BYA)(Y +=ξ′ under the substitution
)(w)(w
)(B1
)(Yξξ′
⋅ξ
−=ξ (3.25)
takes on the form of a linear second order ordinary differential equation
0)(wBA)(wBB
)(w =ξ+ξ′
′−ξ′′ (3.26)
with unknown function )(w ξ .
We now transform equation (3.26) under the substitution
)(y)(B)(w ξ⋅ξ=ξ (3.27)
The derivatives of the function )(uξ transform as
′+′=ξ′ yBBy21
B
1)(w (3.28)
′′+′′+
′⋅−′′=ξ′′ yByBy
B)B(
41
B21
B
1)(w
2 (3.29)
Equation (3.26), because of (3.22) and (3.27)-(3.29), takes on the form
+
′+′
′−
′′+′′+
′⋅−′′ yBBy
21
BB
B
1yByBy
B)B(
41
B21
B
1 2
10
0yBMBB
41
BB
21 2
=
+
′+
′
′−+
which gives us upon multiplying by B , the simple equation (some miraculous
cancellations take place)
0)(yM)(y =ξ⋅+ξ′′ (3.30)
Considering various forms of the constant M, we can determine )(y ξ and then
)(w ξ from (3.27). For 2mM = , equation (3.30) admits the general solution
)msin(C)mcos(C)(y 21 ξ+ξ=ξ , whereas for 2mM −= admits the general
solution )msinh(C)mcosh(C)(y 21 ξ+ξ=ξ . If 0M = then 21 CC)(y +ξ=ξ .
The function )(Yξ can be determined from (3.25) and then )(uξ from (3.7), using
the expressions 0a , 1a and 2a from (3.23), (3.15) and (3.14) respectively.
We thus find that
=++=ξ 2210 YaYaa)(u
+
ξξ′
⋅ξ
−′−+
−
′−ω=
)(w)(w
)(B1
)Bka12(Mka8BB
ka3 222
2
2
22
)(w)(w
)(B1
)Bka12(
ξξ′
⋅ξ
−−+
or
−
ξξ′′
+
−
′−ω=ξ
)(w)(w
BB
ka12Mka8BB
ka3)(u 222
2
2
2
)(w)(w
ka12
ξξ′
− (3.31)
Since )(y)(B)(w ξ⋅ξ=ξ , we obtain that
)(y)(y
)(B)(B
21
)(w)(w
ξξ′
+ξξ′
⋅=ξξ′
(3.32)
11
Using the above expression, we obtain from (3.31) that
2
22
)(y)(y
ka12Mka8)(u
ξξ′
−−ω=ξ (3.33)
This is quite a remarkable result : no matter what the coefficients of Riccati’s
equation are, we arrive at the equation (3.33) where )(y ξ satisfies equation (3.30).
We thus obtain the following three solutions, depending on the values of the
constant M (see equation (3.30))
(I) 2
2222
)mtan(C)mtan(C1
mka12mka8)(u
ξ+ξ⋅−−−ω=ξ for 2mM = (3.34)
where ω satisfies equation (3.24): 04422 cmka8
21 =+ω−
(II) 2
2222
)mtanh(C)mtanh(C1
mka12mka8)(u
ξ+ξ⋅+−+ω=ξ for 2mM −= (3.35)
where ω satisfies equation (3.24): 04422 cmka8
21 =+ω−
(III) 2
2
C1C
ka12)(u
ξ+−ω=ξ for 0M = (3.36)
where ω satisfies equation (3.24): 02 c
21 =ω−
C is an arbitrary constant, the ratio of the two constants appearing in the general
solution of equation (3.30).
3.2. Second Case. The Extended Riccati Method.
In this case we consider the expansion
∑∑==
+=ξn
1kkk
n
0k
kk
Y
bYa)(u
and balance the second order derivative term with the second order nonlinear term
of (3.4). We then find 2n = and thus
12
2212
210Y
b
Y
bYaYaa)(u ++++=ξ (3.37)
where again all the coefficients 0a , 1a , 2a and 1b , 2b depend on ξ , and Y
satisfies Riccati’s equation 2BYAY +=′ . From equation (3.37) we obtain, taking
into account 2BYAY +=′
+−′−+′+′−+′=ξ′′ }B)bB2b(A)aA2a()BbAaa({)(u 2121110
++′+′+′+ Y}A)Baa(2)aA2a({ 1221
++′+′++′+ 221221 Y}BAa6)Baa(B)aA2a({
422
3212 Y)Ba(6Y})Ba(2B)Baa(2{ +′++′+
−′−+′−′
+Y
B)bbA(2)bB2b( 2121
222121
Y
)bBA(6)bbA(A)bB2b( −′′−+−′−
42
2
3221
Y
bA6
Y
)bA(2A)bbA(2 +′−′−+ (3.38)
Therefore equation (3.4), under the substitution (3.42) and (3.43), becomes
+
+++++
++++ω−2
2212
2102212
210Y
b
Y
bYaYaa
21
Y
b
Y
bYaYaa)(
+−′−+′+′−+′+ }B)bB2b(A)aA2a()BbAaa([{ka 21211102
++′+′+′+ Y}A)Baa(2)aA2a({ 1221
++′+′++′+ 221221 Y}BAa6)Baa(B)aA2a({
422
3212 Y)Ba(6Y})Ba(2B)Baa(2{ +′++′+
−′−+′−′
+Y
B)bbA(2)bB2b( 2121
222121
Y
)bBA(6)bbA(A)bB2b( −′′−+−′−
13
042
2
3221 c]
Y
bA6
Y
)bA(2A)bbA(2 =+′−′−+
Equating the coefficients of Y to zero, we obtain a system of differential
equations from which we can determine the various expansion coefficients.
coefficient of 0Y :
++++ω− )ba2ba2a(21
a 2211200
021211102 c]B)bB2b(A)aA2a()BbAaa([ka =−′−+′+′−+′+ (3.39)
coefficient of Y:
+++ω− 10121 aabaa)(
0]A)Baa(2)aA2a([ka 12212 =+′+′+′+ (3.40)
coefficient of 2Y :
+++ω− )aa2a(21
a)( 20212
0]BAa6)Baa(B)aA2a([ka 212212 =+′+′++′+ (3.41)
coefficient of 3Y :
0])Ba(B)Baa[(ka2aa 2122
21 =′++′+ (3.42)
coefficient of 4Y :
0)Ba6(kaa21 2
222
2 =+ (3.43)
coefficient of 1Y − :
0]B)bbA(2)bB2b[(kababab)( 21212
21101 =′−+′−′+++ω− (3.44)
coefficient of 2Y − :
−++ω− )ba2b(21
b)( 20212
0)]bBA(6)bbA(A)bB2b[(ka 221212 =−′′−+−′− (3.45)
14
coefficient of 3Y − :
0])bA(A)bbA[(ka2bb 2212
21 =′−′−+ (3.46)
coefficient of 4Y − :
0)Ab6(kab21 2
222
2 =+ (3.47)
Solving the system of equations (3.47) and (3.46) (ignoring the trivial solutions)
we obtain
222 Aka12b −= and Aka12b 2
1 ′= (3.48)
Solving the system of equations (3.43) and (3.42) (ignoring the trivial solutions)
we obtain
222 Bka12a −= and Bka12a 2
1 ′−= (3.49)
From (3.41), using equations (3.49), we obtain
+
′−
′′−ω= AB8
BB
3BB
4kaa2
20 (3.50)
From (3.45), using equations (3.48), we obtain
+
′−
′′−ω= AB8
AA
3AA
4kaa2
20 (3.51)
From (3.44), using the coefficients (3.48) and (3.49), we get
+′
′+
′′′′
−ω= AB10A
BA14
AA
kaa2
20 (3.52)
From (3.40), using the coefficients (3.48) and (3.49), we get
+′
′+
′′′′
−ω= AB10BAB
14BB
kaa2
20 (3.53)
Finally from equation (3.39), using again (3.48) and (3.49), we get
+−′′−+ω−+ )BABA()ka144(aa21
ka 22420
20
2
15
02242 c)BA2ABBABA()ka24( =+′′+′′+′′−+ (3.54)
Equating the two expressions (3.50) and (3.51) for 0a we conclude that A and B
are proportional each other:
AsB 2−= (3.55)
with 2s− being the proportionality factor where s is real. The choice AsB 2=
leads to imaginary solutions (see below, eqns (3.59) and (3.60)). Let us now prove
that the functions B and A are proportional each other. Equating the two
expressions (3.50) and (3.51) for 0a , we obtain the relation
22
A
A3
A
A4
B
B3
B
B4
′−
′′=
′−
′′, which is equivalent to
′+
′
′−
′=
′′−
′′BB
AA
BB
AA
3BB
AA
4 and under the substitution BABAF ′−′= ,
taking into account the identity BABA)BABA( ′′−′′=′′−′ , we obtain the relation
AB)AB(
3FF
4′
=′
which upon integration gives 0AB
FF
3
=
or
0B
B
A
A)BABA(
3
=
′−
′′−′ , where we have set the integration constant equal to
zero. From the last relation, equating to zero either factor, i.e. 0BABA =′−′ or
0BB
AA =
′−
′, we conclude that A and B are proportional each other.
Equating again the two expressions (3.51) and (3.52) for 0a and taking into
account (3.55), we obtain the equation
0)(A)(As16)(A))(A(2))(A(3)(A)(A 42232 =ξ′ξ⋅−ξ′′′ξ−ξ′+ξ′′′⋅ξ (3.56)
This equation has been solved in Appendix A.
We have to determine the function )(w ξ from equation (3.16)
0)(wBA)(wBB
)(w =ξ+ξ′
′−ξ′′ (3.57)
16
This equation written in terms of A, using (3.47), takes on the form
0)(wAs)(wAA
)(w 22 =ξ−ξ′
′−ξ′′ (3.58)
The previous equation can be written as 0wAsA
AwAw 2
2=−
′′−
′′ which is
equivalent to 0wAsAw 2 =−
′
′. Multiplying by
Aw′
, we obtain
0wwsAw
Aw 2 =′−
′
′′, which is equivalent to 0)w(s
Aw 22
2
=′−′
′
and from this, by integration,
0wsA
w 222
=−
′ (3.59)
where we have put the constant of integration equal to zero. We thus get
)(As)(w)(w ξ±=
ξξ′
(3.60)
Another method of solution of equation (3.51) is provided in Paper I.
Therefore
s1
)(w)(w
)(As
1)(w)(w
)(B1
)(Y2
±=ξξ′
⋅ξ
=ξξ′
⋅ξ
−=ξ (3.61)
We then obtain the following expression for the function )(u ξ using (3.42), (3.46),
(3.47) and (3.53):
)(Aska4)(A)(A
ka)(u ξ±ξξ′
+ω=ξ (3.62)
Using the solution of the function )(Aξ calculated in Appendix A, we are able to
find the function )(u ξ . Using (A.15), i.e.
D4]eKa)ama(2[
eDaK4)(A
2Ds4101
Ds41
−++ρ=ξ
ξ⋅−
ξ⋅−
we obtain from (3.62) that
17
−++ρ
⋅++ρ⋅−−ω=ξξ⋅−
ξ⋅−ξ⋅−
D4]eKa)ama(2[
e]eKa)ama(2[K21Dska4)(u
2Ds4101
Ds4Ds4101
−++ρ±
ξ⋅−
ξ⋅−
D4]eKa)ama(2[
eaDKska16
2Ds4101
Ds41 (3.63)
where K is an arbitrary constant, and D, ρ , m, n are given by
na)ama(D 221
201 ρ−+ρ= ,
130
2 aas64
1=ρ , 40
22
204
21 as16babam +−=
21
2042
41
24
40
22
21
404
2602
280
4 aabb2ababaabs224abs32as256n −++−−=
The solution (3.63) contains 0a , 1a , 2b , 4b as free parameters.
4. The G'/G method with variable expansion coefficients
Using the expansion
2
210 G
G)(a
G
G)(a)(a)(u
′ξ+
′ξ+ξ=ξ (4.1)
equation (3.4) becomes
22
210
2
210 GG
)(aGG
)(a)(a21
GG
)(aGG
)(a)(a)(
′ξ+
′ξ+ξ+
′ξ+
′ξ+ξω−
′′
′−
′′′+
′′−′′′+
′′′+′′+
GG
GG
a3GG
aGG
a2GG
a2GG
aaka 11
2
11102
2
2
3
22
2
2
3
1 G
Ga2
G
Ga4
G
G
G
Ga4
G
Ga
G
Ga2
′′+
′′−
′′
′′+
′′′+
′+
0
4
2
2
22 cGG
a6GG
GG
a10GG
GG
a2 =
′+
′′
′−
′′′
′+ (4.2)
We expand the previous equation and we equate the coefficients of G to zero. We
find
Coefficient of 0G :
18
0022
00 cakaa21
a =′′++ω− (4.3)
Coefficient of 1G− :
0Ga)GaGa2Ga(kaGaa 11112
10 =′ω−′′′+′′′+′′′+′ (4.4)
Coefficient of 2G− :
})G(aGGa2)G(a2)G(a2GGa3GGa4{ka 222
21
2212
2 ′′′+′′′′+′′−′′+′′′−′′′′
0)G(a)G(a21
)G(aa 22
221
220 =′ω−′+′+ (4.5)
Coefficient of 3G− :
0)G(aa}G)G(a10)G(a2)G(a4{ka 321
22
31
32
2 =′+′′′−′+′′− (4.6)
Coefficient of 4G− :
0)G(a21
)G(aka6 422
42
2 =′+′ (4.7)
From equation (4.7) we find
22 ka12a −= (4.8)
Using the value of 2a into (4.6), we determine the coefficient 1a :
GG
ka12a 21 ′
′′= (4.9)
The values of 1a and 2a are substituted into (4.5) and we obtain the equation
′
′′′
+′′′′
−
′′′
+ω=GG
GG
ka2GG
kaa 22
20 (4.10)
The value of 1a is substituted into (4.4) and we obtain the equation
′′′′
+′
′′′
+
′′′
″
′′′
−ω=GG
GG
2
GGGG
kaa 20 (4.11)
Combining (4.10) and (4.11), we derive the equation
19
0
GGGG
GG
GG 2
=
′′′
″
′′′
+′′′′
−
′′′
(4.12)
The above equation determines the function )(G ξ . Using the substitution
GG
F′′′
= (4.13)
equation (4.13) transforms into 0FFF =′−′′ , which by integration gives
MF21
F 2 =−′ (4.14)
where M is the constant of integration.
(I) If 22M λ= , λ real, then the solution of (4.14) is given by
)tan(2F µ+ξλλ= , where µ is a constant. The differential equation FG
G =′′′
admits the solution [ ]
µ+ξλ−µ+λξ
λ+ξ+= )()(tan
1CCG 21 . We also obtain
])tan([CC
)](tan1[C
GG
21
22
µ−µ+ξλ+λµ+ξλλ+λ=
′, an expression reserved for later use in (4.15).
(II) If 22M λ−= , λ real, then the solution of (4.14) is given by
ξλ
ξλ
µ+µ−λ=
2
2
e1
e12F where µ is a constant. The differential equation F
GG =
′′′
admits the solution ξλµ++=
22
1e1
CCG . We also obtain
)]e1(CC[)e1(
eC2
GG
212
2
22
ξλξλ
ξλ
µ++µ+µλ=
′, an expression reserved for later use in
(4.15). (III) If 0M = , then the solution of (4.14) is given by ξ−µ
= 2F . The
differential equation FGG =
′′′
admits the solution ξ−µ
+=ξ 21
CC)(G where µ is a
20
constant. We also obtain )]B(CC[)B(
C
GG
12
2
ξ−+ξ−=
′, an expression reserved for
later use in (4.15).
Using the values of 1a and 2a into (4.1) we obtain
′−
′′+=ξ
22
0 GG
GG
ka12a)(u
and using (4.10), we get
′−
′′+
′
′′′
+′′′′
−
′′′
+ω=ξ2
222
2
GG
GG
ka12GG
GG
ka2GG
ka)(u
or, in simplified form,
′
′+
′
′′′
+
′′′
−ω=ξGG
ka12GG
4GG
ka)(u 22
2 (4.15)
So far we have not used equation (4.3). This equation, combined with (4.10), gives
us a compatibility condition between the constants and the various parameters. It
might also be considered as the equation determining the ω parameter.
Using the three solutions for G, we can find the following three expressions of
)(u ξ , using (4.15). We obtain
(I) +λ+µ+ξλλ−ω=ξ ]2)(tan3[ka4)(u 2222
×µ+ξλ+µ−λ
µ+ξλ+λ+2
221
2222
)](tanC)CC([
)](tan1[Cka12
}C)(tan)](tanC)CC(2[{ 2221 −µ+ξλµ+ξλ+µ−λλ× (4.16)
The compatibility condition, equation (4.3), reads
04422 cka8
21 =λ+ω− (4.17)
(II) +
µ+µ−
µ+µ−λ−ω=ξ ξλ
ξλ
ξλ
ξλ
22
22
2
222
)e1(
e8
e1
e1ka4)(u
21
22121
22
41
221
222
]eCCC[)e1(
]eCCC[Cka48 ξλξλ
ξλ
µ++µ+µ−+µλ+ (4.18)
The compatibility condition, equation (4.3), reads
04422 cka8
21 =λ+ω− (4.19)
(III)
ξ−µ+ξ−µ+−
ξ−µ−ω=ξ
212
1222
2
)](CC[
)](C2C[C1
)(
1ka12)(u (4.20)
The compatibility condition, equation (4.3), reads
02 c
21 =ω− (4.21)
Appendix A.
In this Appendix we solve equation (3.56):
0)(A)(As16)(A))(A(2))(A(3)(A)(A 42232 =ξ′ξ⋅−ξ′′′ξ−ξ′+ξ′′′⋅ξ (A.1)
We consider an expansion of the form
∑=
ξϕ=ξn
0k
kk )(a)(A (A.2)
where )(ξϕ satisfies Jacobi's differential equation
40
31
2234 bbbbb)(
dd ϕ+ϕ+ϕ+ϕ+=ξϕξ
(A.3)
Upon substitution of (A.2) into (A.1) and balancing )(A)(A 2 ξ′′′⋅ξ with
)(A)(A 4 ξ′ξ , and taking into account (A.3), we obtain 2n = . We thus substitute
)(a)(aa)(A 2210 ξϕ+ξϕ+=ξ (A.4)
Since
=ξϕ′ξϕ+=ξ′ )()](a2a[)(A 21
40
31
223421 bbbbb)](a2a[ ϕ+ϕ+ϕ+ϕ+ξϕ+= (A.5)
22
+ξϕ′ϕ+ϕ+ϕ+ϕ+=ξ′′ )(bbbbba2)(A 40
31
22342
)(bbbbb2
)b4b3b2b()a2a(4
03
12
234
30
212321 ξϕ′
ϕ+ϕ+ϕ+ϕ+
ϕ+ϕ+ϕ+ϕ++
+ϕ+ϕ+ϕ+ϕ+= )bbbbb(a2 40
31
22342
)b4b3b2b()a2a(21 3
02
12321 ϕ+ϕ+ϕ+ϕ+⋅+ (A.6)
+ξϕ′ϕ+ϕ+ϕ+=ξ′′′ )()b4b3b2b(a2)(A 30
21232
)()b4b3b2b(a 30
21232 ξϕ′ϕ+ϕ+ϕ++
)()b12b6b2()a2a(21 2
01221 ξϕ′ϕ+ϕ+ϕ+⋅+
+ϕ+ϕ+ϕ+= )b4b3b2b(a2{ 30
21232
)b4b3b2b(a 30
21232 ϕ+ϕ+ϕ++
×ϕ+ϕ+ϕ+⋅+ })b12b6b2()a2a(21 2
01221
40
31
2234 bbbbb ϕ+ϕ+ϕ+ϕ+× (A.7)
)()a2a()aaa(2))(A( 212
2102 ξϕ′ϕ+ϕ+ϕ+=′ξ
×ϕ+ϕ+ϕ+= )a2a()aaa(2 212
210
40
31
2234 bbbbb ϕ+ϕ+ϕ+ϕ+× (A.8)
Upon substituting (A.4)-(A.8) into (A.1) we obtain an equation which when
multiplied by 40
31
2234 bbbbb ϕ+ϕ+ϕ+ϕ+ , results in an equation which can
be brought into an equation of 13-th degree in ϕ . Equating all the coefficients to
zero, we obtain fourteen equations. Solving this system of simultaneous equations,
we obtain four solutions, from which we select only the non-trivial one:
00 aa = , 11 aa = , 0a2 = , 21
20 as16b = ,
30
421
40
22
201
1a2
)baas48ba(ab
−+=
23
22 bb = , 10
2204
21
40
2
3 aa2
baba3as16b
−−−= , 44 bb = (A.9)
Equation (A.3) then becomes
−ϕ+ϕ−+
+ϕ=
ξϕξ
22
330
421
40
22
20142
12
2
ba2
)baas48ba(aas16)(
dd
410
2204
21
40
2
baa2
baba3as16 +ϕ−−−
The above equation can be brought into the form
)()(a
aas16)(
dd
21
2
1
021
22
θρ−ϕθρ−ϕ
+ϕ=
ξϕξ
(A.10)
where
130
2 aas64
1=ρ , nm2,1 ±=θ , 40
22
204
21 as16babam +−=
21
2042
41
24
40
22
21
404
2602
280
4 aabb2ababaabs224abs32as256n −++−−= (A.11)
We then have (taking the plus sign only)
)()(a
aas4)(
dd
211
01 θρ−ϕθρ−ϕ
+ϕ=ξϕ
ξ
and by integration
=
+
+ϕ−
+ϕ+
+ϕ⋅−
+ϕ211
02
1
0
11
021
1
0 a
Da
ap
a
a
aD2
a
ap
a
D2
a
a1
ξ⋅−⋅= Ds4eK (A.12)
where
na)ama(D 221
201 ρ−+ρ= ,
1
01
a
)ama(2p
+ρ= (A.13)
and K is an arbitrary constant.
Equation (A.12), fortunately enough, can be solved with respect to ϕ , giving
24
1
02Ds42
1
Ds4
a
a
D4)eKp(a
eDK4)( −
−+=ξϕ
ξ⋅−
ξ⋅− (A.14)
We thus obtain that, using the above expression and (A.4),
D4)eKp(a
eDaK4)(A
2Ds421
Ds41
−+=ξ
ξ⋅−
ξ⋅− (A.15)
References
[1] P. C. Abbott, E. J. Parkes and B. R. Duffy:“The Jacobi elliptic-function
method for finding periodic-wave solutions to nonlinear evolution
equations”. Available online at
http//physics.uwa.edu.au/pub/Mathematica/Solitons
[2] M. A. Abdou: “Adomian decomposition method for solving the telegraph
equation in charged particle transport”.
J. Quant. Spectro. Rad. Trans 95 (2005) 407-414
[3] M. A. Abdou: “Exact solutions for nonlinear evolution equations via
the extended projective Riccati equation expansion method”.
Electron. J. Theor. Physics 4 (2007) 17-30.
[4] M. A. Abdou: “The extended F-expansion method and its applications
for a class of nonlinear evolution equations”.
Chaos, Solitons and Fractals 31 (2007) 95-104.
[5] M. A. Abdou: “On the variational iteration method”.
Phys. Lett. A 366 (2007) 61-68
[6] M. A. Abdou:”A generalized auxiliary equation method and its
applications”. J. Nonlin. Dyn. 52 (2008) 95-102
[7] M. A. Abdou: “An improved generalized F-expansion method and its
applicatuions”. J. Comput. and Appl. Math. 214 (2008) 202-208
[8] M. A. Abdou: “Generalized solitary and periodic solutions for nonlinear
partial differential equations by the Exp-function method”.
J. Nonlin. Dyn. 52 (2008) 1-9
25
[9] M. A. Abdou and A. A. Soliman: “New applications of Variational
Iteration Method”. Physica D 211 (2005) 1-8
[10] M. A. Abdou and A. Elhanbaly: “Construction of periodic and solitary
wave solutions by the extended Jacobi elliptic function expansion method".
Comm. in Nonlin. Science and Num. Sim. 12 (2007) 1229-1241
[11] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur: “The Inverse
Scattering Transform. Fourier Analysis for Nonlinear Problems”.
Stud. Appl. Math. 53 (1974) 249-315
[12] M. J. Ablowitz and P. A. Clarkson: “Solitons, Nonlinear Evolution
Equations and Inverse Scattering Transform”.
Cambridge University Press, Cambridge, 1991.
[13] M. J. Ablowitz and H. Segur: “Solitons and the Inverse Scattering
Transform”. SIAM, 1981.
[14] G. Adomian: “Nonlinear Stochastic Operator Equations”.
Academic Press, San Diego, (1986).
[15] S. Antoniou: “The Riccati equation method with variable expansion
coefficients. I. Solving the Burgers equation”. submitted for publication
[16] I. Aslan: “Application of the exp-function method to nonlinear lattice
differential equations for multi-wave and rational solutions”.
Mathem. Methods in the Applied Sciences 60 (2011) 1707-1710.
[17] D. Baldwin, Ü. Göktaş, W. Hereman, L. Hong, R.S. Martino and
J.C. Miller : “Symbolic computation of exact solutions expressible in
hyperbolic and elliptic functions for nonlinear PDEs”.
J. Symb. Comp. 37 (2004) 669-705
[18] A. Bekir and A. Boz: “Exact Solutions for Nonlinear Evolution Equations
using Exp-Function Method”. Phys. Lett. A 372 (2008) 1619-1625.
[19] E. D. Belokolos, A. . Bobenko, V. Z. Enolskii, A. R. Its and V. Matveev:
"Algebro-Geometric Approach to Nonlinear Integral Equations"
Springer-Verlag 1994
26
[20] G. Bluman and S. Kumei: “Symmetries and Differential Equations”.
Springer-Verlag 1989
[21] A. Borhanifar and A. Z. Moghanlu: “Application of the −′ )G/G(
expansion method for the Zhiber-Sabat equation and other related
equations”. Mathem. and Comp. Mod. 54 (2011) 2109-2116.
[22] H. T. Chen and H. Q. Zhang: “Improved Jacobian elliptic function
method and its applications”.
Chaos, Solitons and Fractals 15 (2003) 585-591
[23] Y. Chen and Q. Wang: “Extended Jacobi elliptic function rational
expansion method and abundant families of Jacobi elliptic function
solutions to (1+1)-dimensional dispersive long wave equation”.
Chaos, Solitons and Fractals 24 (2005) 745-757.
[24] A. E. Ebaid: “Generalization of He’s Exp-Function Method and New
Exact Solutions for Burgers Equation”. Z. Naturforsch. 64a (2009) 604–608
[25] S. A. El-Wakil, E. M. Abulwafa, A. Elhanbaly and M. A. Abdou:“The
extended homogeneous balance method and its applications”.
Chaos, Solitons and Fractals 33 (2007) 1512-1522
[26] S. A. El-Wakil and M. A. Abdou: “New exact travelling wave solutions
using modified extended tanh-function method”.
Chaos, Solitons and Fractals 31 (2007) 840-852
[27] S. A. El-Wakil, M. A. Abdou and A. Hendi: “New periodic wave
solutions via Exp-function method”. Physics Letters A 372 (2008) 830-840
[28] S. A. El-Wakil and M. A. Abdou: "New applications of the homotopy
analysis method". Zeitschrift für Naturforschung A (2008)
[29] E. Fan: “Two new applications of the homogeneous balance method”
Physics Letters A 265 (2000) 353-357
[30] E. Fan: “Extended tanh-function method and its applications to nonlinear
equations” . Physics Letters A 277 (2000) 212-218.
27
[31] E. Fan and Y. C. Hon: “Applications of extended tanh-method to
“special” types of nonlinear equations”.
Appl. Math. and Comp. 141 (2003) 351-358.
[32] E. Fan and H. Zhang: “Applications of the Jacobi elliptic function method
to special-type nonlinear equations”. Phys. Lett. A 305 (2002) 383-392.
[33] J. Feng, W. Li and Q. Wan: “Using −′ )G/G( expansion method to seek
traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation”
Appl. Math. Comp. 217 (2011) 5860-5865.
[34] Z. S. Feng: "The first integral method to study the Burgers-Korteweg de
Vries equation". Phys. Lett. A: Math Gen. A 302 (2002) 343-349
[35] E. Fermi, J. Pasta, S. Ulam and M. Tsingou: “Studies of Nonlinear
Problems. I". Los Alamos report LA-1940. May 1955
[36] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura:
“Method for solving the Korteweg-de Vries equation”.
Phys. Rev. Lett. 19 (1967) 1095-1097
[37] P. Gray and S. Scott: “Chemical Oscillations and Instabilities”
Clarendon Press, Oxford, 1990.
[38] P. Griffiths and W. E. Sciesser: “Traveling Wave Analysis of Partial
Differential Equations”. Academic Press 2012.
[39] J. H. He: “A variational iteration method-a kind of nonlinear analytical
technique: Some examples”. Int. J. Nonlin. Mech. 34 (1999) 699–708
[40] J. H. He and M. A. Abdou: “New periodic solutions for nonlinear
evolution equations using Exp function method”.
Chaos, Solitons and Fractals 34 (2007) 1421-1429
[41] J. H. He and X. H. Wu: “Exp-Function method for nonlinear wave
equations”. Chaos, Solitons and Fractals 30 (2006) 700-708.
[42] W. Hereman and W. Malfliet: “The tanh method: A Tool to Solve
Nonlinear Partial Differential Equations with Symbolic Software”.
Available online, Colorado School of Mines.
28
[43] R. Hirota: “The Direct Method in Soliton Theory”
Cambridge University Press 2004.
[44] R. Hirota: “Exact Solution of the KdV Equation for Multiple Collisions of
Solitons”. Physics Review Letters 27 (1971) 1192-1194.
[45] P. E. Hydon: “Symmetry Methods for Differential Equations”
Cambridge University Press 2000
[46] M. Inc and M. Ergüt : “Periodic wave solutions for the generalized
shallow water wave equation by the improved Jacobi elliptic function
method”. Appl. Math. E-Notes 5 (2005) 89-96
[47] A. Jabbari, H. Kheiri and A. Bekir : “Exact solutions of the coupled
Higgs equation and the Maccari system using He’s semi-inverse method
and −′ )G/G( expansion method”.
Comp. and Math. with Applic. 62 (2011) 2177-2186.
[48] A. J. M. Jawad, M. D. Petkovich and A. Biswas : “Modified simple
equation method for nonlinear evolution equations”.
Appl. Math. Comput. 217 (2010) 869–877
[49] Y. Keskin and G. Oturanc: “Reduced Differential Transform Method
for Partial Differential Equations".
Int. J. Nonl. Sci. and Num. Simul. 10 (2009) 741-749
[50] D. J. Korteweg and G. de Vries: "On the change of form of long waves
advancing in a rectangular channel and on a new type of long stationary
waves". Phil. Mag. 39 (1895) 422-443
[51] N. A. Kudryashov: “Exact Solutions of the Generalized Kuramoto-
Sivashinsky Equation”. Physics Letters A 147 (1990) 287-291.
[52] N. A. Kudryashov: “Simplest equation method to look for exact solutions
of nonlinear differential equations”. arXiv:nlin/0406007v1, 4 Jun 2004
[53] N. A. Kudryashov: “Nonlinear differential equations with exact solutions
expressed via the Weirstrass function”. arXiv:nlin/0312035v1,16 Dec 2003
29
[54] S. Liao: “Homotopy Analysis Method in Nonlinear Differential
Equations”. Springer 2012
[55] S. K. Liu, Z. Fu, S. Liu and Q. Zhao : “Jacobi elliptic function
expansion method and periodic wave solutions of nonlinear wave
equations”. Physics Letters A 289 (2001) 69-74.
[56] S. K. Liu, Z. Fu, S. Liu and Q. Zhao : “Expansion about the Jacobi
Elliptic Function and its applications to Nonlinear Wave Equations”.
Acta Phys. Sinica 50 (2001) 2068-2072.
[57] D. Lu and Q. Shi: “New Jacobi elliptic functions solutions for the
combined KdV-mKdV Equation”. Int. J. Nonlin. Sci. 10 (2010) 320-325
[58] W. Malfliet :“Solitary wave solutions of nonlinear wave equations”.
American Journal of Physics 60 (1992) 650-654.
[59] W. Malfliet : “The tanh method: a tool for solving certain classes of
nonlinear evolution and wave equations”
J. Comp. Appl. Math. 164-165 (2004) 529-541
[60] W. Malfliet and W. Hereman: “The tanh method: I. Exact solutions of
nonlinear evolution and wave equations”.
Physica Scripta 54 (1996) 563-568
[61] W. Malfliet and W. Hereman: “The tanh method: II. Perturbation
technique for conservative systems”. Physica Scripta 54 (1996) 569-575.
[62] R. M. Miura : “Korteweg-de Vries equations and generalizations. I.
A remarkable explicit nonlinear transformation”.
J. Math. Phys. 9 (1968) 1202-1204
[63] R. M. Miura, C. S. Gardner and M.D. Kruskal: “Korteweg-de Vries
equations and generalizations. II. Existence of conservation laws and
constants of motion”. J. Math. Phys. 9 (1968) 1204-1209.
[64] R. M. Miura : “The Korteweg de Vries equation: a survey of results".
SIAM Rev. 18 (1976) 412-459.
30
[65] J. Murray : “Mathematical Biology”. Springer-Verlag, Berlin, 1989.
[66] H. Naher, F. Abdullah and M. A. Akbar: “The Exp-function method for
new exact solutions of the nonlinear partial differential equations”.
Intern. J. Phys. Sciences 6 (2011) 6706- 6716.
[67] H. Naher, F. A. Abdullah and M. A. Akbar: “New travelling wave
solutions of the higher dimensional nonlinear partial differential
equation by the Exp-function method”. J. Appl. Math. (2012)
[68] H. Naher, F. A. Abdullah and M. A. Akbar:“The −′ )G/G( expansion
method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon
equation”. Math. Prob. in Engin. (2011)
[69] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov:
“Theory of Solitons: The Inverse Scattering Method”. Plenum, NY 1984
[70] P. J. Olver: “Applications of Lie Groups to Differential Equations ”
Graduate Texts in Mathematics, vol.107, Springer Verlag, N.Y. 1993
[71] L. V. Ovsiannikov: “Group Analysis of Differential Equations”
Academic Press, New York 1982
[72] T. Ozis and I. Aslan: “Application of the −′ )G/G( expansion method
to Kawahara type equations using symbolic computation”.
Applied Mathematics and Computation 216 (2010) 2360-2365.
[73] E. J. Parkes and B. R. Duffy: “An automated tanh-function method for
finding solitary wave solutions to nonlinear evolution equations”.
Comp. Phys. Comm. 98 (1996) 288-300
[74] E. J. Parkes, B. R. Duffy and P. C. Abbott: “The Jacobi elliptic function
method for finding periodic-wave solutions to nonlinear evolution
equations”. Phys. Lett. A 295 (2002) 280-286
[75] E. J. Parkes, E. J. Zhu, B. R. Duffy and H. C. Huang: “Sech-polynomial
traveling solitary-wave solutions of odd-order generalized KdV equations”
Phys. Lett. A 248 (1998) 219-224
31
[76] K. R. Raslan: “The first integral method for solving some important
nonlinear partial differential equations”. Nonlin. Dynamics 2007
[77] C. Rogers and W. F. Shadwick: “Bäcklund Transformations”.
Academic Press, New York, 1982.
[78] W. Rui, S. Xie, Y. Long and B. He: "Integral Bifurcation Method and its
Applications for solving the modified Equal Width Wave equation and its
variants". Rostock Math. Kolloq. 62 (2007) 87-106
[79] J. S. Russell:"Report on Waves"
The British Association for the Advancement of Science, London 1845
[80] A. H. Salas, and C. A. Gomez : “Application of the Cole-Hopf
transformation for finding exact solutions to several forms of the
seventh-order KdV equation”. Math. Prob. in Eng. (2010)
[81] A. A. Soliman and H. A. Abdou: “New exact solutions of nonlinear
variants of the RLW, the phi-four and Boussinesq equations based on
modified extended direct algebraic method”.
Intern. Journ. of Nonl. Sci. 7 (2009) 274-282.
[82] H. Stephani: “Differential Equations: Their Solutions Using Symmetries”.
Cambridge University Press, 1989
[83] N. Taghizadeh, M. Akbari and A. Ghelichzadeh: “Exact solution of
Burgers equations by homotopy perturbation method and reduced
differential transformation method”.
Austr. J. of Basic and Applied Sciences 5 (2011) 580-589
[84] J. L. Tuck and M. T. Menzel: “The Superposition of the Nonlinear
Weighted String (FPU) Problem”.
Advances in Mathematics 9 (1972) 399-407
[85] N. K. Vitanov: “Application of simplest equations of Bernoulli and Riccati
kind for obtaining exact traveling-wave solutions for a class of PDEs
with polynomial nonlinearity”.
Comm. in Nonlin. Sci. and Num. Simulation 15 (2010) 2050–2060
32
[86] M. L. Wang, Y. B. Zhou and Z. B. Li: “Application of a homogeneous
balance method to exact solutions of nonlinear equations in
Mathematical Physics”. Physics Letters A 216 (1996) 67-75.
[87] M. L. Wang and X. Z. Li: “Applications of F-Expansion to periodic wave
solutions for a new Hamiltonian amplitude equation”.
Chaos, Solitons and Fractals 24 (2005) 1257-1268.
[88] M. Wang, X. Li and J. Zhang: “The −′ )G/G( expansion method and
travelling wave solutions of nonlinear evolution equations in
[89] A.M. Wazwaz: “A reliable modification of Adomian’s decomposition
Method”. Appl. Math.Comput. 92 (1998) 1–7.
[90] A. M. Wazwaz: “The tanh-coth method for solitons and kink solutions
for nonlinear parabolic equations”.
Appl. Math. Comput. 188 (2007) 1467-1475.
[91] A. M. Wazwaz: “Partial Differential Equations and Solitary Waves
Theory”. Springer-Verlag Berlin Heidelberg 2009
[92] J. Weiss, M. Tabor and G. Carnevale: “The Painleve Property for Partial
Differential Equations”. Journal of Math. Physics 24 (1982) 522-526.
[93] J. Weiss, M. Tabor and G. Carnevale: “The Painleve Property ”.
J. of Math. Phys. 24 (1983) 1405-
[94] G. Whitham: “Linear and Nonlinear Waves”. Wiley, NY 1974
[95] K. Yahya, J. Biafar, H. Azari and P. R. Fard: “Homotopy Perturbation
Method for Image Restoration and Denoising”. Available online.
[96] O. Yu. Yefimova: “The modified simplest equation method to look for
exact solutions of nonlinear partial differential equations”.
arXiv:1011.4606v1 [nlin.SI] 20 Nov 2010
[97] N. J. Zabusky and M. D. Kruskal: "Interaction of "solitons" in a
collisionless plasma and the recurrence of initial states"
Phys. Rev. Letters 15 (1965) 240-243
33
[98] E. M. E. Zayed: “Traveling wave solutions for higher dimensional
nonlinear evolution equations using the −′ )G/G( expansion method”.
J. of Appl. Math. & Inform. 28 (2010) 383- 395
[99] E. M. E. Zayed: “A note on the modified simple equation method applied
to Sharma-Tasso-Olver equation”.
Appl. Math. Comp. 218 (2011) 3962-3964
[100] E. M. E. Zayed and K. A. Gepreel: “The (G′/G)-expansion method for
finding traveling wave solutions of nonlinear partial differential equations
in mathematical physics”. J. Math. Phys. 50 (2009) 013502
[101] X. L. Zhang and H. Q. Zhang: “A new generalized Riccati equation
rational expansion method to a class of nonlinear evolution equations with
nonlinear terms of any order”. Appl. Math. and Comp.186 (2007) 705-714