Finding General Solutions of Nonlinear Evolution
Transcript Of Finding General Solutions of Nonlinear Evolution
Math. Sci. Lett. 3, No. 1, 1-8 (2014)
1
Mathematical Sciences Letters
An International Journal
http://dx.doi.org/10.12785/msl/030101
Finding General Solutions of Nonlinear Evolution ′
Equations by Improved (GG )-expansion Method
E. Osman1, M. Khalfallah2 and H. Sapoor1,∗ 1 Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt 2 Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
Received: 6 Jun. 2013, Revised: 2 Oct. 2013, Accepted: 3 Oct. 2013 Published online: 1 Jan. 2014
Abstract: In this paper, an improved method named the improved ( GG′ )-expansion is introduced. Then we studied the modified equal width (MEW) and modified Benjamin-Bona-Mahony (MBBM) equations. The periodic and solitary wave solutions are constructed by using improved ( GG′ )-expansion method. Some new traveling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions. The solitary wave solutions are derived from the hyperbolic function solutions.
Keywords: Improved ( GG′ )-expansion method, modified equal width wave equation, modified Benjamin–Bona–Mahony equation,Solitary wave solution.
1 Introduction
Nonlinear wave phenomena play a major role in sciences such as plasma physics, optical fibers, fluid mechanics, chemical physics and geo-chemistry. Many powerful methods have been proposed to obtain exact solutions of nonlinear evolution equations, such as sine–cosine method [1, 2, 3, 4, 5], extended tanh method [6, 7, 8, 9], Hirota’s bilinear scheme [10], homogeneous balance method [11], Riccati equation rational expansion method [12, 13], and so on.
In recent years, with the development of symbolic
computation packages like Maple and Mathematica,
which enable us to perform the tedious and complex
′
computations
on
computer
The
(
G G
)-expansion
method
proposed by Wang et al. [14], is one of the most effective
direct methods to obtain travelling wave solutions of a
large number of NLEEs,such as the KdV equation,the
mKdV equation, the variant Boussinesq equations,the
Hirota–Satsuma equations,and so on .Later, the further
developed methods named the generalized
′
′
(
G G
)-expansion
method,the
modified
(
G G
)-expansion
∗ Corresponding author e-mail: hussien [email protected]
′
method
and
the
extended
(
G G
)-expansion
method
have
been proposed in Refs. [15, 16, 17], respectively. As we
know, when using the direct method, the choice of an
appropriate ansatz is great importance . In this paper, by
introducing a new general ansatze, we propose the ′
improved ( GG )-expansion method, which can be used to obtain travelling wave solutions of NLEEs.
2 Description of the improved ( GG′ )-expansion method
Suppose that a nonlinear evolution equation, say in two independent variables x and t, is given by
N(u, ut , ux, utt , uxx, uxt , ...) = 0,
(2.1)
where u = u(x,t) is an unknown function, N is a polynomial in u = u(x,t) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. To determine u explicitly, we take the following five steps:
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
Step 1: Use the travelling wave transformation:
u(x,t) = u(ξ ),
ξ = x − wt, (2.2)
where w is a constant to be determined latter. Then, the NLEE (2.1) is reduced to a nonlinear ordinary differential equation (NLODE) for u = u(ξ )
N(u, −wu′ , u′ , w2u′′ , u′′ , −wu′′ , ...) = 0. (2.3)
Step 2: We suppose that the NLODE (2.3) has the following solution:
∑ ∑ ′
′
−1
u(ξ ) =
ai(
G G
)i
m
+ a0 +
ai( GG )i , (2.4)
(1 + β ( G′ ))i
i=1 (1 + β ( G′ ))i
i=−m
G
G
where β , ai (i = −m, −m + 1, ..., m − 1, m) are constants to be determined later, m is a positive integer, and G =
G(ξ ) satisfies the following second order linear ordinary differential equation(LODE) :
G′′ + µG = 0, (2.5)
where µ is a real constant. The general solutions of Eq. (2.5) can be listed as follows.When µ < 0, we obtain the hyperbolic function solution of Eq. (2.5)
G(ξ ) = A1 cosh √−µξ + A2 sinh √−µξ , (2.6)
where A1 and A2 are arbitrary constants.When µ > 0, we obtain the trigonometric function solution of Eq. (2.5)
G(ξ ) = A1 sin √µξ + A2 cos √µξ ,
(2.7)
where A1 and A2 are arbitrary constants.When µ = 0, we obtain the rational function solution of Eq. (2.5)
G(ξ ) = A1 + A2ξ
(2.8)
where A1 and A2 are arbitrary constants.
Step 3:
Determine the positive integer m by balancing the highest
order derivative and nonlinear terms in Eq. (2.3).
Step 4:
Substituting (2.4) along with Eq. (2.5) into Eq. (2.3) and ′
then setting all the coefficients of ( GG ) of the resulting system’s numerator to zero, yields a set of
over-determined nonlinear algebraic equations for w, β
and ai(i = −m, −m + 1, ..., m − 1, m).
Step 5:
Assuming
that
the
constants
w, β ,
ai(i = −m, −m + 1, ..., m − 1, m) can be obtained by
solving the algebraic equations in Step 4, then
substituting these constants and the known general
solutions of Eq. (2.5) into (2.4), we can obtain the explicit
solutions of Eq. (2.1) immediately.
3 General traveling wave solutions of the MEW equation
We first consider the modified equal width equation in its normalized form [18, 19]
ut + a(u3)x + buxxt = 0,
(3.1)
where a and b are arbitrary constants. Making the transformationu(x,t) = u(x − ct) = u(ξ ),
then substituting u(x − ct) into (3.1) we obtain the
following nonlinear ODE equation
−cu′ + a(u3)′ − bcu′′′ = 0,
(3.2)
where”′” is the derivative with respect to ξ (i.e. u′ = uξ ) and c is wave speed.
Integrating (3.2) once and setting the integral constant
as zero, we obtain wave equation
−cu + a(u3) − bcu′′ = 0.
(3.3)
By balancing the highest order derivative terms and nonlinear terms in Eq. (3.3), we find that Eq. (3.3) own the solutions in the form
′
′
u(ξ ) = a0 +
a1(
G G
)
+
b1(1
+
β
(
G G
))
,
+ β ( G′ )
( G′ )
1
G
G
(3.4)
where G = G(ξ ) satisfies Eq. (2.5),β , a0, a1,b1, a and b are
constants to be determined latter.
Substituting (3.4) along with Eq. (2.5) into Eq. (3.3)
′
and
then
setting
all
the
coefficients
of
(
G G
)
of
the
resulting system’s numerator to zero, yields a set of
over-determined nonlinear algebraic equations for
β , a0, a1,b1, a and b. Solving the over-determined
algebraic equations by Maple or Mathematica, we can
obtain the following sets of solutions
When µ < 0, we obtain the hyperbolic function solutions
of Eq. (3.3)
Case 1: a0 = 0, a1 = 0, b1 = ∓
2acb , µ = −2b1 , β = 0
u(ξ ) = ∓ c × 2ab A1 cosh √−µξ + A2 sinh √−µξ )
√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,
(3.5)
where A1, A2 are arbitrary constants In particular, when setting A1 = 0 , A2 = 0, the solutions
(3.5) can be written as
u(ξ ) = ∓
c tanh
1 ξ.
(3.6)
a
2b
Setting again A1 > 0, A21 > A22 the following kink-shaped solution of Eq. (3.3)
u(ξ ) = ± c tanh( 1 ξ + ξ0),
(3.7)
a
2b
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where ξ0 = tanh−1 AA12 .
Case 2: a0 = 0, a1 = ∓
2abc , b1 = 0, µ = −2b1 , β = 0
u(ξ ) = ∓ 2bc ×
√a
√
√
( −µ(A1 sinh −µξ + A2 cosh −µξ ))
A1 cosh √−µξ + A2 sinh √−µξ ,
(3.8)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.8) can be written as
u(ξ ) = ±
c coth
1 ξ.
(3.9)
a
2b
Setting again A1 > 0, A21 > A22 the following singular soliton solution of Eq. (3.3)
u(ξ ) = ± where ξ0 = tanh−1 AA12 .
c coth( a
1 ξ + ξ0), 2b
(3.10)
the 3D graphs of kink and anti-kink wave solutions are shown in Fig. 1. In the graphs, the abscissa axis is t, the cordinate axis is x and the vertical axis is u.
Case 3: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , µ = −8b1 , β =0
u(ξ ) = ∓ 2bc ×
√a
√
√
( −µ(A1 sinh −µξ + A2 cosh −µξ ))
A1 cosh √−µξ + A2 sinh √−µξ ∓
1 c× 4 2ab
(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,
(3.11)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.11) can be written as
u(ξ ) = ∓
c tanh
1 ξ∓
c coth
1 ξ . (3.12)
4a
8b
4a
8b
Fig. 1: The 3D graphs of Eq. (3.6) and Eq.(3.9) as a=2.5, b=2, c=4 : (a)kink wave 1, (b) anti-kink wave 1, (c) kink wave 2, (d) anti-kink wave 2
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0
u(ξ ) = ∓ 2bc ×
√a
√
√
( −µ(A1 sinh −µξ + A2 cosh −µξ ))
A1 cosh √−µξ + A2 sinh √−µξ ∓
1 c× 2 2ab
(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,
(3.13)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.13) can be written as
u(ξ ) = ∓I
c tanh
−1 ξ ± I coth
−1 ξ ,
(3.14)
2a
4b
4b
√ where I = −1.
When µ > 0 ,b ⋖ 0 we get the trigonometric function
solutions of Eq. (3.3)
Case 1: a0 = 0, a1 = 0, b1 = ∓
2acb , µ = −2b1 , β = 0
u(ξ ) = ∓ c × 2ab
A1 sin √µξ + A2 cos √µξ ) √µ(A1 cos √µξ − A2 sin √µξ ) ,
(3.15)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.15) can be written as
u(ξ ) = ∓ −c cot −1 ξ .
a
2b
(3.16)
Case 2: a0 = 0, a1 = ∓
2abc , b1 = 0, µ = −2b1 , β = 0
u(ξ ) = ∓ 2bc ×
√a √
√
µ(A1 cos µξ − A2 sin µξ )
A1 sin √µξ + A2 cos √µξ ) ,
(3.17)
where A1, A2 are arbitrary constants. Setting A1 = 0 , A2 = 0, we get travelling wave
solutions of the type of tangent function
u(ξ ) = ± −c tan −1 ξ .
a
2b
(3.18)
The 3D graphs of travelling wave solutions of the type of tangent function are shown in Fig. 2. In the graphs, the abscissa axis is t, the ordinate axis is x and the vertical axis is u.
Fig. 2: The 3D graphs of (3.18) as a=-2,b=-3,c=4,,x∈(-0.5,0.5)
Case 3:
a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , β =0
u(ξ ) = ∓ 2bc ×
√a √
√
( µ(A1 cos µξ − A2 sin µξ ))
A1 sin √µξ + A2 cos √µξ ∓
1 c× 4 2ab
(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,
µ = −8b1 , (3.19)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.19) can be written as
u(ξ ) = ±I
c tan
−1 ξ ∓ I
c cot
−1 ξ , (3.20)
4a
8b
4a
8b
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√ where I = −1.
Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0
u(ξ ) = ∓ 2bc ×
√a √
√
( µ(A1 cos µξ − A2 sin µξ ))
A1 sin √µξ + A2 cos √µξ ∓
1 c× 2 2ab
(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,
(3.21)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.21) can be written as
u(ξ ) = ±
c tan
1 ξ±
c cot
1 ξ , (3.22)
2a
4b
2a
4b
When µ = 0 we get the rational function solutions of Eq. (3.3)
u(ξ ) = ∓ 2bc ( A2 ), a A1 + A2ξ
(3.23)
where A1, A2 are arbitrary constants.
4 General traveling wave solutions of the MBBM equation
′
We
now
employ
the
improved
(
G G
)
-expansion
to
the
modified Benjamin–Bona–Mahony equation [20]
ut + ux + u2ux + uxxt = 0
(4.1)
As described in Section 3, the wave variable
u(x,t) = u(x − ct) = u(ξ )
carries (4.1) into the ODE (1 − c)u′ + u2u′ − cu′′′ = 0,
(4.2)
Integrating (4.2) and setting the constant of integration to be zero we obtain.
(1 − c)u + u3 − cu′′ = 0, 3
(4.3)
According to equaion (3.4) in section 3 we have When µ < 0, we obtain the hyperbolic function
solutions of Eq. (4.3)
Case 1:
√
(c−1)
a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0
√√
√
√
u(ξ ) = ±
6c
−µ(A1 sinh √
−µξ + A2 cosh √
−µξ ) ,
A1 cosh −µξ + A2 sinh −µξ )
(4.4)
where A1, A2 are arbitrary constants
In particular, when setting A2 = 0 , A1 = 0, the solutions
(4.4) can be written as
u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ . (4.5) 2c
setting again A1 = 0 , A2 = 0, the solutions (4.4) can be written as
u(ξ ) = ± 3(c − 1) coth (c − 1) ξ . (4.6) 2c
Case 2: a0 = 0,
β =0
√ a1 = ± 6c,
b1 = ∓ (1−4 c)
6c ,
µ
=
(c−1)
4c ,
√√
√
√
u(ξ ) = ±
6c
−µ(A1 sinh √
−µξ + A2 cosh √
−µξ )
A1 cosh −µξ + A2 sinh −µξ )
∓ (1 − c) 6 ×
4 √c
√
A1 cosh −µξ + A2 sinh −µξ )
√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .
(4.7)
In particular, when setting A2 = 0 , A1 = 0, the solutions (4.7) can be written as
u(ξ ) = ± ∓
3(1 − c) tanh 2
3(1 − c) coth 2
(1 − c) ξ 4c
(1 − c) ξ . 4c
(4.8)
Case 3: a0 = 0,
µ = − (c8−c1) ,
√ a1 = ± 6c, β =0
b1 = ∓(1 − c) 332c ,
√√
√
√
u(ξ ) = ±
6c
−µ(A1 sinh √
−µξ + A2 cosh √
−µξ )
A1 cosh −µξ + A2 sinh −µξ )
∓(1 − c) 3
√32c
√
A1 cosh −µξ + A2 sinh −µξ )
√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .
(4.9)
In particular, when setting A2 = 0 , A1 = 0, the solutions (4.9) can be written as
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ
4
8c
± 3(c − 1) coth (c − 1) ξ .
4
8c
(4.10)
When µ > 0 we get the trigonometric function solutions
of Eq. (4.3)
Case 1:
√
(c−1)
a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0
√√
√
√
u(ξ ) = ±
6c
µ(A1 cos √
µξ − A2 sin √
µξ) .
A1 sin µξ + A2 cos µξ )
(4.11)
In particular, when setting A2 = 0 , A1 = 0, the solutions (4.11) can be written as
u(ξ ) = ± 3(1 − c) cot (1 − c) ξ . 2c
(4.12)
setting again A1 = 0 , A2 = 0, the solutions (4.11) can be written as
u(ξ ) = ± 3(1 − c) tan (1 − c) ξ . 2c
(4.13)
The 3D graphs of travelling wave solutions of the type of
tangent function are shown in Fig. 3. In the graphs, the
abscissa axis is t, the ordinate axis is x and the vertical
axis is u.
Case 2:
√
a0 = 0, a1 = ± 6c, b1 = ∓ (1−4 c)
6c ,
µ
=
(c−1)
4c ,
β =0
√√
√
√
u(ξ ) = ±
6c(
µ(A1 cos √
µξ − A2 sin √
µξ ))
A1 sin µξ + A2 cos µξ
∓ (1 − c) 6
4 √c
√
(A1 sin µξ + A2 cos µξ )
× √µ ((A1 cos √µξ − A2 sin √µξ )) .
(4.14)
setting again A1 = 0 , A2 = 0, the solutions (4.14) can be written as
u(ξ ) = ∓ ∓
3(c − 1) tan 2
3(c − 1) cot 2
(c − 1) ξ 4c
(c − 1) ξ . 4c
(4.15)
Fig. 3: The 3D graphs of (4.13) as c=1/2, x∈(-0.5,0.5)
Case 3: a0 = 0,
µ = − (c8−c1) ,
√ a1 = ± 6c, β =0
b1 = ∓(1 − c) 332c ,
√√
√
√
u(ξ ) = ±
6c(
µ(A1 cos √
µξ − A2 sin √
µξ ))
A1 sin µξ + A2 cos µξ
∓(1 − c) 3
3√2c
√
(A1 sin µξ + A2 cos µξ )
× √µ ((A1 cos √µξ − A2 sin √µξ )) .
(4.16)
setting A1 = 0 , A2 = 0, the solutions (4.16) can be written as
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u(ξ ) = ∓ ±
3(1 − c) tan 4
3(1 − c) cot 2
(1 − c) ξ 8c
(1 − c) ξ . 8c
(4.17)
When µ = 0 we get the rational function solutions of Eq.
(4.3)
√
A2
u(ξ ) = ± 6( A1 + A2ξ ),
(4.18)
where A1, A2 are arbitrary constants.
5 Conclusions
′
The
improved
(
G G
)-expansion
method
is
applied
successfully for solving the modified equal width (MEW)
and modified Benjamin–Bona–Mahony (MBBM)
equations.These exact solutions include the hyperbolic
function solutions, trigonometric function solutions and
rational function solutions. When the parameters are
taken as special values, the solitary wave solutions are
derived from the hyperbolic function solutions. This
method has more advantages: it is direct and concise.
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
El-sayed
Osman
received the PhD degree
in Mathematics at Clarkson
University
(USA).His
research interests are in the
areas of applied mathematics
including Soliton theory
and mathematical methods
and models. He has published
research articles in Egyptian
and international journals of mathematical sciences.
Mohamed Khalfallah received the PhD degree in applied mathematics in ”The theory of Soliton”, Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt. He has a number of papers published in many of scientific journals.
Hussien
Sapoor
received the Master degree
in applied mathematics. His
research interests are in the
areas of applied mathematics
”The theory of Soliton”,
Mathematics Department,
Faculty of Science,Sohage
University, Sohage, Egypt.
c 2014 NSP Natural Sciences Publishing Cor.
1
Mathematical Sciences Letters
An International Journal
http://dx.doi.org/10.12785/msl/030101
Finding General Solutions of Nonlinear Evolution ′
Equations by Improved (GG )-expansion Method
E. Osman1, M. Khalfallah2 and H. Sapoor1,∗ 1 Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt 2 Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
Received: 6 Jun. 2013, Revised: 2 Oct. 2013, Accepted: 3 Oct. 2013 Published online: 1 Jan. 2014
Abstract: In this paper, an improved method named the improved ( GG′ )-expansion is introduced. Then we studied the modified equal width (MEW) and modified Benjamin-Bona-Mahony (MBBM) equations. The periodic and solitary wave solutions are constructed by using improved ( GG′ )-expansion method. Some new traveling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions. The solitary wave solutions are derived from the hyperbolic function solutions.
Keywords: Improved ( GG′ )-expansion method, modified equal width wave equation, modified Benjamin–Bona–Mahony equation,Solitary wave solution.
1 Introduction
Nonlinear wave phenomena play a major role in sciences such as plasma physics, optical fibers, fluid mechanics, chemical physics and geo-chemistry. Many powerful methods have been proposed to obtain exact solutions of nonlinear evolution equations, such as sine–cosine method [1, 2, 3, 4, 5], extended tanh method [6, 7, 8, 9], Hirota’s bilinear scheme [10], homogeneous balance method [11], Riccati equation rational expansion method [12, 13], and so on.
In recent years, with the development of symbolic
computation packages like Maple and Mathematica,
which enable us to perform the tedious and complex
′
computations
on
computer
The
(
G G
)-expansion
method
proposed by Wang et al. [14], is one of the most effective
direct methods to obtain travelling wave solutions of a
large number of NLEEs,such as the KdV equation,the
mKdV equation, the variant Boussinesq equations,the
Hirota–Satsuma equations,and so on .Later, the further
developed methods named the generalized
′
′
(
G G
)-expansion
method,the
modified
(
G G
)-expansion
∗ Corresponding author e-mail: hussien [email protected]
′
method
and
the
extended
(
G G
)-expansion
method
have
been proposed in Refs. [15, 16, 17], respectively. As we
know, when using the direct method, the choice of an
appropriate ansatz is great importance . In this paper, by
introducing a new general ansatze, we propose the ′
improved ( GG )-expansion method, which can be used to obtain travelling wave solutions of NLEEs.
2 Description of the improved ( GG′ )-expansion method
Suppose that a nonlinear evolution equation, say in two independent variables x and t, is given by
N(u, ut , ux, utt , uxx, uxt , ...) = 0,
(2.1)
where u = u(x,t) is an unknown function, N is a polynomial in u = u(x,t) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. To determine u explicitly, we take the following five steps:
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
Step 1: Use the travelling wave transformation:
u(x,t) = u(ξ ),
ξ = x − wt, (2.2)
where w is a constant to be determined latter. Then, the NLEE (2.1) is reduced to a nonlinear ordinary differential equation (NLODE) for u = u(ξ )
N(u, −wu′ , u′ , w2u′′ , u′′ , −wu′′ , ...) = 0. (2.3)
Step 2: We suppose that the NLODE (2.3) has the following solution:
∑ ∑ ′
′
−1
u(ξ ) =
ai(
G G
)i
m
+ a0 +
ai( GG )i , (2.4)
(1 + β ( G′ ))i
i=1 (1 + β ( G′ ))i
i=−m
G
G
where β , ai (i = −m, −m + 1, ..., m − 1, m) are constants to be determined later, m is a positive integer, and G =
G(ξ ) satisfies the following second order linear ordinary differential equation(LODE) :
G′′ + µG = 0, (2.5)
where µ is a real constant. The general solutions of Eq. (2.5) can be listed as follows.When µ < 0, we obtain the hyperbolic function solution of Eq. (2.5)
G(ξ ) = A1 cosh √−µξ + A2 sinh √−µξ , (2.6)
where A1 and A2 are arbitrary constants.When µ > 0, we obtain the trigonometric function solution of Eq. (2.5)
G(ξ ) = A1 sin √µξ + A2 cos √µξ ,
(2.7)
where A1 and A2 are arbitrary constants.When µ = 0, we obtain the rational function solution of Eq. (2.5)
G(ξ ) = A1 + A2ξ
(2.8)
where A1 and A2 are arbitrary constants.
Step 3:
Determine the positive integer m by balancing the highest
order derivative and nonlinear terms in Eq. (2.3).
Step 4:
Substituting (2.4) along with Eq. (2.5) into Eq. (2.3) and ′
then setting all the coefficients of ( GG ) of the resulting system’s numerator to zero, yields a set of
over-determined nonlinear algebraic equations for w, β
and ai(i = −m, −m + 1, ..., m − 1, m).
Step 5:
Assuming
that
the
constants
w, β ,
ai(i = −m, −m + 1, ..., m − 1, m) can be obtained by
solving the algebraic equations in Step 4, then
substituting these constants and the known general
solutions of Eq. (2.5) into (2.4), we can obtain the explicit
solutions of Eq. (2.1) immediately.
3 General traveling wave solutions of the MEW equation
We first consider the modified equal width equation in its normalized form [18, 19]
ut + a(u3)x + buxxt = 0,
(3.1)
where a and b are arbitrary constants. Making the transformationu(x,t) = u(x − ct) = u(ξ ),
then substituting u(x − ct) into (3.1) we obtain the
following nonlinear ODE equation
−cu′ + a(u3)′ − bcu′′′ = 0,
(3.2)
where”′” is the derivative with respect to ξ (i.e. u′ = uξ ) and c is wave speed.
Integrating (3.2) once and setting the integral constant
as zero, we obtain wave equation
−cu + a(u3) − bcu′′ = 0.
(3.3)
By balancing the highest order derivative terms and nonlinear terms in Eq. (3.3), we find that Eq. (3.3) own the solutions in the form
′
′
u(ξ ) = a0 +
a1(
G G
)
+
b1(1
+
β
(
G G
))
,
+ β ( G′ )
( G′ )
1
G
G
(3.4)
where G = G(ξ ) satisfies Eq. (2.5),β , a0, a1,b1, a and b are
constants to be determined latter.
Substituting (3.4) along with Eq. (2.5) into Eq. (3.3)
′
and
then
setting
all
the
coefficients
of
(
G G
)
of
the
resulting system’s numerator to zero, yields a set of
over-determined nonlinear algebraic equations for
β , a0, a1,b1, a and b. Solving the over-determined
algebraic equations by Maple or Mathematica, we can
obtain the following sets of solutions
When µ < 0, we obtain the hyperbolic function solutions
of Eq. (3.3)
Case 1: a0 = 0, a1 = 0, b1 = ∓
2acb , µ = −2b1 , β = 0
u(ξ ) = ∓ c × 2ab A1 cosh √−µξ + A2 sinh √−µξ )
√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,
(3.5)
where A1, A2 are arbitrary constants In particular, when setting A1 = 0 , A2 = 0, the solutions
(3.5) can be written as
u(ξ ) = ∓
c tanh
1 ξ.
(3.6)
a
2b
Setting again A1 > 0, A21 > A22 the following kink-shaped solution of Eq. (3.3)
u(ξ ) = ± c tanh( 1 ξ + ξ0),
(3.7)
a
2b
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3
where ξ0 = tanh−1 AA12 .
Case 2: a0 = 0, a1 = ∓
2abc , b1 = 0, µ = −2b1 , β = 0
u(ξ ) = ∓ 2bc ×
√a
√
√
( −µ(A1 sinh −µξ + A2 cosh −µξ ))
A1 cosh √−µξ + A2 sinh √−µξ ,
(3.8)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.8) can be written as
u(ξ ) = ±
c coth
1 ξ.
(3.9)
a
2b
Setting again A1 > 0, A21 > A22 the following singular soliton solution of Eq. (3.3)
u(ξ ) = ± where ξ0 = tanh−1 AA12 .
c coth( a
1 ξ + ξ0), 2b
(3.10)
the 3D graphs of kink and anti-kink wave solutions are shown in Fig. 1. In the graphs, the abscissa axis is t, the cordinate axis is x and the vertical axis is u.
Case 3: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , µ = −8b1 , β =0
u(ξ ) = ∓ 2bc ×
√a
√
√
( −µ(A1 sinh −µξ + A2 cosh −µξ ))
A1 cosh √−µξ + A2 sinh √−µξ ∓
1 c× 4 2ab
(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,
(3.11)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.11) can be written as
u(ξ ) = ∓
c tanh
1 ξ∓
c coth
1 ξ . (3.12)
4a
8b
4a
8b
Fig. 1: The 3D graphs of Eq. (3.6) and Eq.(3.9) as a=2.5, b=2, c=4 : (a)kink wave 1, (b) anti-kink wave 1, (c) kink wave 2, (d) anti-kink wave 2
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0
u(ξ ) = ∓ 2bc ×
√a
√
√
( −µ(A1 sinh −µξ + A2 cosh −µξ ))
A1 cosh √−µξ + A2 sinh √−µξ ∓
1 c× 2 2ab
(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,
(3.13)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.13) can be written as
u(ξ ) = ∓I
c tanh
−1 ξ ± I coth
−1 ξ ,
(3.14)
2a
4b
4b
√ where I = −1.
When µ > 0 ,b ⋖ 0 we get the trigonometric function
solutions of Eq. (3.3)
Case 1: a0 = 0, a1 = 0, b1 = ∓
2acb , µ = −2b1 , β = 0
u(ξ ) = ∓ c × 2ab
A1 sin √µξ + A2 cos √µξ ) √µ(A1 cos √µξ − A2 sin √µξ ) ,
(3.15)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.15) can be written as
u(ξ ) = ∓ −c cot −1 ξ .
a
2b
(3.16)
Case 2: a0 = 0, a1 = ∓
2abc , b1 = 0, µ = −2b1 , β = 0
u(ξ ) = ∓ 2bc ×
√a √
√
µ(A1 cos µξ − A2 sin µξ )
A1 sin √µξ + A2 cos √µξ ) ,
(3.17)
where A1, A2 are arbitrary constants. Setting A1 = 0 , A2 = 0, we get travelling wave
solutions of the type of tangent function
u(ξ ) = ± −c tan −1 ξ .
a
2b
(3.18)
The 3D graphs of travelling wave solutions of the type of tangent function are shown in Fig. 2. In the graphs, the abscissa axis is t, the ordinate axis is x and the vertical axis is u.
Fig. 2: The 3D graphs of (3.18) as a=-2,b=-3,c=4,,x∈(-0.5,0.5)
Case 3:
a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , β =0
u(ξ ) = ∓ 2bc ×
√a √
√
( µ(A1 cos µξ − A2 sin µξ ))
A1 sin √µξ + A2 cos √µξ ∓
1 c× 4 2ab
(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,
µ = −8b1 , (3.19)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.19) can be written as
u(ξ ) = ±I
c tan
−1 ξ ∓ I
c cot
−1 ξ , (3.20)
4a
8b
4a
8b
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√ where I = −1.
Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0
u(ξ ) = ∓ 2bc ×
√a √
√
( µ(A1 cos µξ − A2 sin µξ ))
A1 sin √µξ + A2 cos √µξ ∓
1 c× 2 2ab
(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,
(3.21)
In particular, when setting A1 = 0 , A2 = 0, the solutions (3.21) can be written as
u(ξ ) = ±
c tan
1 ξ±
c cot
1 ξ , (3.22)
2a
4b
2a
4b
When µ = 0 we get the rational function solutions of Eq. (3.3)
u(ξ ) = ∓ 2bc ( A2 ), a A1 + A2ξ
(3.23)
where A1, A2 are arbitrary constants.
4 General traveling wave solutions of the MBBM equation
′
We
now
employ
the
improved
(
G G
)
-expansion
to
the
modified Benjamin–Bona–Mahony equation [20]
ut + ux + u2ux + uxxt = 0
(4.1)
As described in Section 3, the wave variable
u(x,t) = u(x − ct) = u(ξ )
carries (4.1) into the ODE (1 − c)u′ + u2u′ − cu′′′ = 0,
(4.2)
Integrating (4.2) and setting the constant of integration to be zero we obtain.
(1 − c)u + u3 − cu′′ = 0, 3
(4.3)
According to equaion (3.4) in section 3 we have When µ < 0, we obtain the hyperbolic function
solutions of Eq. (4.3)
Case 1:
√
(c−1)
a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0
√√
√
√
u(ξ ) = ±
6c
−µ(A1 sinh √
−µξ + A2 cosh √
−µξ ) ,
A1 cosh −µξ + A2 sinh −µξ )
(4.4)
where A1, A2 are arbitrary constants
In particular, when setting A2 = 0 , A1 = 0, the solutions
(4.4) can be written as
u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ . (4.5) 2c
setting again A1 = 0 , A2 = 0, the solutions (4.4) can be written as
u(ξ ) = ± 3(c − 1) coth (c − 1) ξ . (4.6) 2c
Case 2: a0 = 0,
β =0
√ a1 = ± 6c,
b1 = ∓ (1−4 c)
6c ,
µ
=
(c−1)
4c ,
√√
√
√
u(ξ ) = ±
6c
−µ(A1 sinh √
−µξ + A2 cosh √
−µξ )
A1 cosh −µξ + A2 sinh −µξ )
∓ (1 − c) 6 ×
4 √c
√
A1 cosh −µξ + A2 sinh −µξ )
√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .
(4.7)
In particular, when setting A2 = 0 , A1 = 0, the solutions (4.7) can be written as
u(ξ ) = ± ∓
3(1 − c) tanh 2
3(1 − c) coth 2
(1 − c) ξ 4c
(1 − c) ξ . 4c
(4.8)
Case 3: a0 = 0,
µ = − (c8−c1) ,
√ a1 = ± 6c, β =0
b1 = ∓(1 − c) 332c ,
√√
√
√
u(ξ ) = ±
6c
−µ(A1 sinh √
−µξ + A2 cosh √
−µξ )
A1 cosh −µξ + A2 sinh −µξ )
∓(1 − c) 3
√32c
√
A1 cosh −µξ + A2 sinh −µξ )
√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .
(4.9)
In particular, when setting A2 = 0 , A1 = 0, the solutions (4.9) can be written as
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ
4
8c
± 3(c − 1) coth (c − 1) ξ .
4
8c
(4.10)
When µ > 0 we get the trigonometric function solutions
of Eq. (4.3)
Case 1:
√
(c−1)
a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0
√√
√
√
u(ξ ) = ±
6c
µ(A1 cos √
µξ − A2 sin √
µξ) .
A1 sin µξ + A2 cos µξ )
(4.11)
In particular, when setting A2 = 0 , A1 = 0, the solutions (4.11) can be written as
u(ξ ) = ± 3(1 − c) cot (1 − c) ξ . 2c
(4.12)
setting again A1 = 0 , A2 = 0, the solutions (4.11) can be written as
u(ξ ) = ± 3(1 − c) tan (1 − c) ξ . 2c
(4.13)
The 3D graphs of travelling wave solutions of the type of
tangent function are shown in Fig. 3. In the graphs, the
abscissa axis is t, the ordinate axis is x and the vertical
axis is u.
Case 2:
√
a0 = 0, a1 = ± 6c, b1 = ∓ (1−4 c)
6c ,
µ
=
(c−1)
4c ,
β =0
√√
√
√
u(ξ ) = ±
6c(
µ(A1 cos √
µξ − A2 sin √
µξ ))
A1 sin µξ + A2 cos µξ
∓ (1 − c) 6
4 √c
√
(A1 sin µξ + A2 cos µξ )
× √µ ((A1 cos √µξ − A2 sin √µξ )) .
(4.14)
setting again A1 = 0 , A2 = 0, the solutions (4.14) can be written as
u(ξ ) = ∓ ∓
3(c − 1) tan 2
3(c − 1) cot 2
(c − 1) ξ 4c
(c − 1) ξ . 4c
(4.15)
Fig. 3: The 3D graphs of (4.13) as c=1/2, x∈(-0.5,0.5)
Case 3: a0 = 0,
µ = − (c8−c1) ,
√ a1 = ± 6c, β =0
b1 = ∓(1 − c) 332c ,
√√
√
√
u(ξ ) = ±
6c(
µ(A1 cos √
µξ − A2 sin √
µξ ))
A1 sin µξ + A2 cos µξ
∓(1 − c) 3
3√2c
√
(A1 sin µξ + A2 cos µξ )
× √µ ((A1 cos √µξ − A2 sin √µξ )) .
(4.16)
setting A1 = 0 , A2 = 0, the solutions (4.16) can be written as
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7
u(ξ ) = ∓ ±
3(1 − c) tan 4
3(1 − c) cot 2
(1 − c) ξ 8c
(1 − c) ξ . 8c
(4.17)
When µ = 0 we get the rational function solutions of Eq.
(4.3)
√
A2
u(ξ ) = ± 6( A1 + A2ξ ),
(4.18)
where A1, A2 are arbitrary constants.
5 Conclusions
′
The
improved
(
G G
)-expansion
method
is
applied
successfully for solving the modified equal width (MEW)
and modified Benjamin–Bona–Mahony (MBBM)
equations.These exact solutions include the hyperbolic
function solutions, trigonometric function solutions and
rational function solutions. When the parameters are
taken as special values, the solitary wave solutions are
derived from the hyperbolic function solutions. This
method has more advantages: it is direct and concise.
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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...
El-sayed
Osman
received the PhD degree
in Mathematics at Clarkson
University
(USA).His
research interests are in the
areas of applied mathematics
including Soliton theory
and mathematical methods
and models. He has published
research articles in Egyptian
and international journals of mathematical sciences.
Mohamed Khalfallah received the PhD degree in applied mathematics in ”The theory of Soliton”, Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt. He has a number of papers published in many of scientific journals.
Hussien
Sapoor
received the Master degree
in applied mathematics. His
research interests are in the
areas of applied mathematics
”The theory of Soliton”,
Mathematics Department,
Faculty of Science,Sohage
University, Sohage, Egypt.
c 2014 NSP Natural Sciences Publishing Cor.