Traveling Wave Solutions of the Fourth Order Boussinesq
Transcript Of Traveling Wave Solutions of the Fourth Order Boussinesq
Physical Review & Research International
4(1): 255-266, 2014
SCIENCEDOMAIN international www.sciencedomain.org
Traveling Wave Solutions of the Fourth Order
Boussinesq Equation via the Improved (G '/ G )
Expansion Method
Ahata Shamul Alam1, M. M. Zaman2 and Sayeda Sultana3*
1Department of Statistics, Dhaka College, Bangladesh. 2Fortunex Limited, Baridhara, Bangladesh.
3Department of Economics, Kalachandpur School and College, Dhaka, Bangladesh.
Authors’ contributions
The work presented here was carried out in collaboration between three authors. Author ASA carried out the main part of this manuscript. Author MMZ discussed and polished the manuscript. Author SS data analyzed and polished the manuscript. All authors have read
and approved the manuscript.
Research Article
Received 17th March 2013 Accepted 11th May 2013
Published 16th October 2013
ABSTRACT
In this article, the fourth order Boussinesq equation is investigated to generate many new
traveling wave solutions via the improved (G '/ G) -expansion method. In the method the
second order linear ordinary differential equation with constant coefficients is used. Also, the resulted solutions are presented in three different families including solitons and periodic solutions. Furthermore, some of our solutions are coincided with published results which gained by other authors and some are new. Aims: The aim of this work is to construct many new exact traveling wave solutions including solitons, periodic and rational solutions of the fourth order Boussinesq equation
by applying the improved (G '/ G) -expansion method. Methodology: The improved (G '/ G) -expansion method is effective and powerful
mathematical tool for solving nonlinear partial differential equations which arise in mathematical physics, engineering sciences and other technical arena. In addition,
____________________________________________________________________________________________ *Corresponding author: Email: [email protected];
Physical Review & Research International, 4(1): 255-266, 2014
p
h
G′′ + λG′ + µG = 0, together with f (ϕ ) = ∑ ah (G '/ G ) , is implemented as traveling wave
h= − p
solutions, where either a− p or ap may be zero, but both a− p and ap cannot be zero at the
same time. Results: The obtained traveling wave solutions are described in terms of the hyperbolic functions, the trigonometric functions and the rational functions. Conclusion: The constructed solutions may express a variety of new features of waves, further, may be valuable in the theoretical and numerical studies of the considered
equation. Moreover, the obtained exact solutions reveal that the improved (G '/ G) -
expansion method is a promising mathematical tool, because, it can establish abundant new traveling wave solutions of different physical structures. Also, some of our solutions are in good agreement with already published results for a special case and others are new.
Keywords: The improved-expansion method; the Boussinesq equation; traveling wave solutions; nonlinear evolution equations.
1. INTRODUCTION
The study of nonlinear evolution equations (NLEEs) has now become one of the most exciting and tremendously active research areas to scientists. In recent time, many authors who are interested in the nonlinear complex physical phenomena generated analytical solutions of NLEEs. They introduced various methods, for instance, the inverse scattering method [1], the homogeneous balance method [2], the Hirota’s bilinear transformation method [3,4], the Jacobi elliptic function expansion method [5-7], the generalized Riccati equation method [8], the tanh-coth method [9-11], the F-expansion method [12], the Expfunction method [13-15] and others [16-18].
Wang et al. [19] presented a powerful method which is called the (G '/ G) -expansion
method. They generated exact traveling wave solutions in three different families including solitons, periodic and rational solutions for some nonlinear partial differential equations
(PDEs) via this method. In the method, u (ξ ) = ∑m ai (G '/ G)i is implemented as traveling
i=0
wave solutions, where am ≠ 0. Later on, several scientists investigated various nonlinear
PDEs to construct traveling wave solutions via this method [20-26]. More recently, Zhang et
al. [27] extended this method and called the improved (G '/ G) -expansion method. They
∑p
h
used f (ϕ ) = ah (G '/ G ) as traveling wave solutions, where either a− p or ap may be
h=− p
zero, but both a− p and ap cannot be zero simultaneously. After that, many researchers
studied different NLEEs to construct exact solutions by using this improved (G '/ G) -
expansion method. For example, Zhao et al. [28] executed the same method to establish exact solutions of the variant Boussinesq equations. Nofel et al. [29] constructed traveling wave solutions for the fifth-order KdV equation by using this method. Hamad et al. [30] studied higher dimensional potential YTSF equation to obtain analytical solutions via the
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Physical Review & Research International, 4(1): 255-266, 2014
same method. Naher and Abdullah [31] applied this powerful method to construct traveling wave solutions of the nonlinear reaction diffusion equation whilst they [32] implemented this method to generate exact solutions of the (2+1)-dimensional Modified Zakharov-Kuznetsov
equation via the improved (G '/ G) -expansion method. In Ref. [33] Naher et al. generated many solutions of the compound KdV-Burgers equation via the improved (G '/ G) -expansion
method and so on.
In this paper, we would like to investigate well established fourth order Boussinesq equation
by applying the improved (G '/ G) -expansion method to construct a rich class of new exact
traveling wave solutions including solitons, periodic and rational solutions.
2. DESCRIPTION OF THE IMPROVED (G '/ G) -EXPANSION METHOD
Let us consider the general nonlinear partial differential equation:
P (u,ut ,ux ,uxt ,ut t ,uxx ,...) = 0,
(1)
where u = u ( x,t ) is an unknown function, P is a polynomial in u ( x,t ) and its partial
derivatives in which the highest order partial derivatives and the nonlinear terms are involved.
The main steps of the method are as follows:
Step 1. Suppose the traveling wave variable:
u ( x,t ) = f (ϕ ), ϕ = x −Vt,
(2)
where V is the speed of the traveling wave. Using Eq. (2), Eq. (1) is transformed into an
ordinary differential equation for f = f (ϕ ) :
Q ( f , f ', f ′′, f ′′′,...) = 0,
(3)
where Q is a function of f (ϕ ) and its total derivatives.
Step 2. Eq. (3) can be integrated term by term one or more times, according to possibility, yields constant(s) of integration. The integral constant may be zero, for simplicity.
Step 3. Suppose that the traveling wave solution of Eq. (3) can be stated in the following
form:
p
h
f (ϕ ) = ∑ ah (G '/ G)
(4)
h= − p
with G = G (ϕ ) satisfies the second order linear ODE:
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Physical Review & Research International, 4(1): 255-266, 2014
G′′ + λG′ + µG = 0,
(5)
where ah (h = 0, ±1, ± 2,..., ± p), λ and µ are constants.
Step 4. The value of the positive integer p can be determined by taking the homogeneous balance between the highest order nonlinear terms and the highest order derivatives appearing in Eq. (3).
Step 5. Substitute Eq. (4) together with Eq. (5) into Eq. (3) with the value of p obtained in Step 4. Equating the coefficients of (G '/ G )r , (r = 0, ±1, ± 2,...), then setting each coefficient to zero, we obtain a set of algebraic equations for ah (h = 0, ±1, ± 2,..., ± p),V , λ and µ.
Step 6. Solve the system of algebraic equations which are obtained in step 5 with the aid of
commercial software Maple and we obtain values for ah (h = 0, ±1, ± 2,..., ± p),V , λ and µ.
Then, substitute obtained values in Eq. (4) along with the general solution of Eq. (5) with the value of p , we can obtain the traveling wave solutions of Eq. (1).
2.1 Application of the Method
In this subsection, we have studied the fourth order Boussinesq equation to construct exact
traveling wave solutions via the improved (G '/ G ) -expansion method.
2.1.1 The fourth order Boussinesq equation
We consider the fourth order Boussinesq equation followed by Naher and Abdullah [36]:
( ) utt − α 2 uxx − β u2 + uxxxx = 0.
(6)
xx
Now, we use the wave transformation Eq. (2) into the Eq. (6), therefore, integrating with
respect to ϕ twice and setting the constants of integration to zero, yields:
(V 2 −α 2 ) f − β f 2 + f ′′ = 0.
(7)
Taking the homogeneous balance between f 2 and f ′′ in Eq. (7), we obtain p = 2.
Therefore, the solution of Eq. (7) is of the form:
f (ϕ ) = a−2 (G '/ )G −2 + a−1 (G '/ )G −1 + a0 + a1 (G '/ G ) + a2 (G '/ G )2 ,
(8)
where a−2 , a−1, a0 , a1 and a2 are constants to be determined.
Substituting Eq. (8) together with Eq. (5) into the Eq. (7), the left-hand side of Eq. (7) is
converted into a polynomial of (G '/ G )r , (r = 0, ±1, ± 2,...). According to Step 5, collecting all terms with the same power of (G '/ G ). Then, setting each coefficient of the resulted
polynomial to zero, yields a set of algebraic equations (for simplicity, which are not
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Physical Review & Research International, 4(1): 255-266, 2014
presented) for a−2 , a−1, a0 , a1, a2 ,V , λ and µ. Solving the system of obtained algebraic
equations with the help of algebraic software Maple 13, we obtain four different values.
Case 1:
( ) a = 0, a = 0, a = 6µ , a = 6λ , a = 6 , V = ± α 2 − λ 2 − 4µ ,
(9)
−2
−1
0 β 1 β 2β
where α, β , λ and µ are free parameters.
Case 2:
( ) a = 0, a = 0, a = 2µ + λ 2 , a = 6λ , a = 6 , V = ± α 2 + λ2 − 4µ ,
(10)
−2
−1
0
β
1 β 2β
where α, β , λ and µ are free parameters.
Case 3:
( ) a = 6µ 2 , a = 6λµ , a = 6µ , a = 0, a = 0, V = ± α 2 − λ 2 − 4µ ,
(11)
−2
β −1
β
0β 1
2
where α, β , λ and µ are free parameters.
Case 4:
( ) a = 6µ2 , a = 6λµ , a = 2µ + λ2 , a = 0, a = 0, V = ± α 2 + λ2 − 4µ , (12)
−2 β −1 β
0
β
1
2
where α, β , λ and µ are free parameters.
Substituting the general solution Eq. (5) into Eq. (8), we obtain three different families of traveling wave solutions of Eq. (7):
Family 1: Hyperbolic function solutions:
When λ2 − 4µ > 0, we obtain
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Physical Review & Research International, 4(1): 255-266, 2014
f (ϕ ) = a
−λ 1 +
−2 2 2
P sinh 1
λ2 − 4µ
2
P cosh 1
2
λ2 − 4µ ϕ + Q cosh 1 2
λ 2 − 4µ ϕ + Q sinh 1 2
λ2
−
4µ
ϕ
−2
λ 2 − 4µ ϕ
−λ 1
P sinh 1
λ 2 − 4µ ϕ + Q cosh 1
λ2
−
4µ
ϕ
−1
+ a−1 + λ 2 − 4µ
2
2 + a0
2 2
P cosh 1 λ2 − 4µ ϕ + Q sinh 1 λ 2 − 4µ ϕ
2
2
(13)
−λ 1 + a1 + 2 2
P sinh 1
λ2 − 4µ
2
P cosh 1
2
λ 2 − 4µ ϕ + Q cosh 1 2
λ 2 − 4µ ϕ + Q sinh 1 2
λ2
− 4µ
ϕ
λ2 − 4µ ϕ
+a
−λ
+1
2 2 2
P sinh 1
λ2 − 4µ
2
P cosh 1
2
λ2 − 4µ ϕ + Q cosh 1 2
λ2 − 4µ ϕ + Q sinh 1 2
λ2
− 4µ
ϕ
2
,
λ 2 − 4µ ϕ
if P and Q are taken particular values, various known solutions can be rediscovered.
Family 2: Trigonometric function solutions:
When λ2 − 4µ < 0, we obtain
f (ϕ ) = a
−λ 1 +
−2 2 2
−P sin 1
4µ − λ2
2
P cos 1
2
4µ − λ 2 ϕ + Q cos 1 2
4µ − λ 2 ϕ + Q sin 1 2
4µ
−
λ
2
ϕ
−2
4µ − λ2ϕ
−λ 1
−P sin 1
4µ − λ2 ϕ + Q cos 1
4µ
−
λ
2
ϕ
−1
+ a−1 + 4µ − λ 2
2
2 + a0
2 2
P cos 1 4µ − λ 2 ϕ + Q sin 1 4µ − λ 2ϕ
2
2
(14)
−λ 1 + a1 + 2 2
−P sin 1
4µ − λ2
2
P cos 1
2
4µ − λ2 ϕ + Q cos 1 2
4µ − λ 2 ϕ + Q sin 1 2
4µ
−
λ
2
ϕ
4µ − λ 2 ϕ
−λ 1 + a2 + 2 2
−P sin 1
4µ − λ2
2
P cos 1
2
4µ − λ2 ϕ + Q cos 1 2
4µ − λ2 ϕ + Q sin 1 2
4µ
−
λ
2
ϕ
2
,
4µ − λ 2ϕ
if P and Q are taken particular values, various known solutions can be rediscovered.
Family 3: Rational solution: When λ2 − 4µ = 0, we obtain
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Physical Review & Research International, 4(1): 255-266, 2014
f (ϕ ) = a
−λ
+
Q
−2
−λ
+a +
Q
−1
−λ
+a +a +
Q
−2 2 P + Qϕ −1 2 P + Qϕ 0 1 2 P + Qϕ
(15)
−λ Q 2
+ a2
2
+
P + Qϕ
.
Substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), we obtain the hyperbolic function solution Eq. (13), then our traveling wave solutions become respectively (if P = 0 but Q ≠ 0 ):
( ) f1 (ϕ ) = 3 λ22β− 4µ coth 12 λ 2 − 4µ ϕ 2 −1 ,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
( ) f2 (ϕ ) = λ22−β4µ 3 coth 12 λ2 − 4µ ϕ 2 −1 ,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
3
β
2
λ2 − 4µ coth 1
2
2
λ2
− 4µ
ϕ
−2
+λ
−λ
+
2
λ2 − 4µ coth 1
2
2
−1 λ2 − 4µ ϕ +1,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
f
(ϕ ) =
1
6µ
−λ
+
4
β
2
λ 2 − 4µ coth 1
2
2
λ2
− 4µ
ϕ
−2
+ λµ
−λ
+
2
λ 2 − 4µ coth 1
2
2
−1
λ 2 − 4µ ϕ
+
2µ
+
λ
2
,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
Again, substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), we obtain the hyperbolic function solution Eq. (13), then we obtain following solutions respectively (if Q = 0 but P ≠ 0 ):
( ) f5 (ϕ ) = 3 λ22β− 4µ tanh 12 λ2 − 4µ ϕ 2 −1,
( ) f6 (ϕ ) = λ22−β4µ 3 tanh 12 λ2 − 4µ ϕ 2 −1,
f
(ϕ
)
=
6µ
µ
−λ
+
7
β
2
λ 2 − 4µ tanh 1
2
2
λ2
− 4µ
−2 ϕ
+λ
−λ
+
2
λ 2 − 4µ tanh 1
2
2
−1 λ 2 − 4µ ϕ +1 ,
f
(ϕ ) =
1
6µ
−λ
+
8
β
2
λ 2 − 4µ tanh 1
2
2
λ2
− 4µ
−2 ϕ
+
λ
µ
−
λ
+
2
λ 2 − 4µ tanh 1
2
2
−1
λ 2 − 4µ ϕ
+
2µ
+
λ
2
,
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Physical Review & Research International, 4(1): 255-266, 2014
Again, substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), yields the trigonometric function solution Eq. (14), then our obtain solutions become respectively (if P = 0 but Q ≠ 0 ):
( ) f9 (ϕ ) = 3 4µ2β− λ2 cot 12 4µ − λ2 ϕ 2 +1,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
( ) f10 (ϕ ) = 4µ2−βλ2 3 cot 12 4µ − λ2 ϕ 2 +1 ,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
11
β
2
4µ − λ2 cot 1
2
2
4µ − λ2
−2 ϕ
+
λ
−
λ
+
2
4µ − λ2 cot 1
2
2
−1 4µ − λ 2 ϕ +1,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
f
(ϕ )
=
1
6µ
−λ
+
12
β
2
4µ − λ2 cot 1
2
2
4µ − λ2
−2 ϕ
+
λµ
−
λ
+
2
4µ − λ 2 cot 1
2
2
−1
4µ − λ 2 ϕ
+
2µ
+
λ
2
,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
Also, substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), yields the trigonometric function solution Eq. (14), our traveling wave solutions become respectively (if Q = 0 but P ≠ 0 ):
( ) f13 (ϕ ) = 3 4µ2β− λ2 tan 12 4µ − λ 2 ϕ 2 +1.
( ) f14 (ϕ ) = 4µ2−βλ2 3 tan 12 4µ − λ2 ϕ 2 +1.
f
(ϕ
)
=
6µ
µ
−λ
+
15
β
2
4µ − λ 2 tan 1
2
2
4µ − λ2
−2 ϕ
+
λ
−λ
+
2
4µ − λ2 tan 1
2
2
−1 4µ − λ2 ϕ +1.
f
(ϕ )
=
1
6µ
−λ
+
16
β
2
4µ − λ 2 tan 1
2
2
4µ − λ2
−2 ϕ
+
λµ
−λ
+
2
4µ − λ 2 tan 1
2
2
−1
4µ − λ 2 ϕ
+
2µ
+
λ
2
.
Family 3: Rational solutions:
Substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), we obtain the rational function solution Eq. (15), and our solutions become respectively (if λ2 − 4µ = 0 ):
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Physical Review & Research International, 4(1): 255-266, 2014
( ) f
(ϕ ) =
3
2Q
2
− λ 2 − 4µ ,
17
2β ( P + Qϕ )
where ϕ = x m α 2 − (λ2 − 4µ ) t.
( ) f
(ϕ ) =
1
3
2Q
2
− λ 2 − 4µ ,
18
2β ( P + Qϕ )
where ϕ = x m α 2 + (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
Q
−2 −λ +λ +
Q
−1 +1,
19
β 2 P + Qϕ 2 P + Qϕ
where ϕ = x m α 2 − (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
Q
−2 −λ +λ +
Q
−1
+
2µ
+
λ2
,
20
β 2 P + Qϕ 2 P + Qϕ β
where ϕ = x m α 2 + (λ2 − 4µ ) t.
3. RESULTS AND DISCUSSION
Many researchers solved the fourth-order Boussinesq equation for obtaining analytical
solutions by using different methods. For instance, Zhang [34] studied this equation by
applying the F-expansion method to obtain exact solutions. Wazwaz [35] executed the
extended tanh method to establish analytical solutions of the same equation while Naher
and Abdullah [36] investigated this equation via the basic (G '/ G) -expansion method to
construct traveling wave solutions. To the best of our knowledge, this equation is not solved
via the improved (G '/ G) -expansion method. We have studied mentioned equation by
applying the improved (G '/ G) -expansion method and abundant solutions are constructed in
this article. Moreover, it is worth declaring that some of our obtained solutions are in good
agreement with already published results and others have not been reported in the literature.
Our solutions f1 (ϕ ) , f2 (ϕ ), f5 (ϕ ) , f6 (ϕ ), f9 (ϕ ), f10 (ϕ ), f13 (ϕ ), f14 (ϕ ), f17 (ϕ ) and f18 (ϕ ) are
coincided with the published results
A1 (ψ ) , A2 (ψ ), A3 (ψ ) ,
A4 (ψ ),
A7 (ψ ), A8 (ψ ) , A9 (ψ ), A10 (ψ ), A13 (ψ ) and A14 (ψ ) respectively, which are gained by Naher
and Abdullah [36]. Beyond this, we obtain new exact traveling wave solutions
f3 (ϕ ), f4 (ϕ ) , f7 (ϕ ), f8 (ϕ ), f11 (ϕ ), f12 (ϕ ), f15 (ϕ ) , f16 (ϕ ), f19 (ϕ ) and f20 (ϕ ) which are not
being established in the previous literature.
4. CONCLUSION
In this article, an improved (G '/ G) -expansion method is applied to generate new traveling
wave solutions of nonlinear partial differential equation, namely, the Boussinesq equation. The obtained solutions are presented through the hyperbolic functions, the trigonometric
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Physical Review & Research International, 4(1): 255-266, 2014
functions and the rational functions. Moreover, our constructed solutions show that the solution procedure of this method is very simple, reliable and straightforward. In addition, this
method can give many new solutions at a time than the basic (G '/ G) -expansion method. We hope, this powerful and effective, improve (G '/ G) -expansion method could lead to find
abundant new traveling wave solutions for various nonlinear partial differential equations which frequently arise in real time technical arena.
ACKNOWLEDGEMENTS
The authors would like to express their sincere thanks to the anonymous referee(s) for their valuable comments and suggestions.
COMPETING INTERESTS
The authors declare that they have no competing interests.
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2. Wang ML, Zhou YB, Li ZB. Application of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Pyss. Lett. A. 1996;216:6775.
3. Hirota R. Exact solution of the KdV equation for multiple collisions of solutions. Physics Review Letters. 1971;27:1192-1194.
4. Yan J. Soliton resonances of the nonisospectral modified Kadomtsev-Petviashvili equation. Applied Mathematics. 2011;2:685-693.
5. Liu S, Fu Z, Liu S, Zhao Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A. 2001;289:69–74.
6. Chen Y, Wang Q. Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation. Chaos, Solit. Fract. 2005;24(3):745-757.
7. Xiang C. Jacobi elliptic function solutions for (2+1) dimensional Boussinesq and Kadomtsev-Petviashvili equation, Applied Mathematics. 2011;2:1313-1316.
8. Yan Z, Zhang HQ. New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water. Phy. Lett. A. 2001;285:355-362.
9. Malfliet W. Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 1992;60:650–654.
10. Wazwaz AM. The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. and Comput. 2007;188:1467-1475.
11. Abdel-All NH, Abdel Razek MA, Seddeek AK. Expanding the tanh-function method for solving nonlinear equations. Applied Mathematics. 2011;2:1096-1104.
12. Wang ML, Li XZ. Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons and Fractals. 2005;24:1257-1268.
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SCIENCEDOMAIN international www.sciencedomain.org
Traveling Wave Solutions of the Fourth Order
Boussinesq Equation via the Improved (G '/ G )
Expansion Method
Ahata Shamul Alam1, M. M. Zaman2 and Sayeda Sultana3*
1Department of Statistics, Dhaka College, Bangladesh. 2Fortunex Limited, Baridhara, Bangladesh.
3Department of Economics, Kalachandpur School and College, Dhaka, Bangladesh.
Authors’ contributions
The work presented here was carried out in collaboration between three authors. Author ASA carried out the main part of this manuscript. Author MMZ discussed and polished the manuscript. Author SS data analyzed and polished the manuscript. All authors have read
and approved the manuscript.
Research Article
Received 17th March 2013 Accepted 11th May 2013
Published 16th October 2013
ABSTRACT
In this article, the fourth order Boussinesq equation is investigated to generate many new
traveling wave solutions via the improved (G '/ G) -expansion method. In the method the
second order linear ordinary differential equation with constant coefficients is used. Also, the resulted solutions are presented in three different families including solitons and periodic solutions. Furthermore, some of our solutions are coincided with published results which gained by other authors and some are new. Aims: The aim of this work is to construct many new exact traveling wave solutions including solitons, periodic and rational solutions of the fourth order Boussinesq equation
by applying the improved (G '/ G) -expansion method. Methodology: The improved (G '/ G) -expansion method is effective and powerful
mathematical tool for solving nonlinear partial differential equations which arise in mathematical physics, engineering sciences and other technical arena. In addition,
____________________________________________________________________________________________ *Corresponding author: Email: [email protected];
Physical Review & Research International, 4(1): 255-266, 2014
p
h
G′′ + λG′ + µG = 0, together with f (ϕ ) = ∑ ah (G '/ G ) , is implemented as traveling wave
h= − p
solutions, where either a− p or ap may be zero, but both a− p and ap cannot be zero at the
same time. Results: The obtained traveling wave solutions are described in terms of the hyperbolic functions, the trigonometric functions and the rational functions. Conclusion: The constructed solutions may express a variety of new features of waves, further, may be valuable in the theoretical and numerical studies of the considered
equation. Moreover, the obtained exact solutions reveal that the improved (G '/ G) -
expansion method is a promising mathematical tool, because, it can establish abundant new traveling wave solutions of different physical structures. Also, some of our solutions are in good agreement with already published results for a special case and others are new.
Keywords: The improved-expansion method; the Boussinesq equation; traveling wave solutions; nonlinear evolution equations.
1. INTRODUCTION
The study of nonlinear evolution equations (NLEEs) has now become one of the most exciting and tremendously active research areas to scientists. In recent time, many authors who are interested in the nonlinear complex physical phenomena generated analytical solutions of NLEEs. They introduced various methods, for instance, the inverse scattering method [1], the homogeneous balance method [2], the Hirota’s bilinear transformation method [3,4], the Jacobi elliptic function expansion method [5-7], the generalized Riccati equation method [8], the tanh-coth method [9-11], the F-expansion method [12], the Expfunction method [13-15] and others [16-18].
Wang et al. [19] presented a powerful method which is called the (G '/ G) -expansion
method. They generated exact traveling wave solutions in three different families including solitons, periodic and rational solutions for some nonlinear partial differential equations
(PDEs) via this method. In the method, u (ξ ) = ∑m ai (G '/ G)i is implemented as traveling
i=0
wave solutions, where am ≠ 0. Later on, several scientists investigated various nonlinear
PDEs to construct traveling wave solutions via this method [20-26]. More recently, Zhang et
al. [27] extended this method and called the improved (G '/ G) -expansion method. They
∑p
h
used f (ϕ ) = ah (G '/ G ) as traveling wave solutions, where either a− p or ap may be
h=− p
zero, but both a− p and ap cannot be zero simultaneously. After that, many researchers
studied different NLEEs to construct exact solutions by using this improved (G '/ G) -
expansion method. For example, Zhao et al. [28] executed the same method to establish exact solutions of the variant Boussinesq equations. Nofel et al. [29] constructed traveling wave solutions for the fifth-order KdV equation by using this method. Hamad et al. [30] studied higher dimensional potential YTSF equation to obtain analytical solutions via the
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same method. Naher and Abdullah [31] applied this powerful method to construct traveling wave solutions of the nonlinear reaction diffusion equation whilst they [32] implemented this method to generate exact solutions of the (2+1)-dimensional Modified Zakharov-Kuznetsov
equation via the improved (G '/ G) -expansion method. In Ref. [33] Naher et al. generated many solutions of the compound KdV-Burgers equation via the improved (G '/ G) -expansion
method and so on.
In this paper, we would like to investigate well established fourth order Boussinesq equation
by applying the improved (G '/ G) -expansion method to construct a rich class of new exact
traveling wave solutions including solitons, periodic and rational solutions.
2. DESCRIPTION OF THE IMPROVED (G '/ G) -EXPANSION METHOD
Let us consider the general nonlinear partial differential equation:
P (u,ut ,ux ,uxt ,ut t ,uxx ,...) = 0,
(1)
where u = u ( x,t ) is an unknown function, P is a polynomial in u ( x,t ) and its partial
derivatives in which the highest order partial derivatives and the nonlinear terms are involved.
The main steps of the method are as follows:
Step 1. Suppose the traveling wave variable:
u ( x,t ) = f (ϕ ), ϕ = x −Vt,
(2)
where V is the speed of the traveling wave. Using Eq. (2), Eq. (1) is transformed into an
ordinary differential equation for f = f (ϕ ) :
Q ( f , f ', f ′′, f ′′′,...) = 0,
(3)
where Q is a function of f (ϕ ) and its total derivatives.
Step 2. Eq. (3) can be integrated term by term one or more times, according to possibility, yields constant(s) of integration. The integral constant may be zero, for simplicity.
Step 3. Suppose that the traveling wave solution of Eq. (3) can be stated in the following
form:
p
h
f (ϕ ) = ∑ ah (G '/ G)
(4)
h= − p
with G = G (ϕ ) satisfies the second order linear ODE:
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G′′ + λG′ + µG = 0,
(5)
where ah (h = 0, ±1, ± 2,..., ± p), λ and µ are constants.
Step 4. The value of the positive integer p can be determined by taking the homogeneous balance between the highest order nonlinear terms and the highest order derivatives appearing in Eq. (3).
Step 5. Substitute Eq. (4) together with Eq. (5) into Eq. (3) with the value of p obtained in Step 4. Equating the coefficients of (G '/ G )r , (r = 0, ±1, ± 2,...), then setting each coefficient to zero, we obtain a set of algebraic equations for ah (h = 0, ±1, ± 2,..., ± p),V , λ and µ.
Step 6. Solve the system of algebraic equations which are obtained in step 5 with the aid of
commercial software Maple and we obtain values for ah (h = 0, ±1, ± 2,..., ± p),V , λ and µ.
Then, substitute obtained values in Eq. (4) along with the general solution of Eq. (5) with the value of p , we can obtain the traveling wave solutions of Eq. (1).
2.1 Application of the Method
In this subsection, we have studied the fourth order Boussinesq equation to construct exact
traveling wave solutions via the improved (G '/ G ) -expansion method.
2.1.1 The fourth order Boussinesq equation
We consider the fourth order Boussinesq equation followed by Naher and Abdullah [36]:
( ) utt − α 2 uxx − β u2 + uxxxx = 0.
(6)
xx
Now, we use the wave transformation Eq. (2) into the Eq. (6), therefore, integrating with
respect to ϕ twice and setting the constants of integration to zero, yields:
(V 2 −α 2 ) f − β f 2 + f ′′ = 0.
(7)
Taking the homogeneous balance between f 2 and f ′′ in Eq. (7), we obtain p = 2.
Therefore, the solution of Eq. (7) is of the form:
f (ϕ ) = a−2 (G '/ )G −2 + a−1 (G '/ )G −1 + a0 + a1 (G '/ G ) + a2 (G '/ G )2 ,
(8)
where a−2 , a−1, a0 , a1 and a2 are constants to be determined.
Substituting Eq. (8) together with Eq. (5) into the Eq. (7), the left-hand side of Eq. (7) is
converted into a polynomial of (G '/ G )r , (r = 0, ±1, ± 2,...). According to Step 5, collecting all terms with the same power of (G '/ G ). Then, setting each coefficient of the resulted
polynomial to zero, yields a set of algebraic equations (for simplicity, which are not
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Physical Review & Research International, 4(1): 255-266, 2014
presented) for a−2 , a−1, a0 , a1, a2 ,V , λ and µ. Solving the system of obtained algebraic
equations with the help of algebraic software Maple 13, we obtain four different values.
Case 1:
( ) a = 0, a = 0, a = 6µ , a = 6λ , a = 6 , V = ± α 2 − λ 2 − 4µ ,
(9)
−2
−1
0 β 1 β 2β
where α, β , λ and µ are free parameters.
Case 2:
( ) a = 0, a = 0, a = 2µ + λ 2 , a = 6λ , a = 6 , V = ± α 2 + λ2 − 4µ ,
(10)
−2
−1
0
β
1 β 2β
where α, β , λ and µ are free parameters.
Case 3:
( ) a = 6µ 2 , a = 6λµ , a = 6µ , a = 0, a = 0, V = ± α 2 − λ 2 − 4µ ,
(11)
−2
β −1
β
0β 1
2
where α, β , λ and µ are free parameters.
Case 4:
( ) a = 6µ2 , a = 6λµ , a = 2µ + λ2 , a = 0, a = 0, V = ± α 2 + λ2 − 4µ , (12)
−2 β −1 β
0
β
1
2
where α, β , λ and µ are free parameters.
Substituting the general solution Eq. (5) into Eq. (8), we obtain three different families of traveling wave solutions of Eq. (7):
Family 1: Hyperbolic function solutions:
When λ2 − 4µ > 0, we obtain
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Physical Review & Research International, 4(1): 255-266, 2014
f (ϕ ) = a
−λ 1 +
−2 2 2
P sinh 1
λ2 − 4µ
2
P cosh 1
2
λ2 − 4µ ϕ + Q cosh 1 2
λ 2 − 4µ ϕ + Q sinh 1 2
λ2
−
4µ
ϕ
−2
λ 2 − 4µ ϕ
−λ 1
P sinh 1
λ 2 − 4µ ϕ + Q cosh 1
λ2
−
4µ
ϕ
−1
+ a−1 + λ 2 − 4µ
2
2 + a0
2 2
P cosh 1 λ2 − 4µ ϕ + Q sinh 1 λ 2 − 4µ ϕ
2
2
(13)
−λ 1 + a1 + 2 2
P sinh 1
λ2 − 4µ
2
P cosh 1
2
λ 2 − 4µ ϕ + Q cosh 1 2
λ 2 − 4µ ϕ + Q sinh 1 2
λ2
− 4µ
ϕ
λ2 − 4µ ϕ
+a
−λ
+1
2 2 2
P sinh 1
λ2 − 4µ
2
P cosh 1
2
λ2 − 4µ ϕ + Q cosh 1 2
λ2 − 4µ ϕ + Q sinh 1 2
λ2
− 4µ
ϕ
2
,
λ 2 − 4µ ϕ
if P and Q are taken particular values, various known solutions can be rediscovered.
Family 2: Trigonometric function solutions:
When λ2 − 4µ < 0, we obtain
f (ϕ ) = a
−λ 1 +
−2 2 2
−P sin 1
4µ − λ2
2
P cos 1
2
4µ − λ 2 ϕ + Q cos 1 2
4µ − λ 2 ϕ + Q sin 1 2
4µ
−
λ
2
ϕ
−2
4µ − λ2ϕ
−λ 1
−P sin 1
4µ − λ2 ϕ + Q cos 1
4µ
−
λ
2
ϕ
−1
+ a−1 + 4µ − λ 2
2
2 + a0
2 2
P cos 1 4µ − λ 2 ϕ + Q sin 1 4µ − λ 2ϕ
2
2
(14)
−λ 1 + a1 + 2 2
−P sin 1
4µ − λ2
2
P cos 1
2
4µ − λ2 ϕ + Q cos 1 2
4µ − λ 2 ϕ + Q sin 1 2
4µ
−
λ
2
ϕ
4µ − λ 2 ϕ
−λ 1 + a2 + 2 2
−P sin 1
4µ − λ2
2
P cos 1
2
4µ − λ2 ϕ + Q cos 1 2
4µ − λ2 ϕ + Q sin 1 2
4µ
−
λ
2
ϕ
2
,
4µ − λ 2ϕ
if P and Q are taken particular values, various known solutions can be rediscovered.
Family 3: Rational solution: When λ2 − 4µ = 0, we obtain
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Physical Review & Research International, 4(1): 255-266, 2014
f (ϕ ) = a
−λ
+
Q
−2
−λ
+a +
Q
−1
−λ
+a +a +
Q
−2 2 P + Qϕ −1 2 P + Qϕ 0 1 2 P + Qϕ
(15)
−λ Q 2
+ a2
2
+
P + Qϕ
.
Substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), we obtain the hyperbolic function solution Eq. (13), then our traveling wave solutions become respectively (if P = 0 but Q ≠ 0 ):
( ) f1 (ϕ ) = 3 λ22β− 4µ coth 12 λ 2 − 4µ ϕ 2 −1 ,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
( ) f2 (ϕ ) = λ22−β4µ 3 coth 12 λ2 − 4µ ϕ 2 −1 ,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
3
β
2
λ2 − 4µ coth 1
2
2
λ2
− 4µ
ϕ
−2
+λ
−λ
+
2
λ2 − 4µ coth 1
2
2
−1 λ2 − 4µ ϕ +1,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
f
(ϕ ) =
1
6µ
−λ
+
4
β
2
λ 2 − 4µ coth 1
2
2
λ2
− 4µ
ϕ
−2
+ λµ
−λ
+
2
λ 2 − 4µ coth 1
2
2
−1
λ 2 − 4µ ϕ
+
2µ
+
λ
2
,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
Again, substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), we obtain the hyperbolic function solution Eq. (13), then we obtain following solutions respectively (if Q = 0 but P ≠ 0 ):
( ) f5 (ϕ ) = 3 λ22β− 4µ tanh 12 λ2 − 4µ ϕ 2 −1,
( ) f6 (ϕ ) = λ22−β4µ 3 tanh 12 λ2 − 4µ ϕ 2 −1,
f
(ϕ
)
=
6µ
µ
−λ
+
7
β
2
λ 2 − 4µ tanh 1
2
2
λ2
− 4µ
−2 ϕ
+λ
−λ
+
2
λ 2 − 4µ tanh 1
2
2
−1 λ 2 − 4µ ϕ +1 ,
f
(ϕ ) =
1
6µ
−λ
+
8
β
2
λ 2 − 4µ tanh 1
2
2
λ2
− 4µ
−2 ϕ
+
λ
µ
−
λ
+
2
λ 2 − 4µ tanh 1
2
2
−1
λ 2 − 4µ ϕ
+
2µ
+
λ
2
,
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Physical Review & Research International, 4(1): 255-266, 2014
Again, substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), yields the trigonometric function solution Eq. (14), then our obtain solutions become respectively (if P = 0 but Q ≠ 0 ):
( ) f9 (ϕ ) = 3 4µ2β− λ2 cot 12 4µ − λ2 ϕ 2 +1,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
( ) f10 (ϕ ) = 4µ2−βλ2 3 cot 12 4µ − λ2 ϕ 2 +1 ,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
11
β
2
4µ − λ2 cot 1
2
2
4µ − λ2
−2 ϕ
+
λ
−
λ
+
2
4µ − λ2 cot 1
2
2
−1 4µ − λ 2 ϕ +1,
where ϕ = x m α 2 − (λ2 − 4µ ) t.
f
(ϕ )
=
1
6µ
−λ
+
12
β
2
4µ − λ2 cot 1
2
2
4µ − λ2
−2 ϕ
+
λµ
−
λ
+
2
4µ − λ 2 cot 1
2
2
−1
4µ − λ 2 ϕ
+
2µ
+
λ
2
,
where ϕ = x m α 2 + (λ2 − 4µ ) t.
Also, substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), yields the trigonometric function solution Eq. (14), our traveling wave solutions become respectively (if Q = 0 but P ≠ 0 ):
( ) f13 (ϕ ) = 3 4µ2β− λ2 tan 12 4µ − λ 2 ϕ 2 +1.
( ) f14 (ϕ ) = 4µ2−βλ2 3 tan 12 4µ − λ2 ϕ 2 +1.
f
(ϕ
)
=
6µ
µ
−λ
+
15
β
2
4µ − λ 2 tan 1
2
2
4µ − λ2
−2 ϕ
+
λ
−λ
+
2
4µ − λ2 tan 1
2
2
−1 4µ − λ2 ϕ +1.
f
(ϕ )
=
1
6µ
−λ
+
16
β
2
4µ − λ 2 tan 1
2
2
4µ − λ2
−2 ϕ
+
λµ
−λ
+
2
4µ − λ 2 tan 1
2
2
−1
4µ − λ 2 ϕ
+
2µ
+
λ
2
.
Family 3: Rational solutions:
Substituting Eqs. (9), (10), (11) and (12) together with the general solution Eq. (5) into the Eq. (8), we obtain the rational function solution Eq. (15), and our solutions become respectively (if λ2 − 4µ = 0 ):
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Physical Review & Research International, 4(1): 255-266, 2014
( ) f
(ϕ ) =
3
2Q
2
− λ 2 − 4µ ,
17
2β ( P + Qϕ )
where ϕ = x m α 2 − (λ2 − 4µ ) t.
( ) f
(ϕ ) =
1
3
2Q
2
− λ 2 − 4µ ,
18
2β ( P + Qϕ )
where ϕ = x m α 2 + (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
Q
−2 −λ +λ +
Q
−1 +1,
19
β 2 P + Qϕ 2 P + Qϕ
where ϕ = x m α 2 − (λ2 − 4µ ) t.
f
(ϕ
)
=
6µ
µ
−λ
+
Q
−2 −λ +λ +
Q
−1
+
2µ
+
λ2
,
20
β 2 P + Qϕ 2 P + Qϕ β
where ϕ = x m α 2 + (λ2 − 4µ ) t.
3. RESULTS AND DISCUSSION
Many researchers solved the fourth-order Boussinesq equation for obtaining analytical
solutions by using different methods. For instance, Zhang [34] studied this equation by
applying the F-expansion method to obtain exact solutions. Wazwaz [35] executed the
extended tanh method to establish analytical solutions of the same equation while Naher
and Abdullah [36] investigated this equation via the basic (G '/ G) -expansion method to
construct traveling wave solutions. To the best of our knowledge, this equation is not solved
via the improved (G '/ G) -expansion method. We have studied mentioned equation by
applying the improved (G '/ G) -expansion method and abundant solutions are constructed in
this article. Moreover, it is worth declaring that some of our obtained solutions are in good
agreement with already published results and others have not been reported in the literature.
Our solutions f1 (ϕ ) , f2 (ϕ ), f5 (ϕ ) , f6 (ϕ ), f9 (ϕ ), f10 (ϕ ), f13 (ϕ ), f14 (ϕ ), f17 (ϕ ) and f18 (ϕ ) are
coincided with the published results
A1 (ψ ) , A2 (ψ ), A3 (ψ ) ,
A4 (ψ ),
A7 (ψ ), A8 (ψ ) , A9 (ψ ), A10 (ψ ), A13 (ψ ) and A14 (ψ ) respectively, which are gained by Naher
and Abdullah [36]. Beyond this, we obtain new exact traveling wave solutions
f3 (ϕ ), f4 (ϕ ) , f7 (ϕ ), f8 (ϕ ), f11 (ϕ ), f12 (ϕ ), f15 (ϕ ) , f16 (ϕ ), f19 (ϕ ) and f20 (ϕ ) which are not
being established in the previous literature.
4. CONCLUSION
In this article, an improved (G '/ G) -expansion method is applied to generate new traveling
wave solutions of nonlinear partial differential equation, namely, the Boussinesq equation. The obtained solutions are presented through the hyperbolic functions, the trigonometric
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Physical Review & Research International, 4(1): 255-266, 2014
functions and the rational functions. Moreover, our constructed solutions show that the solution procedure of this method is very simple, reliable and straightforward. In addition, this
method can give many new solutions at a time than the basic (G '/ G) -expansion method. We hope, this powerful and effective, improve (G '/ G) -expansion method could lead to find
abundant new traveling wave solutions for various nonlinear partial differential equations which frequently arise in real time technical arena.
ACKNOWLEDGEMENTS
The authors would like to express their sincere thanks to the anonymous referee(s) for their valuable comments and suggestions.
COMPETING INTERESTS
The authors declare that they have no competing interests.
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