Trial Equation Method for Solving the Improved Boussinesq Equation ()
In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours. The objective of this paper is to apply Liu’s method and the theory of complete discrimination system for polynomial [12-16] to find the exact solutions of the nonlinear differential equation.
2. Description of Trial Equation Method
The objective of this section is to outline the use of trial equation method for solving a nonlinear partial differential equation (PDE). Suppose we have a nonlinear PDE for 
in the form
(1)
where 
is a polynomial, which includes nonlinear terms and the highest order derivatives and so on.
Step 1 Taking the wave transformation
, reduces Equation (1) to the ordinary differential equation (ODE).
(2)
Step 2 Take trial equation method
(3)
Integrating the Equation (3) with respect to 
once, we get
(4)
where m, 
and integration constant 
are to be determined. Substituting Equations (3), (4) and other derivative terms into Equation (2) yields a polynomial 
of μ. According to the balance principle we can determine the value of m. Setting the coefficients of 
to zero, we get a system of algebraic equations. Solving this system, we can determine values of 
and integration constant.
Step 3 Rewrite Equation (4) by the integral form
(5)
According to the complete discrimination system of the polynomial, we classify the roots of 
and solve the integral equation (5). Thus we obtain the exact solutions to Equation (1).
3. Application of Trial Equation Method
The improved Boussinesq equation [17,18] reads as
(6)
Taking the traveling wave transformation 
and
, we can obtain the corresponding reduced ODE.
(7)
we take the trial equation as follows
(8)
According to the trial equation method of rank homogeneous equation, balancing 
with 
(or
) gets
, so Equation (8) has the following specific form
(9)
Integrating Equation (9) with respect to 
once, we yield
(10)
where values of 
and the integration constant d are to be determined latter. By Equation (9) and Equation (10), we derive the following formula
(11)
Substituting Equations (9), (10) and (11) into Equation (7), we have
(12)
where
(13)
(14)
(15)
(16)
Let the coefficient 
be zero, we will yield nonlinear algebraic equations. Solving the equations, we will determine the values of
. We get 
and d are two arbitrary constants. When the above conditions are satisfied, we use the complete discrimination system for the third order polynomial and have the following solving process.
Let
(17)
Then Equation (10) becomes
(18)
where 
is a function of
. The integral form of Equation (18) is
(19)
Denote
(20)
(21)
According to the complete discrimination system, we give the corresponding single traveling wave solutions to Equation (6).
Case1. 
has a double real root and a simple real root. Then we have
(22)
When
, the corresponding solutions are
(23)
(24)
(25)
Case2. 
has a triple root. Then we have
(26)
The corresponding solution is
(27)
Case 3. 
has three different real roots. Then we have
(28)
When
, we take the transformation as follows
(29)
According to the Equation (19), we have
(30)
where
. On the basis of Equation (30) and the definition of the Jacobi elliptic sine function, we have
(31)
The corresponding solutions is
(32)
when
, we take the transformation as follows
(33)
The corresponding solutions is
(34)
where
.
Case 4. 
has only a real root. Then we have
(35)
when
, we take the transformation as follows
(36)
According to the Equation (19), we have
where
(37)
On the basis of Equation (37) and the definition of the Jacobi elliptic cosine function, we have
(38)
The corresponding solutions is
(39)
In Equations (23), (24), (25), (27), (32), (34), and (39), the integration constant 
has been rewritten, but we still use it. The solutions 
are all possible exact traveling wave solutions to Equation (6). We can see it is easy to write the corresponding solutions to the improved Boussinesq equation.
4. Conclusion
Trial equation method is a systematic method to solve nonlinear differential equations. The advantage of this method is that we can deal with nonlinear equations with linear methods. This method has the characteristics of simple steps and clear effectivity. Based on the idea of the trial equation method and the aid of the computerized symbolic computation, some exact traveling wave solutions to the improved Boussinesq equation have been obtained. With the same method, some of other equations can be dealt with.
Acknowledgements
I would like to thank the referees for their valuable suggestions.
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