Abstract
In this paper, we consider some nonlinear pseudoparabolic BenjaminBonaMahonyBurgers
(BBMB) equations. These equations are of a class of nonlinear pseudoparabolic or
Sobolevtype equations
Keywords:
nonlinear pseudoparabolic equation; BenjaminBonaMahonyBurgers (BBMB)type equation; Sobolevtype equation; tanh method1 Introduction
The partial differential equations of the form
arise in many areas of mathematics and physics, where
An important special case of (1) is the BenjaminBonaMahonyBurgers (BBMB) equation
It has been proposed in [10] as a model to study the unidirectional long waves of small amplitudes in water, which is an alternative to the Kortewegde Vries equation of the form
The BBMB equation has been tackled and investigated by many authors. For more details, we refer the reader to [1115] and the references therein.
In [16], a generalized BenjaminBonaMahonyBurgers equation
has been considered and a set of new solitons, kinks, antikinks, compactons, and Wadati
solitons have been derived using by the classical Lie method, where α is a positive constant,
Peregrine [17] and Benjamin, Bona, and Mahony [10] have proposed equation (4) with the parameters
Khaled, Momani, and Alawneh obtained explicit and numerical solutions of BBMB equation (4) by using the Adomian’s decomposition method [18] .
Tari and Ganji implemented variational iteration and homotopy perturbation methods
obtaining approximate explicit solutions for (4) with
In addition, we consider
The aim of this work is twofold. First, it is to obtain the exact solutions of the
BenjaminBonaMahonyBurgers (BBMB) equation and the generalized BenjaminBonaMahonyBurgers
equation with
2 Outline of the tanh method
Wazwaz has summarized the tanh method [21] in the following manner:
(i) First, consider a general form of the nonlinear equation
(ii) To find the traveling wave solution of equation (5), the wave variable
Based on this, one may use the following changes:
and so on for other derivatives. Using (7) changes PDE (5) to an ODE
(iii) If all terms of the resulting ODE contain derivatives in ξ, then by integrating this equation and by considering the constant of integration to be zero, one obtains a simplified ODE.
(iv) A new independent variable
is introduced that leads to the change of derivatives:
where other derivatives can be derived in a similar manner.
(v) The ansatz of the form
is introduced, where M is a positive integer, in most cases, that will be determined. If M is not an integer, then a transformation formula is used to overcome this difficulty. Substituting (10) and (11) into ODE (8) yields an equation in powers of Y.
(vi) To determine the parameter M, the linear terms of highest order in the resulting equation with the highestorder
nonlinear terms are balanced. With M determined, one collects all the coefficients of powers of Y in the resulting equation where these coefficients have to vanish. This will give
a system of algebraic equations involving the
Throughout the work, Mathematica or Maple is used to deal with the tedious algebraic operations.
3 The BenjaminBonaMahonyBurgers (BBMB) equation
The BenjaminBonaMahonyBurgers (BBMB) equation is given by
where α is a positive constant. Using the wave variable
Balancing
where
Using Maple gives nine sets of solutions
These sets give the following solutions respectively:
If we accept
4 The generalized BenjaminBonaMahonyBurgers equation
We consider the generalized BenjaminBonaMahonyBurgers equation
where α is a positive constant and
Case 1.
Using the wave variable
Balancing
we find the system of equations
Maple gives three sets of solutions
where k is left as a free parameter. These give the following solutions:
Case 2.
Using the wave variable
Balancing the second term with the last term in (25) gives
we find the system of equations
Using Maple, we obtain nine sets of solutions
These sets give the solutions
Case 3.
Using the wave variable
Balancing
we find the system of equations
Solving the resulting system, we find the following sets of solutions with
These in turn give the solutions
5 Conclusion
In summary, we implemented the tanh method to solve some nonlinear pseudoparabolic BenjaminBonaMahonyBurgers equations and obtained new solutions which could not be attained in the past. Besides, we have seen that the tanh method is easy to apply and reliable to solve the pseudoparabolic and the Sobolevtype equations.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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