In this paper, we consider some nonlinear pseudo-parabolic Benjamin-Bona-Mahony-Burgers (BBMB) equations. These equations are of a class of nonlinear pseudo-parabolic or Sobolev-type equations , α is a fixed positive constant, arising from the mathematical physics. The tanh method with the aid of symbolic computational system is employed to investigate exact solutions of BBMB-type equations and the exact solutions are found. The results obtained can be viewed as verification and improvement of the previously known data.
Keywords:nonlinear pseudo-parabolic equation; Benjamin-Bona-Mahony-Burgers (BBMB)-type equation; Sobolev-type equation; tanh method
The partial differential equations of the form
arise in many areas of mathematics and physics, where , , , η and α are non-negative constants, Δ denotes the Laplace operator acting on the space variables x. Equations of type (1) with only one time derivative appearing in the highest-order term are called pseudo-parabolic and they are a special case of Sobolev equations. They are characterized by derivatives of mixed type (i.e., time and space derivatives together) appearing in the highest-order terms of the equation and were studied by Sobolev . Sobolev equations have been used to describe many physical phenomena [2-8]. Equation (1) arises as a mathematical model for the unidirectional propagation of nonlinear, dispersive, long waves. In applications, u is typically the amplitude or velocity, x is proportional to the distance in the direction of propagation, and t is proportional to elapsed time .
An important special case of (1) is the Benjamin-Bona-Mahony-Burgers (BBMB) equation
It has been proposed in  as a model to study the unidirectional long waves of small amplitudes in water, which is an alternative to the Korteweg-de Vries equation of the form
In , a generalized Benjamin-Bona-Mahony-Burgers 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, , and is a -smooth nonlinear function. Equation (4) with the dissipative term arises in the phenomena for both the bore propagation and the water waves.
Peregrine  and Benjamin, Bona, and Mahony  have proposed equation (4) with the parameters , , and . Furthermore, Benjamin, Bona, and Mahony proposed equation (4) as an alternative regularized long-wave equation with the same parameters.
Khaled, Momani, and Alawneh obtained explicit and numerical solutions of BBMB equation (4) by using the Adomian’s decomposition method  .
Tari and Ganji implemented variational iteration and homotopy perturbation methods obtaining approximate explicit solutions for (4) with  and El-Wakil, Abdou, and Hendi used another method (the exp-function) to obtain the generalized solitary solutions and periodic solutions of this equation .
In addition, we consider and obtain analytic solutions in a closed form.
The aim of this work is twofold. First, it is to obtain the exact solutions of the Benjamin-Bona-Mahony-Burgers (BBMB) equation and the generalized Benjamin-Bona-Mahony-Burgers equation with , , ; and second, it is to show that the tanh method can be applied to obtain the solutions of pseudo-parabolic equations.
2 Outline of the tanh method
Wazwaz has summarized the tanh method  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 is introduced so that
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 highest-order 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 and ( ), V, and μ. Having determined these parameters, knowing that M is a positive integer in most cases, and using (11), one obtains an analytic solution in a closed form.
Throughout the work, Mathematica or Maple is used to deal with the tedious algebraic operations.
3 The Benjamin-Bona-Mahony-Burgers (BBMB) equation
The Benjamin-Bona-Mahony-Burgers (BBMB) equation is given by
where α is a positive constant. Using the wave variable carries (12) into the ODE
Balancing with in (13) gives . The tanh method admits the use of the finite expansion
where . Substituting (14) into (13), collecting the coefficients of Y, and setting it equal to zero, we find the system of equations
Using Maple gives nine sets of solutions
These sets give the following solutions respectively:
If we accept , then we obtain solutions
4 The generalized Benjamin-Bona-Mahony-Burgers equation
We consider the generalized Benjamin-Bona-Mahony-Burgers equation
where α is a positive constant and .
Case 1. .
Using the wave variable carries (19) into the ODE
Balancing with in (20) gives . Using the finite expansion
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 , then by integrating this equation and considering the constant of integration to be zero, we obtain
Balancing the second term with the last term in (25) gives . Using the finite expansion
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 , then by integrating this equation once and considering the constant of integration to be zero, we obtain
Balancing with in (30) gives . Using the finite expansion
we find the system of equations
Solving the resulting system, we find the following sets of solutions with :
These in turn give the solutions
In summary, we implemented the tanh method to solve some nonlinear pseudo-parabolic Benjamin-Bona-Mahony-Burgers 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 pseudo-parabolic and the Sobolev-type equations.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
Chen, PJ, Gurtin, ME: On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys.. 19, 614–627 (1968). Publisher Full Text
Barenblat, G, Zheltov, I, Kochina, I: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech.. 24, 1286–1303 (1960). Publisher Full Text
Coleman, BD, Noll, W: An approximation theorem for functionals with applications to continuum mechanics. Arch. Ration. Mech. Anal.. 6, 355–370 (1960). Publisher Full Text
Huilgol, R: A second order fluid of the differential type. Int. J. Non-Linear Mech.. 3, 471–482 (1968). Publisher Full Text
Barenblat, GI, Zheltov, IP, Kochina, IN: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech.. 24, 1286–1303 (1960). Publisher Full Text
Karch, G: Asymptotic behaviour of solutions to some pseudoparabolic equations. Math. Methods Appl. Sci.. 20, 271–289 (1997). Publisher Full Text
Benjamin, TB, Bona, JL, Mahony, JJ: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A. 272, 47–78 (1972). Publisher Full Text
Raupp, MA: Galerkin methods applied to the Benjamin-Bona-Mahony equation. Bull. Braz. Math. Soc.. 6, 65–77 (1975). Publisher Full Text
Ewing, RE: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation. SIAM J. Numer. Anal.. 15, 1125–1150 (1978). Publisher Full Text
Manickam, SAV, Pani, AK, Chang, SK: A second-order splitting combined with orthogonal cubic spline collocation method for the Roseneau equation. Numer. Methods Partial Differ. Equ.. 14, 695–716 (1998). Publisher Full Text
Al-Khaled, K, Momani, S, Alawneh, A: Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations. Appl. Math. Comput.. 171, 281–292 (2005). Publisher Full Text
Tari, H, Ganji, DD: Approximate explicit solutions of nonlinear BBMB equations by He’s methods and comparison with the exact solution. Phys. Lett. A. 367, 95–101 (2007). Publisher Full Text
El-Wakil, SA, Abdou, MA, Hendi, A: New periodic wave solutions via Exp-function method. Phys. Lett. A. 372, 830–840 (2008). Publisher Full Text
Wazwaz, AM: The Hirota’s bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equation. Appl. Math. Comput.. 200, 160–166 (2008). Publisher Full Text