- Research
- Open Access

# Exact solutions of Benjamin-Bona-Mahony-Burgers-type nonlinear pseudo-parabolic equations

- Ömer Faruk Gözükızıl
^{1}Email author and - Şamil Akçağıl
^{1}

**2012**:144

https://doi.org/10.1186/1687-2770-2012-144

© Gözükızıl and Akçağıl; licensee Springer 2012

**Received: **16 August 2012

**Accepted: **22 November 2012

**Published: **10 December 2012

## Abstract

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 ${u}_{t}-\mathrm{\Delta}{u}_{t}-\alpha \mathrm{\Delta}u=f(x,u,\mathrm{\nabla}u)$, *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

## 1 Introduction

arise in many areas of mathematics and physics, where $u=u(x,t)$, $x\in \mathrm{\Omega}\subset {\mathbb{R}}^{n}$, $t\ge 0$, *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 [1]. 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 [9].

The BBMB equation has been tackled and investigated by many authors. For more details, we refer the reader to [11–15] and the references therein.

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, $\beta \in \mathbb{R}$, and $g(u)$ is a ${C}^{2}$-smooth nonlinear function. Equation (4) with the dissipative term $\alpha {u}_{xx}$ arises in the phenomena for both the bore propagation and the water waves.

Peregrine [17] and Benjamin, Bona, and Mahony [10] have proposed equation (4) with the parameters $g(u)=u{u}_{x}$, $\alpha =0$, and $\beta =1$. 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 [18] .

Tari and Ganji implemented variational iteration and homotopy perturbation methods obtaining approximate explicit solutions for (4) with $g(u)=\frac{{u}^{2}}{2}$ [19] and El-Wakil, Abdou, and Hendi used another method (the exp-function) to obtain the generalized solitary solutions and periodic solutions of this equation [20].

In addition, we consider $g(u)=\frac{{u}^{3}}{3}$ 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 $g(u)=u{u}_{x}$, $g(u)=\frac{{u}^{2}}{2}$, $g(u)=\frac{{u}^{3}}{3}$; 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

- (i)First, consider a general form of the nonlinear equation$P(u,{u}_{t},{u}_{x},{u}_{xx},\dots )=0.$(5)
- (ii)To find the traveling wave solution of equation (5), the wave variable $\xi =x-Vt$ is introduced so that$u(x,t)=U(\mu \xi ).$(6)

- (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$Y=tanh(\mu \xi )$(9)

- (v)The
*ansatz*of the form$U(\mu \xi )=S(Y)=\sum _{k=0}^{M}{a}_{k}{Y}^{k}+\sum _{k=1}^{M}{b}_{k}{Y}^{-k}$(11)

*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 ${a}_{k}$ and ${b}_{k}$ ($k=0,\dots ,M$),*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

*α*is a positive constant. Using the wave variable $\xi =x-Vt$ carries (12) into the ODE

*Y*, and setting it equal to zero, we find the system of equations

## 4 The generalized Benjamin-Bona-Mahony-Burgers equation

where *α* is a positive constant and $\beta \in \mathbb{R}$.

Case 1. $g(u)=u{u}_{x}$.

Case 2. $g(u)=\frac{{u}^{2}}{2}$.

Case 3. $g(u)=\frac{{u}^{3}}{3}$.

## 5 Conclusion

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.

## Declarations

## Authors’ Affiliations

## References

- Sobolev SL: Some new problems in mathematical physics.
*Izv. Akad. Nauk SSSR, Ser. Mat.*1954, 18: 3-50.MathSciNetGoogle Scholar - Chen PJ, Gurtin ME: On a theory of heat conduction involving two temperatures.
*Z. Angew. Math. Phys.*1968, 19: 614-627. 10.1007/BF01594969View ArticleGoogle Scholar - Barenblat G, Zheltov I, Kochina I: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks.
*J. Appl. Math. Mech.*1960, 24: 1286-1303. 10.1016/0021-8928(60)90107-6View ArticleGoogle Scholar - Taylor D:
*Research of Consolidation of Clays*. Massachusetts Institute of Technology Press, Cambridge; 1952.Google Scholar - Coleman BD, Noll W: An approximation theorem for functionals with applications to continuum mechanics.
*Arch. Ration. Mech. Anal.*1960, 6: 355-370. 10.1007/BF00276168MathSciNetView ArticleGoogle Scholar - Huilgol R: A second order fluid of the differential type.
*Int. J. Non-Linear Mech.*1968, 3: 471-482. 10.1016/0020-7462(68)90032-2MathSciNetView ArticleGoogle Scholar - Ting TW: Certain nonsteady flows of second-order fluids.
*Arch. Ration. Mech. Anal.*1963, 14: 1-26.View ArticleGoogle Scholar - Barenblat GI, Zheltov IP, Kochina IN: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks.
*J. Appl. Math. Mech.*1960, 24: 1286-1303. 10.1016/0021-8928(60)90107-6View ArticleGoogle Scholar - Karch G: Asymptotic behaviour of solutions to some pseudoparabolic equations.
*Math. Methods Appl. Sci.*1997, 20: 271-289. 10.1002/(SICI)1099-1476(199702)20:3<271::AID-MMA859>3.0.CO;2-FMathSciNetView ArticleGoogle Scholar - Benjamin TB, Bona JL, Mahony JJ: Model equations for long waves in nonlinear dispersive systems.
*Philos. Trans. R. Soc. Lond. Ser. A*1972, 272: 47-78. 10.1098/rsta.1972.0032MathSciNetView ArticleGoogle Scholar - Raupp MA: Galerkin methods applied to the Benjamin-Bona-Mahony equation.
*Bull. Braz. Math. Soc.*1975, 6: 65-77. 10.1007/BF02584873MathSciNetView ArticleGoogle Scholar - Wahlbin L: Error estimates for a Galerkin method for a class of model equations for long waves.
*Numer. Math.*1975, 23: 289-303.MathSciNetView ArticleGoogle Scholar - Ewing RE: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation.
*SIAM J. Numer. Anal.*1978, 15: 1125-1150. 10.1137/0715075MathSciNetView ArticleGoogle Scholar - Arnold DN, Douglas J Jr., Thomée V: Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable.
*Math. Comput.*1981, 27: 737-743.Google Scholar - 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.*1998, 14: 695-716. 10.1002/(SICI)1098-2426(199811)14:6<695::AID-NUM1>3.0.CO;2-LView ArticleGoogle Scholar - Bruzon MS, Gandarias ML: Travelling wave solutions for a generalized benjamin-bona-mahony-burgers equation.
*Int. J. Math. Models Methods Appl. Sci.*2008, 2: 103-108.Google Scholar - Peregrine DH: Calculations of the development of an undular bore.
*J. Fluid Mech.*1996, 25: 321-330.MathSciNetView ArticleGoogle Scholar - Al-Khaled K, Momani S, Alawneh A: Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations.
*Appl. Math. Comput.*2005, 171: 281-292. 10.1016/j.amc.2005.01.056MathSciNetView ArticleGoogle Scholar - Tari H, Ganji DD: Approximate explicit solutions of nonlinear BBMB equations by He’s methods and comparison with the exact solution.
*Phys. Lett. A*2007, 367: 95-101. 10.1016/j.physleta.2007.02.085View ArticleGoogle Scholar - El-Wakil SA, Abdou MA, Hendi A: New periodic wave solutions via Exp-function method.
*Phys. Lett. A*2008, 372: 830-840. 10.1016/j.physleta.2007.08.033MathSciNetView ArticleGoogle Scholar - 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.*2008, 200: 160-166. 10.1016/j.amc.2007.11.001MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.