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This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

Open Access Research

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

Ömer Faruk Gözükızıl* and Şamil Akçağıl

Author Affiliations

Department of Mathematics, Sakarya University, Sakarya, Turkey

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Boundary Value Problems 2012, 2012:144  doi:10.1186/1687-2770-2012-144


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/144


Received:16 August 2012
Accepted:22 November 2012
Published:10 December 2012

© 2012 Gözükızıl and Akçağıl; licensee Springer

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.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M1">View MathML</a>, 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

1 Introduction

The partial differential equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M2">View MathML</a>

(1)

arise in many areas of mathematics and physics, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M5">View MathML</a>, 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].

An important special case of (1) is the Benjamin-Bona-Mahony-Burgers (BBMB) equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M6">View MathML</a>

(2)

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 Korteweg-de Vries equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M7">View MathML</a>

(3)

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.

In [16], a generalized Benjamin-Bona-Mahony-Burgers equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M8">View MathML</a>

(4)

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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M9">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M10">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M11">View MathML</a>-smooth nonlinear function. Equation (4) with the dissipative term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M12">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M14">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M15">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M16">View MathML</a>[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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M17">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M17">View MathML</a>; 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 [21] in the following manner:

(i) First, consider a general form of the nonlinear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M21">View MathML</a>

(5)

(ii) To find the traveling wave solution of equation (5), the wave variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22">View MathML</a> is introduced so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M23">View MathML</a>

(6)

Based on this, one may use the following changes:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M24">View MathML</a>

(7)

and so on for other derivatives. Using (7) changes PDE (5) to an ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M25">View MathML</a>

(8)

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M26">View MathML</a>

(9)

is introduced that leads to the change of derivatives:

(10)

where other derivatives can be derived in a similar manner.

(v) The ansatz of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M28">View MathML</a>

(11)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M30">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M31">View MathML</a>), 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M32">View MathML</a>

(12)

where α is a positive constant. Using the wave variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22">View MathML</a> carries (12) into the ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M34">View MathML</a>

(13)

Balancing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M35">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M36">View MathML</a> in (13) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M37">View MathML</a>. The tanh method admits the use of the finite expansion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M38">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M39">View MathML</a>. Substituting (14) into (13), collecting the coefficients of Y, and setting it equal to zero, we find the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M40">View MathML</a>

(15)

Using Maple gives nine sets of solutions

(16)

These sets give the following solutions respectively:

(17)

If we accept <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M43">View MathML</a>, then we obtain solutions

(18)

4 The generalized Benjamin-Bona-Mahony-Burgers equation

We consider the generalized Benjamin-Bona-Mahony-Burgers equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M45">View MathML</a>

(19)

where α is a positive constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M46">View MathML</a>.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M13">View MathML</a>.

Using the wave variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22">View MathML</a> carries (19) into the ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M49">View MathML</a>

(20)

Balancing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M50">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M51">View MathML</a> in (20) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M52">View MathML</a>. Using the finite expansion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M53">View MathML</a>

(21)

we find the system of equations

(22)

Maple gives three sets of solutions

(23)

where k is left as a free parameter. These give the following solutions:

(24)

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M57">View MathML</a>.

Using the wave variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22">View MathML</a>, then by integrating this equation and considering the constant of integration to be zero, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M59">View MathML</a>

(25)

Balancing the second term with the last term in (25) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M37">View MathML</a>. Using the finite expansion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M61">View MathML</a>

(26)

we find the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M62">View MathML</a>

(27)

Using Maple, we obtain nine sets of solutions

(28)

These sets give the solutions

(29)

Case 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M17">View MathML</a>.

Using the wave variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M22">View MathML</a>, then by integrating this equation once and considering the constant of integration to be zero, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M67">View MathML</a>

(30)

Balancing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M36">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M69">View MathML</a> in (30) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M52">View MathML</a>. Using the finite expansion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M71">View MathML</a>

(31)

we find the system of equations

(32)

Solving the resulting system, we find the following sets of solutions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M73">View MathML</a>:

(33)

These in turn give the solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/144/mathml/M75">View MathML</a>

(34)

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.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

References

  1. Sobolev, SL: Some new problems in mathematical physics. Izv. Akad. Nauk SSSR, Ser. Mat.. 18, 3–50 (1954)

  2. Chen, PJ, Gurtin, ME: On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys.. 19, 614–627 (1968). Publisher Full Text OpenURL

  3. 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 OpenURL

  4. Taylor, D: Research of Consolidation of Clays, Massachusetts Institute of Technology Press, Cambridge (1952)

  5. 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 OpenURL

  6. Huilgol, R: A second order fluid of the differential type. Int. J. Non-Linear Mech.. 3, 471–482 (1968). Publisher Full Text OpenURL

  7. Ting, TW: Certain nonsteady flows of second-order fluids. Arch. Ration. Mech. Anal.. 14, 1–26 (1963)

  8. 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 OpenURL

  9. Karch, G: Asymptotic behaviour of solutions to some pseudoparabolic equations. Math. Methods Appl. Sci.. 20, 271–289 (1997). Publisher Full Text OpenURL

  10. 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 OpenURL

  11. Raupp, MA: Galerkin methods applied to the Benjamin-Bona-Mahony equation. Bull. Braz. Math. Soc.. 6, 65–77 (1975). Publisher Full Text OpenURL

  12. Wahlbin, L: Error estimates for a Galerkin method for a class of model equations for long waves. Numer. Math.. 23, 289–303 (1975)

  13. Ewing, RE: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation. SIAM J. Numer. Anal.. 15, 1125–1150 (1978). Publisher Full Text OpenURL

  14. 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.. 27, 737–743 (1981)

  15. 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 OpenURL

  16. Bruzon, MS, Gandarias, ML: Travelling wave solutions for a generalized benjamin-bona-mahony-burgers equation. Int. J. Math. Models Methods Appl. Sci.. 2, 103–108 (2008)

  17. Peregrine, DH: Calculations of the development of an undular bore. J. Fluid Mech.. 25, 321–330 (1996)

  18. 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 OpenURL

  19. 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 OpenURL

  20. 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 OpenURL

  21. 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 OpenURL