Exact solutions of Benjamin-Bona-Mahony-Burgers-type nonlinear pseudo-parabolic equations
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Exact solutions of Benjamin-Bona-Mahony-Burgers-type nonlinear pseudo-parabolic equations ˘ Ömer Faruk Gözükızıl* and S¸ amil Akçagıl *
Correspondence: [email protected] Department of Mathematics, Sakarya University, Sakarya, Turkey
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 ut – ut – αu = f (x, u, ∇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: 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 ut – ηut – αu = f (x, u, ∇u)
()
arise in many areas of mathematics and physics, where u = u(x, t), x ∈ ⊂ Rn , t ≥ , η and α are non-negative constants, denotes the Laplace operator acting on the space variables x. Equations of type () 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 [–]. Equation () 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 () is the Benjamin-Bona-Mahony-Burgers (BBMB) equation –uxxt + ut – αuxx + ( + u)ux = .
()
˘ licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons © 2012 Gözükızıl and Akçagıl; 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.
˘ Boundary Value Problems 2012, 2012:144 Gözükızıl and Akçagıl http://www.boundaryvalueproblems.com/content/2012/1/144
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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
uxxx + ut – uxx + uux = .
()
The BBMB equation has been tackled and investigated by many authors. For more details, we refer the reader to [–] and the references therein. In [], a generalized Benjamin-Bona-Mahony-Burgers equation –uxxt + ut – αuxx + βux + g(u) x =
()
has been considered and a set of new solitons, kinks, antikinks, compac
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