The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions
© Gözükızıl and Akçağıl; licensee Springer 2013
Received: 3 December 2012
Accepted: 7 May 2013
Published: 23 May 2013
We studied mostly important four nonlinear pseudoparabolic physical models: the Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation, the Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation, the one-dimensional Oskolkov equation and the generalised hyperelastic-rod wave equation. By using the tanh-coth method and symbolic computation system Maple, we have obtained abundant new solutions of these equations. The exact solutions show that the tanh-coth method is a powerful mathematical tool for solving nonlinear pseudoparabolic equations.
Equations with a one-time derivative appearing in the highest order term are called pseudoparabolic and arise in many areas of mathematics and physics. They have been used, for instance, for fluid flow in fissured rock, consolidation of clay, shear in second-order fluids, thermodynamics and propagation of long waves of small amplitude. For more details, we refer the reader to [1–5] and references therein.
where γ, θ are constants and . Equation (3) includes several types of the BBM equation as seen in the literature. For more details, we refer the reader to [6–15]. We will study the general form of Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation (3) using the tanh-coth method. We aim to extend the previous works especially in [6, 7] to make further progress for obtaining abundant new travelling wave solutions.
This nonlinear, one-dimensional and pseudoparabolic equation describes nonlinear surface waves that spread along the axis Ox and is the viscosity term [16, 17]. In the literature the inverse scattering method has been thoroughly used to derive the multiple soliton solutions of equation (5) [8, 18–22]. In this work we developed these solutions in a way that can be easily applied by using the tanh-coth method, which is less sophisticated than the inverse scattering method.
This system describes the dynamics of an incompressible viscoelastic Kelvin-Voigt fluid. It was indicated in [23, 24] that the parameter λ can be negative and the negativeness of the parameter λ does not contradict the physical meaning of equation (7). We implemented the tanh-coth method to solve equation (6) and obtained new solutions which could not be attained in the past.
was first introduced in , in which the global existence of dissipative solutions were established, where α, β, θ and γ are constant parameters. This equation includes many important physical models in mathematical physics.
where u is the fluid velocity in the direction x (or, equivalently, the height of the water’s free surface above a flat bottom), α is a constant related to the critical shallow water wave speed. Camassa-Holm equation has been studied in [8, 18] and explicit travelling-wave solutions were sought . Besides, solitary wave solutions for modified forms of this equation were developed by Wazwaz .
The FW equation was used to study the qualitative behaviour of wave-breaking. A peaked solitary wave solution of this type of equation was obtained by Fornberg and Whitham [27, 28]. Using the tanh-coth method, we consider equation (8), which is a combined form of CH, DP and FW equations, and obtain new exact solutions. These solutions can be seen as an improvement of the previously known data.
As stated before, pseudoparabolic-type equations arise in many areas of mathematics and physics to describe many physical phenomena. In recent years considerable attention has been paid to the study of pseudoparabolic-type equations, and to construct exact solutions for this type of equations, several methods, for instance, the tanh-coth method, have been developed. In [29, 30], we discussed some well-known Sobolev-type equations and pseudoparabolic equations and obtained new travelling wave solutions by using the tanh-coth method. Motivated by these studies, we employed the tanh-coth method to investigate new travelling wave solutions for the equations that were previously mentioned.
2 Outline of the tanh-coth method
(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.
where other derivatives can be derived in a similar manner.
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 (17) and (18) into ODE (15) 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 and (), V, and μ. Having determined these parameters, knowing that M is a positive integer in most cases and using (18), one obtains an analytic solution in a closed form.
3 The Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation
4 The Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation
As can be seen easily, these solutions can be obtained by taking in the solutions of previous equations.
5 The one-dimensional Oskolkov equation
6 The generalised hyperelastic-rod wave equation
The last system gives the three sets of solutions as follows.
In this paper, we focused on travelling wave solutions of the general form of Benjamin-Bona-Mahony-Peregrine-Burgers equation, the general form of the Oskolkov-Benjamin-Bona-Mahony-Burgers equation, the one-dimensional Oskolkov equation and the generalised hyperelastic-rod wave equation. We derived various exact travelling wave solutions of these physical structures by using the tanh-coth method. Throughout the work, Maple was used to deal with the tedious algebraic operations.
Dedicated to Professor Hari M Srivastava.
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