Journal of Mathematics

Volume 2013 (2013), Article ID 201276, 5 pages

http://dx.doi.org/10.1155/2013/201276

## New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

Department of Mathematics, Faculty of Science, Bozok University, 66100 Yozgat, Turkey

Received 2 October 2012; Accepted 30 October 2012

Academic Editor: Anjan Biswas

Copyright © 2013 Yusuf Pandir and Halime Ulusoy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

#### 1. Introduction

To find exact solutions to nonlinear evolution equations, some new techniques have been introduced in recent years such as the Hirota method, the extended Jacobi elliptic function expansion method, the exponential function method, -expansion method, the simplest equation method, the trial equation method, Kudryashov’s method, and solitary ansatz method [1–19]. Also some authors define new functions which are named generalized hyperbolic functions for constructing new solutions [20–22]. There are a lot of nonlinear evolution equations that are integrated using the various mathematical methods. Soliton solutions, compactons, singular solitons, and other solutions have been found by using these approaches. These types of solutions are very important and appear in various areas of applied mathematics.

In Section 2, we give the definition and properties of generalized hyperbolic functions. In Section 3, as applications, we obtain exact solutions of the generalized KdV equation and the coupled equal width wave equations.

#### 2. The Definition and Properties of the Symmetrical Hyperbolic Fibonacci and Lucas Functions

In this section, we will define new functions which are named the symmetrical hyperbolic Fibonacci and Lucas functions for constructing new exact solutions of NPDEs and then study the properties of these functions.

*Definition 1. *Suppose that is an independent variable; , , and are all constants. The generalized hyperbolic sine function is
generalized hyperbolic cosine function is
generalized hyperbolic tangent function is
generalized hyperbolic cotangent function is
generalized hyperbolic secant function is
generalized hyperbolic cosecant function is
and the above six kinds of functions are said to be generalized new hyperbolic functions. Thus we can prove the following theory of generalized hyperbolic functions on the basis of Definition 1.

Theorem 2. *The generalized hyperbolic functions satisfy the following relations:
**
The following just part of them are proved here for simplification.*

Theorem 3. *The derivative formulae of generalized hyperbolic functions are as follows:
*

*Proof of (8). *According to (7) and (8), we can get
Similarly, we can prove other differential coefficient formulae in Theorem 3.

*Remark 4. *We see that when , , , and in (1)–(6), new generalized hyperbolic functions , , , , , and degenerate as hyperbolic functions , , , , , and , respectively. In addition, when or in (1)–(6), , , , , , and degenerate as exponential function , , , , and , respectively.

#### 3. Applications

*Example 5. *Application to the generalized form of KdV equation is presented.

The generalized KdV equation that will be studied in this paper are given by [23]
where the first term represents the generalized evolution. The special case with is the regular evolution term. The coefficients of are the nonlinear terms while the coefficients of and are the nonlinear dispersion terms. This equation with has been already studied by [18], where, in addition to soliton solution, compactons and periodic solutions were also obtained. The hypothesis for solving this equation is
where
Here, in (11) and (12), represents the amplitude of the soliton while is the inverse width of the soliton and is the velocity of the soliton. The exponent is unknown at this point and will be evaluated during the course of the derivation of the solutions to (10). From (11), it is possible to obtain
These results will now be substituted in (10) to obtain the 1-soliton solution of the generalized KdV equation. Equation (10) by virtue of (13) reduces to
From (14), equating the exponents and gives
that leads to
Now from (14), the two linearly independent functions are for . Thus setting their coefficients to zero gives
Thus the 1-soliton solution of the generalized KdV equation with generalized evolution is given by
This shows that the restriction on the nonlinear exponents of (10) must be

*Remark 6. *If we take the corresponding values for some parameters, the solution (19) can be reduced to the solution mentioned in [23].

*Example 7. *Application to the coupled equal width wave equations is presented.

We consider the coupled equal width wave equations [24] The hypothesis for solving these equations is where Here in (22)–(23) and are the amplitudes of the -soliton and -soliton, respectively, while and are the velocity of the soliton and and are the inverse widths of the solitons. The exponents and are unknown at this point and their values will fall out in the process of deriving the solution of this equation. From (22), it is possible to obtain Substituting these into (21) yields to Now from (28) and (29), equating the exponents and and also and gives, respectively, that leads to So from (28) and (29), the four linearly independent functions are and for . Therefore, setting their respective coefficients to zero we obtain , , and as follow.

*Case 1. *One has
where , , , and are arbitrary constants. Substituting (32) into (22), we obtain new exact solution to (21),
where .

*Case 2. *One has
where , , , and are arbitrary constants. Substituting (34) into (22), we obtain new exact solution to (21),
where .

*Remark 8. *If we search the corresponding values for some parameters and make some transformations, in particular the solution (35) can be reduced to the solution mentioned in [24].

#### 4. Conclusions and Remarks

We consider generalized hyperbolic functions and new kinds of generalized hyperbolic function transformation to construct new exact solutions of nonlinear partial differential equations. This paper obtains 1-soliton solution to the generalized KdV equation and the coupled equal width wave equations. Our methods also can be applied to construct new exact solutions of other nonlinear partial differential equations.

#### References

- R. Hirota, “Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices,”
*Journal of Mathematical Physics*, vol. 14, no. 7, pp. 810–814, 1973. View at Google Scholar · View at Scopus - Y. Gurefe, Y. Pandir, and E. Misirli, “New exact solutions of stochastic KdV equation,”
*Applied Mathematical Sciences*, vol. 6, no. 65-68, pp. 3225–3234, 2012. View at Google Scholar - X. Wen, “Extended Jacobi elliptic function expansion solutions of variant Boussinesq equations,”
*Applied Mathematics and Computation*, vol. 217, no. 6, pp. 2808–2820, 2010. View at Publisher · View at Google Scholar · View at Scopus - Y. Gurefe and E. Misirli, “Exp-function method for solving nonlinear evolution equations with higher order nonlinearity,”
*Computers and Mathematics with Applications*, vol. 61, no. 8, pp. 2025–2030, 2011. View at Publisher · View at Google Scholar · View at Scopus - G. Ebadi and A. Biswas, “The (${G}^{\prime}/G$) method and 1-soliton solution of the Davey-Stewartson equation,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 694–698, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Gurefe and E. Misirli, “New variable separation solutions of two-dimensional Burgers system,”
*Applied Mathematics and Computation*, vol. 217, no. 22, pp. 9189–9197, 2011. View at Publisher · View at Google Scholar · View at Scopus - N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations,”
*Chaos, Solitons and Fractals*, vol. 24, no. 5, pp. 1217–1231, 2005. View at Publisher · View at Google Scholar · View at Scopus - N. K. Vitanov, “Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 3, pp. 1176–1185, 2011. View at Publisher · View at Google Scholar · View at Scopus - C. S. Liu, “Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations,”
*Computer Physics Communications*, vol. 181, no. 2, pp. 317–324, 2010. View at Publisher · View at Google Scholar · View at Scopus - Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of an irrational trial equation method to high-dimensional nonlinear evolution equations,”
*Journal of Advanced Mathematical Studies*, vol. 5, no. 1, pp. 41–47, 2012. View at Google Scholar - Y. Pandir, Y. Gurefe, U. Kadak, and E. Misirli, “Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 478531, 16 pages, 2012. View at Publisher · View at Google Scholar - Y. Xie and J. Tang, “The superposition method in seeking the solitary wave solutions to the KdV-Burgers equation,”
*Pramana*, vol. 66, no. 3, pp. 479–483, 2006. View at Publisher · View at Google Scholar · View at Scopus - N. A. Kudryashov, “One method for finding exact solutions of nonlinear differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 6, pp. 2248–2253, 2012. View at Publisher · View at Google Scholar - Y. Pandir, Y. Gurefe, and E. Misirli, “A new approach to Kudryashovs method for solving some nonlinear physical models,”
*International Journal of Physical Sciences*, vol. 7, pp. 2860–2866, 2012. View at Google Scholar - N. A. Kudryashov and S. G. Prilipko, “Exact solutions of the generalized K(m,m) equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 3, pp. 1107–1113, 2011. View at Publisher · View at Google Scholar · View at Scopus - G. Ebadi and A. Biswas, “Application of the (${G}^{\prime}/G$)-expansion method for nonlinear diffusion equations with nonlinear source,”
*Journal of the Franklin Institute*, vol. 347, no. 7, pp. 1391–1398, 2010. View at Publisher · View at Google Scholar · View at Scopus - G. Ebadi and A. Biswas, “The (${G}^{\prime}/G$) method and topological soliton solution of the K(m,n) equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 6, pp. 2377–2382, 2011. View at Publisher · View at Google Scholar · View at Scopus - G. Ebadi, A. H. Kara, M. D. Petkovic, A. Yildirim, and A. Biswas, “1-Soliton solution of the generalized KdV equation with generalized evolution,”
*Proceedings of the Romanian Academy A*, vol. 13, pp. 215–224, 2012. View at Google Scholar - A. Biswas, M. D. Petković, and D. Milović, “Topological exact soliton solution of the power law KdV equation,”
*Applied Mathematics and Computation*, vol. 217, no. 4, pp. 1780–1784, 2010. View at Publisher · View at Google Scholar · View at Scopus - Y. J. Ren and H. Q. Zhang, “New generalized hyperbolic functions and auto-Bäcklund transformation to find new exact solutions of the (2+1)-dimensional NNV equation,”
*Physics Letters A*, vol. 357, no. 6, pp. 438–448, 2006. View at Publisher · View at Google Scholar · View at Scopus - E. Yomba, “Generalized hyperbolic functions to find soliton-like solutions for a system of coupled nonlinear Schrödinger equations,”
*Physics Letters A*, vol. 372, no. 10, pp. 1612–1618, 2008. View at Publisher · View at Google Scholar · View at Scopus - Z. I. A. Al-Muhiameed and E. A.-B. Abdel-Salam, “Generalized hyperbolic function solution to a class of nonlinear Schrödinger-type equations,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 265348, 15 pages, 2012. View at Publisher · View at Google Scholar - M. S. Ismail and A. Biswas, “1-Soliton solution of the generalized KdV equation with generalized evolution,”
*Applied Mathematics and Computation*, vol. 216, no. 5, pp. 1673–1679, 2010. View at Publisher · View at Google Scholar · View at Scopus - A. H. A. Ali, A. A. Soliman, and K. R. Raslan, “Soliton solution for nonlinear partial differential equations by cosine-function method,”
*Physics Letters A*, vol. 368, no. 3-4, pp. 299–304, 2007. View at Publisher · View at Google Scholar · View at Scopus