Table of Contents
ISRN Applied Mathematics

Volume 2014, Article ID 948072, 12 pages

http://dx.doi.org/10.1155/2014/948072
Research Article

Exact Solutions for the KdV Equation with Forcing Term by the Generalized tanh-coth Method and the -Expansion Method

1Department of Mathematics, College of Mathematics, Ahar Branch, Islamic Azad University, Ahar 5451116714, Iran

2School of Hejab, Education and Training of Osku, Osku 5351715656, Iran

Received 5 February 2014; Accepted 27 February 2014; Published 3 June 2014

Academic Editors: I. Hashim and W. Yeih

Copyright © 2014 Jalil Manafian and Monireh Nasrollahpour. 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

An application of the generalized tanh-coth method and the -expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. These methods are used for the KdV equation with forcing term. The generalized tanh-coth method and the -expansion method were used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. It is shown that the generalized tanh-coth method and the -expansion method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear problems.

1. Introduction

Nonlinear phenomena play a fundamental role in applied mathematics and physics. Recently, the study of nonlinear partial differential equations in modelling physical phenomena has become an important tool. The investigation of the travelling wave solutions plays an important role in nonlinear sciences. A variety of powerful methods have been presented, such as the inverse scattering transform [1], Hirota’s bilinear method [2], sine-cosine method [3], homotopy perturbation method [4], homotopy analysis method [5, 6], variational iteration method [7, 8], tanh-function method [9], Bäcklund transformation [10], Exp-function method [1116], tanh-coth method [1720], -expansion method [2123], Laplace Adomian decomposition method [24], and differential transform method [25]. Here, we use an effective method for constructing a range of exact solutions for following nonlinear partial differential equations that in this paper we developed solutions as well. The standard tanh method is well-known analytical method which is first presented by Malfliet’s [26] and developed in [26, 27]. In this paper, we explain method which is called the generalized tanh-coth method to look for travelling wave solutions of nonlinear evolution equations. The KdV equation with forcing term has been investigated by Zhang [28] is in the form where is an external forcing function varying with time , and are constants. Our aim of this paper is to obtain analytical solutions of the KdV equation with forcing term and to determine the accuracy of the aforementioned methods in solving these kinds of problems. Symmetries of differential equations are one of the most important concepts in theory of differential equations and physics. One of the most prominent equations is KdV (Kortwege-de Vries) equation with application in shallow water theory. There are diverse methods for finding symmetries of differential equations. One of the most important ones is Lie method. A symmetry is a mapping of one mathematical object into itself or into another mathematical object that preserves some property of the object.

Lie Group Symmetries. A symmetry is a mapping of one mathematical object into itself or into another mathematical object that preserves some property of the object. The easiest symmetries to see are the discrete symmetries of geometrical objects, such as the rotational symmetries of the objects. Note that the sphere in the middle is invariant under a continuous group of rotational symmetries, not just a discrete group. We have been speaking implicitly about groups of transformations of families of curves in the plane. Let and be points in the Euclidean plane, and for in , let be a transformation, depending on the parameter that takes points to points . Usually, we would give a rule defining the composition of two parameters and , but we can always reparameterize the group, so that the composition is additive; that is, . With this parameterization, the identity element becomes . We say the set of transformations is a (additive) transformation group if the following conditions are satisfied:(1) is one-to-one and onto;(2) ; that is, ;(3) (i.e., );(4)for each there exists a unique such that ; that is, .

If, in addition to these four group properties, is infinitely differentiable with respect to and analytic with respect to , we say that is a one-parameter Lie group (or a Lie point transformation) [29]. A point transformation maps points in the Euclidean plane into other points in the plane. There are more general transformations, such as contact transformations and Lie-Bäcklund transformations. The paper is organized as follows. In Sections 2 and 3, we briefly give the steps of the methods and apply the methods to solve the nonlinear partial differential equations. In Section 4, we examine KdV equation with forcing term using the before sections of methods. Also a conclusion is given in Section 5. Finally some references are given at the end of this paper.

2. Basic Idea of the -Expansion Method

Another powerful analytical method is called -expansion method; we give the detailed description of method which is first presented by Wang et al. [30].

Step 1. For a given NLPDE with independent variables and dependent variable can be converted to on ODE at which transformation is wave variable. Also, is constant to be determined later.

Step 2. We seek its solutions in the more general polynomial form as follows: where satisfies the second order LODE in the form where , , , and are constants to be determined later, , but the degree of which is generally equal to or less than , and the positive integer can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (3).

Step 3. Substitute (4) and (5) into (3) with the value of obtained in Step 1. Collecting the coefficients of and then setting each coefficient to zero, we can get a set of overdetermined partial differential equations for , , , , and with the aid of symbolic computation Maple.

Step 4. Solving the algebraic equations in Step 3 and then substituting and general solutions of (5) into (4), we can obtain a series of fundamental solutions of (2) depending of the solution of (5).

3. Basic Idea of the Generalized tanh-coth Method

We now describe the generalized tanh-coth method for the given partial differential equations. We give the detailed description of method which to use this method, we take following steps.

Step 1. For a given NLPDE with independent variables and dependent variable , we consider a general form of nonlinear equation which can be converted to on ODE at which transformation is wave variable. Also, is constant to be determined later.

Step 2. We introduce the Riccati equation as follows: that leads to the change of derivatives which admits the use of a finite series of functions of the form where , , , , and are constants to be determined later. But the positive integer can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (7). If is not an integer, then a transformation formula should be used to overcome this difficulty. For aforementioned method, expansion (10) reduces to the standard tanh method [26] for , .

Step 3. Substitute (8) and (9) into (7) with the value of obtained in Step 2. Collecting the coefficients of and then setting each coefficient to zero, we can get a set of overdetermined partial differential equations for , , , , and with the aid of symbolic computation Maple.

Step 4. Solving the algebraic equations in Step 3 and then substituting in (10).

We will consider the following special solutions of the Riccati equation (8).

Case 1. For each , , and , (8) has the following solution: or or or where is constant.

Case 2. For and , (8) has the following solution:

Case 3. For , , and , (8) has the following solution: but, we know

Case 4. For , , and , (8) has the following solution:

4. The KdV Equation with Forcing Term

Case I (application of the -expansion method). In this section, we employ the KdV equation with forcing term by using the -expansion method as follows: we suppose that and then (21) reduces to Using the transformation as follows where is a constant and is a function of to be determined later, (23) is carried to ODE where prime denotes the differential with respect to . In order to determine value of , we balance the linear term of the highest order with the highest order nonlinear term ; in (25), we have where we conclude and we find . We can suppose that the solution of (21) is of the form and therefore

Substituting (28) and (29) and by using the well-known software Maple, we obtain the system of following results: or or where and are arbitrary constants. Substituting (30)–(32) into expression (28), can be written as or or Substituting the general solutions of (5) into (33)–(35), we have three types of exact solutions of (21) as follows.

Case 1. (33). Consider

Case 2. , (34).

When , we obtain hyperbolic function solution where . When , we get rational solution where . If , , , , then (37) gives

In particular, if , , , , then (37) gives

Case 3. , (35).

When , we obtain hyperbolic function solution where .

When , we have trigonometric function solution where . When , we get rational solution where . If , , , , then (41) gives But, if , , , , then (41) gives In particular, if , , , , then (42) can be written as And if , , , , then (42) can be written as But if , , , , then (41) gets Also if , , , , then (41) gets For instance if , , we get which are the solitary wave solutions of the KdV equation with forcing term. It can be seen that the results are the same, with comparing results [28].

Case II (application of the generalized method). We now employ the KdV equation with forcing term by using the generalized tanh-coth method as follows: We suppose that and then (21) reduces to Using the transformation as follows where is a constant and is a function of to be determined later, (23) is carried to a ODE where prime denotes the differential with respect to .

In order to determine value of , we balance with in (55), and by using (10), we obtain . We can suppose that the solution of (51) is of the form Substituting (56) into (55) and using the well-known software Maple, we obtain the system of following results: or where , , , and are arbitrary constants. Substituting (57) and (58) into expression (56), we obtain where and where .

By the manipulation as explained in the previous section, we have the following.

(I) The First Set for (59). Using the relations (11)–(14), we have where . By using Case 2 in Section 3, relation (15) gets and by using Case 3 and using relations (18) and (19), we obtain where .

Using Case 3 in Section 3 and using or , we get and by using Case 3 and using or , we get and by using Case 4 in Section 3 and by using relations (20), we obtain

(II) The Second Set for (60). Using relations (11)–(14), we have where . By using Case 3 and using relations (18) and (19), we obtain where .

Using Case 3 in Section 3 and using or , we get and by using Case 3 and using or , we get and by using Case 4 in Section 3 and by using relations (20), we obtain where . And above solutions are the exact solutions of the KdV equation with forcing term. We obtain solitary wave and periodic wave solution for the KdV equation with forcing term. It can be seen that the results are the same, with comparing results [28].

5. Conclusion

In this paper, we obtained exact solutions for the KdV equation with forcing term by using the generalized tanh-coth and -expansion methods. Generalized tanh-coth and -expansion methods are useful methods for finding travelling wave solutions of nonlinear evolution equations. These methods have been successfully applied to obtain some new solitary wave and periodic wave solutions to the KdV equation with forcing term. Both generalized tanh-coth method and -expansion method are more powerful in searching for exact solutions of NLPDEs. Some of these results are in agreement with the results reported in the literature. Comparing our results and Zhang’s [28] results, then it can be seen that the results are the same. Also, new results are formally developed in this paper. It can be concluded that this method is a very powerful and efficient technique in finding exact solutions for wide classes of problems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for their valuable suggestions and comments.

References

  1. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. Hirota, The Direct Method in Soliton Theory, vol. 155, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  3. A.-M. Wazwaz, “Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 755–760, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Dehghan and J. Manafian, “The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method,” Zeitschrift fur Naturforschung A, vol. 64, no. 7-8, pp. 420–430, 2009. View at Google Scholar · View at Scopus
  5. M. Dehghan, J. Manafian, and A. Saadatmandi, “The solution of the linear fractional partial differential equations using the homotopy analysis method,” Zeitschrift fur Naturforschung A, vol. 65, pp. 935–949, 2010. View at Google Scholar
  6. M. Dehghan, J. Manafian, and A. Saadatmandi, “Solving nonlinear fractional partial differential equations using the homotopy analysis method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 448–479, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999. View at Google Scholar · View at Scopus
  8. M. Dehghan, J. M. Heris, and A. Saadatmandi, “Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses,” Mathematical Methods in the Applied Sciences, vol. 33, no. 11, pp. 1384–1398, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. X.-H. Meng, W.-J. Liu, H.-W. Zhu, C.-Y. Zhang, and B. Tian, “Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation,” Physica A, vol. 387, no. 1, pp. 97–107, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  11. J. M. Heris and M. Bagheri, “Exact solutions for the modified KdV and the generalized KdV equations via Exp-function method,” Journal of Mathematical Extension, vol. 4, no. 2, pp. 77–98, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. M. Heris and I. Zamanpour, “Analytical treatment of the coupled Higgs equation and the Maccari system via Exp-function method,” Acta Universitatis Apulensis, vol. 33, pp. 203–216, 2013. View at Google Scholar
  13. J. Manafian and I. Zamanpour, “Exact travelling wave solutions of the symmetric regularized long wave (SRLW) using analytical methods,” Statistics, Optimization & Information Computing, vol. 2, pp. 47–55, 2014. View at Google Scholar
  14. M. Dehghan, J. M. Heris, and A. Saadatmandi, “Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21, no. 6-7, pp. 736–753, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. Dehghan, J. M. Heris, and A. Saadatmandi, “Analytical treatment of some partial differential equations arising in mathematical physics by using the Exp-function method,” International Journal of Modern Physics B, vol. 25, no. 22, pp. 2965–2981, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J. Manafianheris and M. Fazli Aghdaei, “Application of the Exp-function method for solving the combined KdV-mKdV and Gardner-KP equations,” Mathematical Sciences, vol. 6, article 68, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. M. Heris and M. Lakestani, “Solitary wave and periodic wave solutions for variants of the KdVBurger and the K(n,n)-Burger equations by the generalized tanh-coth method,” Communications in Numerical Analysis, vol. 2013, Article ID cna-00155, 18 pages, 2013. View at Publisher · View at Google Scholar
  18. A. Bekir and A. C. Cevikel, “The tanh-coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics,” Journal of King Saud University-Science, vol. 23, no. 2, pp. 127–132, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. C. A. G. Sierra and A. H. Salas, “The generalized tanh-coth method to special types of the fifth-order KdV equation,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 873–880, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. Manafian and I. Zamanpour, “Exact travelling wave solutions of the symmetric regularized long wave (SRLW) using analytical methods,” Statistics, Optimization & Information Computing, vol. 2, pp. 47–55, 2014. View at Google Scholar
  21. M. Fazli Aghdaei and J. Manafianheris, “Exact solutions of the couple Boiti-Leon-Pempinelli system by the generalized (G/G)-expansion method,” Journal of Mathematical Extension, vol. 5, no. 2, part 2, pp. 91–104, 2011. View at Google Scholar · View at MathSciNet
  22. A. Bekir, “Application of the (G/G)-expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  23. B. Ayhan and A. Bekir, “The (G/G)-expansion method for the nonlinear lattice equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 9, pp. 3490–3498, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. Manafianheris, “Solving the integro-differential equations using the modified Laplace Adomian decomposition method,” Journal of Mathematical Extension, vol. 6, pp. 41–55, 2012. View at Google Scholar
  25. M. Bagheri and J. M. Heris, “Differential transform method for solving the linear and nonlinear Westervelt equation,” Journal of Mathematical Extension, vol. 6, pp. 81–91, 2012. View at Google Scholar
  26. W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” The American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. W. Malfliet and W. Hereman, “The tanh method. II. Perturbation technique for conservative systems,” Physica Scripta for Experimental and Theoretical Physics, vol. 54, no. 6, pp. 569–575, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. S. Zhang, “Exp-function method exactly solving the KdV equation with forcing term,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 128–134, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  29. M. Nadjafikhah and S.-R. Hejazi, “Lie symmetries and solutions of KdV equation,” International Mathematical Forum, vol. 4, no. 1–4, pp. 165–176, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. Wang, X. Li, and J. Zhang, “The (G/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher · View at Google Scholar · View at MathSciNet