Applied and Computational Mathematics 2015; 4(3): 122-129 Published online April 27, 2015 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.20150403.14 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online) Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method M. Ashrafuzzaman Khan, M. Ali Akbar Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh Email address: akhanmath@yahoo.com (M. A. Khan), ali_math74@yahoo.com (M. A. Akbar) To cite this article: M. Ashrafuzzaman Khan, M. Ali Akbar. Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method. Applied and Computational Mathematics. Vol. 4, No. 3, 2015, pp. 122-129. doi: 10.11648/j.acm.20150403.14 Abstract: Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process. Keywords: MSE Method, Nonlinear Evolution Equations, Solitary Wave Solutions, Exact Solutions, Generalized Fifth-Order Kdv Equation 1. Introduction Nonlinear evolution equations occur not only from many fields of mathematics, but also from other branches of science such as physics, material science, mechanics etc. Intricacy of NLEEs and challenges in their theoretical study has attracted lots of attention from numerous mathematicians and scientists who are concern with nonlinear sciences. Therefore, the studies of exact solutions to NLEEs play a very important role to know the inner structure of the nonlinear phenomena. But the basic problem is, it is not easy to attain their exact solutions. Therefore, in order to examine exact solutions, different groups of mathematicians and physicist are working jointly. In the recent years, considerable developments have been made for searching exact solutions to NLEEs. They established several methods, such as,the inverse scattering transformation method [1], the Hirota’s bilinear method [2], the Backlund transformation method ([3][4]), the Darboux transformation method [5], the Painleve expansion method [6], the Adomian decomposition method ([7][8]), the He’s homotopy perturbation method ([9][10]), the Jacobi elliptic function method ([11][12]), the Miura transformation method [13], the sine-cosine method ([14][15]), the homogeneous balance method [16], the tanh- function method ([17][18]), the extended tanh-function method ([19] [20]), the first integration method [21], the F- expansion method [22],the auxiliary equation method [23], the Lie group symmetry method [24], the variational iteration method [25], the ansatz method ([26][27]), the Exp-function method ([28][29]), the ( / ) G G ′ -expansion method ([30]- [35]), the modified simple equation method ([36]-[40]),the exp( ( )) φη - -expansion method ([41][42]), etc. The modified simple equation method ([36]-[40]) is a recently developed rising method. Its computation is straightforward, systematic, and no need the symbolic computation software to manipulate the algebraic equations. But, the method has some shortcoming, when the balance number is greater than one, usually the method does not give any solution. To the best of our knowledge, till now only two articles are available in the literature concerning higher balance number (for balance number two). Salam [43] used the MSE method to the modified Liouville equation (wherein the balance number is two) and write-down a solution to this equation. However, unfortunately the obtained solution does