American Journal of Computational and Applied Mathematics 2017, 7(1): 1-10 DOI: 10.5923/j.ajcam.20170701.01 Numerical Solutions of Rosenau-RLW Equation Using Galerkin Cubic B-Spline Finite Element Method N. M. Yagmurlu, B. Karaagac, S. Kutluay Department of Mathematics, Inonu University, Malatya, Turkey Abstract In this study, numerical solutions of Rosenau-RLW equation which is one of Rosenau type equations have been obtained by using Galerkin cubic B-spline finite element method. The fourth order Runge-Kutta technique has been used to solve the resulting ordinary differential equation system occured by the application of the method. The accuracy and efficiency of the present method have been tested by calculating the error norms 2 L and L ∞ . Moreover, the computed results have been compared with exact and numerical ones existing in the literature. Keywords Finite element method, Rosenau-RLW equation, Galerkin method, Runge-Kutta, Cubic B-spline, Solitary wave, Interaction 1. Introduction Nonlinear evolution equations play an important role for the studies appeared in nonlinear sciences. These equations can be seen in many studies on nonlinear evolutions such as plasma physics, solid state physics, fluid mechanics, water wave mechanics, meteorology and nonlinear optics. Two important equations belonging to the class of nonlinear evolution equations are =0 t x xxx u uu u µ + + (1) =0 t x x xxt u u uu u ε µ + + − (2) namely KdV and RLW equations, respectively. While KdV equation (1) is a nonlinear model to study the change forms long waves advancing in a rectangular channel, RLW equation (2) is used to simulate wave motion in media with nonlinear wave steeping and dispersion, such as shallow water waves and ion acoustic plasma waves. From the studies on KdV equation, it is well known that the KdV equation has a number of shortcomings. Firstly, it describes an unidirectional propagation of waves. Thus wave-wave and wave-wall interactions can not be treated by the KdV equation. Secondly, both shape and the behavior of high-amplitude waves can not be well predicted by the KdV equation since it was derived under the assumption of weak an harmonicity. In order to overcome these shortcomings of KdV equation, Rosenau [1, 2] has introduced an equation in the form * Corresponding author: murat.yagmurlu@inonu.edu.tr (N. M. Yagmurlu) Published online at http://journal.sapub.org/ajcam Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved = 0. t xxxxt x x u u u uu + + + (3) which is called Rosenau equation. The existence and uniquness of Rosenau equation (3) was proved by Park [3]. Later on, to make more advanced studies on nonlinear waves and to understand other nonlinear behaviours of the waves, the term xxt u − was added to Rosenau equation (3) and the following form has been obtained ( ) =0 p t x xxt xxxt x u u u u u κ σ α β + − + + (4) where κ , σ , α , β are real constants and 2 p ≥ is an integer. This equation is called as generalized Rosenau-RLW equation [4]. There are miscellaneous studies about Rosenau-RLW equation. For example; Zuo et. al [5] have proposed a new conservative difference scheme and also proved the corresponding convergence of the scheme. Pan et. al [6] have studied the initial-boundary problem of the usual Rosenau-RLW equation by finite difference method designing a conservative numerical scheme preserving the original conservative properties for the equation. In Ref. [7], Mittal and Jain have applied B-spline collocation method to the generalized Rosenau-RLW equation to obtain the numerical solutions with the aid of quintic B-spline base functions. Pan et al. [8] have considered the numerical solutions of the Rosenau-RLW equation using Crank-Nicolson type finite difference method and derived the existence of numerical solutions by Brouwer fixed point theorem. Hu and Wang [9] have studied the initial-boundary value problem for Rosenau-RLW equation by proposing a three-level linear finite difference scheme and also obtained the existence, uniqueness of difference solution, and a priori estimates in infinite norm. In Ref. [10], Wongsaijai and Poochinapan have proposed a mathematical model to obtain