%0 Journal Article
%T Numerical Solution of the Modified Equal Width Wave Equation
%A Seydi Battal Gazi Karako£¿
%A Turabi Geyikli
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/587208
%X Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws and error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated. 1. Introduction The modified equal width wave equation (MEW) based upon the equal width wave (EW) equation [1, 2] which was suggested by Morrison et al. [3] is used as a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. This equation is related with the modified regularized long wave (MRLW) equation [4] and modified Korteweg-de Vries (MKdV) equation [5]. All the modified equations are nonlinear wave equations with cubic nonlinearities and all of them have solitary wave solutions, which are wave packets or pulses. These waves propagate in non-linear media by keeping wave forms and velocity even after interaction occurs. Few analytical solutions of the MEW equation are known. Thus numerical solutions of the MEW equation can be important and comparison between analytic solution can be made. Geyikli and Battal Gazi Karako£¿ [6, 7] solved the MEW equation by a collocation method using septic B-spline finite elements and using a Petrov-Galerkin finite element method with weight functions quadratic and element shape functions which are cubic B-splines. Esen applied a lumped Galerkin method based on quadratic B-spline finite elements which have been used for solving the EW and MEW equation [8, 9]. Saka proposed algorithms for the numerical solution of the MEW equation using quintic B-spline collocation method [10]. Zaki considered the solitary wave interactions for the MEW equation by collocation method using quintic B-spline finite elements [11] and obtained the numerical solution of the EW equation by using least-squares method [12]. Wazwaz investigated the MEW equation and two of its variants by the tanh and the sine-cosine methods [13]. A solution based on a collocation method incorporated cubic B-splines is investigated by and Saka and Da£¿ [14]. Variational iteration method is introduced to solve the MEW equation by Lu [15]. Evans and Raslan [16] studied the generalized EW equation by using collocation method based on quadratic B-splines to obtain the numerical solutions of a single solitary
%U http://www.hindawi.com/journals/ijde/2012/587208/