In the present study, the coupled Burgers’ equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L₂ and L_{∞}. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

The basic idea of this article is to investigate the numerical solutions of Gardner Kawahara equation, a particular case of extended Korteweg-de Vries (KdV) equation, by means of finite element method.

In this paper, a numerical scheme based on quintic B-spline collocation method using the Strang splitting technique is presented for the solution of the generalized Rosenau-RLW equation given by appropriate initial-boundary values. For our purpose, firstly the problem is split into two subproblems such that each one includes the derivative in the direction of time. Secondly, each subproblem using collocation finite element method with quintic B-splines for spatial integration is reduced to a system of ordinary differential equations (ODEs). Then, the resulting ODEs for time integration are solved using the Strang time-splitting technique with the second order via the usual Runge-Kutta (RK-4) algorithm with the fourth order. To measure the accuracy and efficiency of the present scheme, a model problem with an exact solution is taken into consideration and investigated for various values of the parameter p. The L₂ and L_{∞} errors with the invariants of discrete mass Q and discrete energy E have been computed and given a comparison with other ones found in the literature. The convergence order of the present numerical scheme has also been computed. Furthermore, the stability analysis of the scheme is numerically examined.