QUADRATIC B-SPLINE FINITE ELEMENT METHOD FOR THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION

Abstract

Abstract. A quadratic B-spline nite element method for the spatialvariable combined with a Newton method for the time variable is proposedto approximate a solution of Benjamin-Bona-Mahony-Burgers (BBMB)equation. Two examples were considered to show the eciency of theproposed scheme. The numerical solutions obtained for various viscositywere compared with the exact solutions. The numerical results show thatthe scheme is ecient and feasible. 1. IntroductionThe mathematical model of propagation of small amplitude long wavesin nonlinear dispersive media is described by the following Benjamin-Bona-Mahony-Burgers equation[1]:8> :u t u xxt u xx + u x + uu = f in [0;L] [0;T];u(0;t) = u(L;t) = 0 on [0;T];u(x;0) = u 0 (x) in [0;L];(1)where >0; are constants, fis a given forcing term. In the physical case, thedispersive e ect of (1) is the same as the Benjamin-Bona-Mahony (BBM) equa-tion, while the dissipative e ect is the same as the Burgers equation, and whichis an alternative model for the Korteweg-de Vries-Burgers (KdVB) equation [2].Numerical methods based on either nite elements[3]-[7], nte di erences[8]-[10],or Adomian decomposition scheme[11, 12]. Quadratic B-spline nite ele-ment method for approximating the solution of Burgers equation can be foundin [13, 14]. Cubic B-spline collocation method for numerical solution of theBBMB equation can be found in [15]. In this paper, we apply the quadraticB-spline nite element method to convert BBMB equation to a nite set of