ϥϮاثالثϭ Ϧلثاملي اϭر الدϤؤتϤ ال لإحصاءلحاسب اϡϮϠعϭ اϬتطبيقاتϭ 8 - 18 أبريل3112 The 38 th International Conference for Statistics, Computer Science and its Applications 8-18 April 2013 1 A comparison between the method of lines and Adomian decomposition method for solving the KdV-Burger equation Mohamed Reda a , Mohamed M. Mousa b , Ibrahim Ahmed Sakr c a,b Department of Basic Engineering Science, Benha Faculty of Engineering, Benha University, Egypt c Department of Engineering Mathematics and Physics, Shoubra Faculty of Engineering, Benha University, Egypt Abstract This paper presents two methods for obtaining the solutions to the nonlinear Korteweg-de Vries–Burgers (KdVB) equation. The first method is a method of lines (MOL). The second method is Adomian decomposition method (ADM). The numerical results of the MOL are compared with the analytical results of the ADM. In order to show the reliability of the considered methods we have compared the obtained solutions with the exact ones. The results reveal that the both methods are effective and convenient for solving such types of partial differential equations but the method of lines gives accurate results over the analytical method. Keywords: KdV-Burger equation; the method of lines; Adomian decomposition method; finite difference scheme; Runge–Kutta method. 1. Introduction This paper is concerned with the initial-boundary value problem associated with the nonlinear dispersive and dissipative wave which was formulated by Korteweg, de Vries and Burgers in the form                       (1) Where  ,  ,  are constant coefficients. It is well known that many physical phenomena can be described by the Korteweg-de Vries– Burgers equation. Eq. (1) can serve as a nonlinear wave model of a fluid in an elastic tube [1],of a liquid with small bubbles [2,] and turbulence [3,4].The coefficients  and  in Eq. (1) represent the damping and the dispersion coefficients, respectively. We note that Eq.(1) is non integrable. Soliton solutions of the KdV equation are known since long time [5,6]. Many problems, however, involve not only dispersion but also dissipation, and these are not governed by the KdV equation. More complicated problems are the flow of liquids containing gas bubbles [7,8],and the propagation of waves in an elastic tube filled with a viscous fluid [9,10]. Other cases regarded the governing evolution equation can be shown to be the so-called Korteweg-de Vries–Burgers equation. In particular, the travelling wave solution to the KdVB equation has been studied extensively. Johnson [11], Demiray [12] and Antar and Demiray [13] derived KdVB equation as the governing evolution equation for waves propagating in fluid-filled elastic or viscoelastic tubes in which the effects of dispersion, dissipation and nonlinearity are present.