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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A New Conservative Difference Scheme for the General Rosenau-RLW Equation | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 516260 13 pages doi 10.1155 2010 516260 Research Article A New Conservative Difference Scheme for the General Rosenau-RLW Equation Jin-Ming Zuo 1 Yao-Ming Zhang 1 Tian-De Zhang 2 and Feng Chang2 1 School of Science Shandong University of Technology Zibo 255049 China 2 School of Mathematics Shandong University Jinan 250100 China Correspondence should be addressed to Jin-Ming Zuo zuojinming@sdut.edu.cn Received 28 May 2010 Accepted 14 October 2010 Academic Editor Colin Rogers Copyright 2010 Jin-Ming Zuo et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable unconditionally stable and second-order convergent. Numerical examples show the efficiency of the scheme. 1. Introduction In this paper we consider the following initial-boundary value problem of the general Rosenau-RLW equation Ut - uxxt Uxxxxt Ux up x 0 xt x Xr 0 t T 1.1 with an initial condition u x 0 u0 x x x xr 1.2 and boundary conditions u xt t u xr t 0 uxx xi t uxx xr t 0 0 t T 1.3 2 Boundary Value Problems where p 2 is a integer and uo x is a known smooth function. When p 2 1.1 is called as usual Rosenau-RLW equation. When p 3 1.1 is called as modified Rosenau-RLW MRosenau-RLW equation. The initial boundary value problem 1.1 - 1.3 possesses the following conservative quantities Q 0 1 f u x t dx 2f uo x t dx Q 0 1.4 2 xi 2 xi Et 2 llulL WuxWL Uxx IIÌ2 1 lluollL uox IIÌ2 lluoxx IlL E 0 . L5 It is known the conservative scheme is better than the nonconservative ones. Zhang et al. 1 .
