Crank-Nicolson Difference Scheme for the Generalized Rosenau-Burgers Equation
In this paper, numerical solution for the generalized Rosenau-Burgers equation is considered and Crank-Nicolson finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence and priori error estimate of the scheme are proved. Numerical results demonstrate that the scheme is efficient and reliable.
[1] F. E. Browder, Existence and uniqueness theorems for solutions of
nonlinear boundary value problems, Proc. Symp. Appl. Math. 17 (1965)
24-49.
[2] Q. Chang, B. Guo, H. Jiang, Finite difference method for the generalized
Zakharov equations, Math. Comput. 64 (1995) 537-553.
[3] Q. Chang, E. Jia, W. Sun, Difference schemes for solving the generalized
nonlinear Schr¨odinger equation, J. Comput. Phys. 148 (1999) 397-415.
[4] S. K. Chung, Finite difference approximate solutions for the Rosenau
equation, Appl. Anal. 69 (1-2) (1998) 149-156.
[5] S. K. Chung, A.K. Pani, Numerical methods for the Rosenau equation,
Appl. Anal. 77 (2001) 351-369.
[6] B. Hu, Y. Xu and J. Hu, Crank-Nicolson finite difference scheme for the
Rosenau-Burges equation, Appl. Math. Comput. 204(2008) 311-316.
[7] L. Liu, M. Mei, A better asymptotic profile of Rosenau-Burgers equation,
Appl. Math. Comput. 131 (2002) 147-170.
[8] L. Liu, M. Mei, Y.S. Wong, Asymptotic behavior of solutions to the
Rosenau-Burgers equation with a periodic initial boundary, Nonlinear
Anal. 67 (2007) 2527-2539.
[9] M. Mei, Long-time behavior of solution for Rosenau-Burgers equation
(I), Appl. Anal. 63 (1996) 315-330.
[10] M. Mei, Long-time behavior of solution for Rosenau-Burgers equation
(II), Appl. Anal. 68 (1998) 333-356.
[11] K. Omrani, F. Abidi, T. Achouri, N. Khiari, A new conservative
finite difference scheme for the Rosenau equation, Appl. Math. Comput.
201(2008) 35-43.
[12] P. Rosenau, Dynamics of dense discrete systems, Progr. Theor. Phys. 79
(1988) 1028-1042.
[13] L. Zhang, Q. Chang, A conservative numerical scheme for nonlinear
Schr¨odinger with wave operator, Appl. Math. Comput. 145 (2003) 603-
612.
[14] Y. Zhou, Application of Discrete Functional Analysis to the Finite
Difference Method, Inter. Acad. Publishers, Beijing, 1990.
[1] F. E. Browder, Existence and uniqueness theorems for solutions of
nonlinear boundary value problems, Proc. Symp. Appl. Math. 17 (1965)
24-49.
[2] Q. Chang, B. Guo, H. Jiang, Finite difference method for the generalized
Zakharov equations, Math. Comput. 64 (1995) 537-553.
[3] Q. Chang, E. Jia, W. Sun, Difference schemes for solving the generalized
nonlinear Schr¨odinger equation, J. Comput. Phys. 148 (1999) 397-415.
[4] S. K. Chung, Finite difference approximate solutions for the Rosenau
equation, Appl. Anal. 69 (1-2) (1998) 149-156.
[5] S. K. Chung, A.K. Pani, Numerical methods for the Rosenau equation,
Appl. Anal. 77 (2001) 351-369.
[6] B. Hu, Y. Xu and J. Hu, Crank-Nicolson finite difference scheme for the
Rosenau-Burges equation, Appl. Math. Comput. 204(2008) 311-316.
[7] L. Liu, M. Mei, A better asymptotic profile of Rosenau-Burgers equation,
Appl. Math. Comput. 131 (2002) 147-170.
[8] L. Liu, M. Mei, Y.S. Wong, Asymptotic behavior of solutions to the
Rosenau-Burgers equation with a periodic initial boundary, Nonlinear
Anal. 67 (2007) 2527-2539.
[9] M. Mei, Long-time behavior of solution for Rosenau-Burgers equation
(I), Appl. Anal. 63 (1996) 315-330.
[10] M. Mei, Long-time behavior of solution for Rosenau-Burgers equation
(II), Appl. Anal. 68 (1998) 333-356.
[11] K. Omrani, F. Abidi, T. Achouri, N. Khiari, A new conservative
finite difference scheme for the Rosenau equation, Appl. Math. Comput.
201(2008) 35-43.
[12] P. Rosenau, Dynamics of dense discrete systems, Progr. Theor. Phys. 79
(1988) 1028-1042.
[13] L. Zhang, Q. Chang, A conservative numerical scheme for nonlinear
Schr¨odinger with wave operator, Appl. Math. Comput. 145 (2003) 603-
612.
[14] Y. Zhou, Application of Discrete Functional Analysis to the Finite
Difference Method, Inter. Acad. Publishers, Beijing, 1990.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54412", author = "Kelong Zheng and Jinsong Hu and ", title = "Crank-Nicolson Difference Scheme for the Generalized Rosenau-Burgers Equation", abstract = "In this paper, numerical solution for the generalized Rosenau-Burgers equation is considered and Crank-Nicolson finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence and priori error estimate of the scheme are proved. Numerical results demonstrate that the scheme is efficient and reliable.
", keywords = "Generalized Rosenau-Burgers equation, difference scheme, stability, convergence.", volume = "3", number = "9", pages = "631-5", }
