Cubic B-spline Collocation Method for Numerical Solution of the Benjamin-Bona-Mahony-Burgers Equation

In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.

• M. Zarebnia
• R. Parvaz

• Benjamin-Bona-Mahony-Burgers equation
• Cubic Bspline
• Collocation method
• Finite difference.

[1] G. Stephenson, Partial Differential Equations for Scientists and Engineers,
Imperial College Press, 1996.
[2] B. Wang, Random attractors for the stochastic BenjaminBonaMahony
equation on unbounded domains, J. Differential Equations 246 (2009)
2506-2537.
[3] K. Al-Khaled, S. Momani, A. Alawneh, Approximate wave solutions for
generalized Benjamin-Bona-Mahony-Burgers equations, Applied Mathematics
and Computation, 171 (2005) 281-292.
[4] M.A. Raupp, Galerkin methods applied to the Benjamin-Bona-Mahony
equation, Bol Soc Brazil Mat 6 (1975), 65-77.
[5] J. Avrin, J.A. Goldstein, Global existence for the Benjamin-Bona-Mahony
equation in arbitrary dimensions, Nonlinear Anal. 9 (1985) 861-865.
[6] A.O. Celebi, V.K. Kalantarov, M. Polat, Attractors for the generalized
Benjamin-Bona-Mahony equation, J. Differential Equations 157 (1999)
439-451.
[7] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, third edition
,Springer-Verlg, 2002.
[8] D. Kincad, W. Cheny, Numerical analysis, Brooks/COLE, 1991.
[9] S.G. Rubin ,R.A. Graves, Cubic Spline Spproximation for Problems in
Fluid Mechanics, NASA TR R-436, Washington, DC; 1975.
[10] G.D. Smith, Numerical Solution of Patial Differential Method, Second
Edition, Oxford University Press, 1978.
[11] R.D Richtmyer, K.W. Morton, Difference Methods for Initial-Value
Problems, Inter science Publishers (John Wiley), New York, (1967).
[12] L. R. T. Gardner, G. A. Gardner and I. Dag, A B-spline finite element
method for the regularised long wave equation, Commun. Numer. Meth.
Eng. 12 (1995) 795-804.