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The Youngnam Mathematical Society (영남수학회)
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QUADRATIC B-SPLINE FINITE ELEMENT METHOD FOR THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION

  • Yin, Yong-Xue (Department of Mathematics, College of Science, YanBian University) ;
  • Piao, Guang-Ri (Department of Mathematics, College of Science, YanBian University)
  • Received : 2013.10.05
  • Accepted : 2013.11.04
  • Published : 2013.11.30
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Abstract

A quadratic B-spline finite element method for the spatial variable combined with a Newton method for the time variable is proposed to approximate a solution of Benjamin-Bona-Mahony-Burgers (BBMB) equation. Two examples were considered to show the efficiency of the proposed scheme. The numerical solutions obtained for various viscosity were compared with the exact solutions. The numerical results show that the scheme is efficient and feasible.

Keywords

References

  1. T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London. 1220 (1972), no. 272, 47-78.
  2. D.J.Korteweg, G.de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philosophical Magazine. (1895), no. 39, 422-443.
  3. M.A.Raupp, Galerkin methods applied to the Benjamin-Bona-Mahony equation, Boletim da Sociedade Brazilian Mathematical. 1 (1975), no. 6, 65-77.
  4. L.Wahlbin, Error estimates for a Galerkin mehtod for a class of model equations for long waves, Numerische Mathematik. 4 (1975), no. 23, 289-303.
  5. R.E.Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation, SIAM Journal on Numerical Analysis. 5 (1978), no. 15, 1125-1150.
  6. D.N.Arnold,J.Douglas Jr.,and V.Thomee, Superconvergence of finite element approxi-mation to the solution of a Sobolev equation in a single space variable, Mathematics of Computation. 153 (1981), no. 36, 737-743.
  7. K.Omrani, The convergence of the fully discrete Galerkin approximations for the Benjamin-Bona-Mahony(BBM)equation, Appl Math Comput. (2006), no. 180, 614-621.
  8. R.Kannan and S.K.Chung, Finite difference approximate solutions for the two-dimensional Burgers system, Comput Math Appl. (2002), no. 44, 193-200.
  9. T.Achouri, N.Khiari and K.Omrani, On the convergence of difference schemes for the Benjamin-Bona-Mahony(BBM)equation, Appl Math Comput. (2006), no. 182, 999-1005.
  10. K.Omrani and M.Ayadi, Finite difference discretization of the Benjamin-Bona-Mahony-Burgers(BBMB)equation, Numer Meth Part Differ Equat. (2008), no. 24, 239-248.
  11. D.Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl Math Comput. (2004), no. 149, 833-841.
  12. K.Al-Khaled, S.Momani, and A.Alawneh, Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations, Appl Math Comput. (2005), no. 171, 281-292.
  13. T.Ozis, A.Esen, S.Kutluay, Numerical solution of Burgers equation by quadratic B-spline finite element, Appl Math Comput.(2005), no. 165, 237-249.
  14. E.N.Aksan, Quadratic B-spline finite element method for numerical solution of the Burgers equation, Appl Math Comput. (2006), no. 174, 884-896.
  15. M. Zarebnia, R. Parvaz, Cubic B-spline collocation method for numerical solution of the Benjamin-Bona-Mahony-Burgers equation, International journal of mathematical sciences. 4 (2013), no. 7, 34-37.
  16. P.M.Prenter, Splines and Variational Methods, Wiley, New York, 1975.

Cited by

  1. Solution BBM-Burger Equation via Quartic Trigonometric B-spline Approach vol.1879, pp.2, 2013, https://doi.org/10.1088/1742-6596/1879/2/022109
  2. Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation vol.14, pp.2, 2013, https://doi.org/10.32513/tmj/19322008120