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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 106343, 22 pages
Optimal Error Estimate of Chebyshev-Legendre Spectral Method for the Generalised Benjamin-Bona-Mahony-Burgers Equations
1School of Mathematics, Lanzhou City University, Lanzhou 730070, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Received 29 December 2011; Revised 6 April 2012; Accepted 15 April 2012
Academic Editor: M. Victoria Otero-Espinar
Copyright © 2012 Tinggang Zhao 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.
- D. H. Peregrine, “Calculations of the development of an undular bore,” The Journal of Fluid Mechanics, vol. 25, no. 2, pp. 321–330, 1996.
- T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philosophical Transactions of the Royal Society of London, vol. 272, no. 1220, pp. 47–78, 1972.
- G. Karch, “Asymptotic behaviour of solutions to some pseudoparabolic equations,” Mathematical Methods in the Applied Sciences, vol. 20, no. 3, pp. 271–289, 1997.
- L. Zhang, “Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions,” Nonlinear Analysis, vol. 25, no. 12, pp. 1343–1369, 1995.
- S. Kinami, M. Mei, and S. Omata, “Convergence to diffusion waves of the solutions for Benjamin-Bona-Mahony-Burgers equations,” Applicable Analysis, vol. 75, no. 3-4, pp. 317–340, 2000.
- N. A. Larkin and M. P. Vishnevskii, “Dissipative initial boundary value problem for the BBM-equation,” Electronic Journal of Differential Equations, vol. 149, pp. 1–10, 2008.
- K. Al-Khaled, S. Momani, and A. Alawneh, “Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations,” Applied Mathematics and Computation, vol. 171, no. 1, pp. 281–292, 2005.
- M. S. Bruzón, M. L. Gandarias, and J.-C. Camacho, “Symmetry for a family of BBM equations,” Journal of Nonlinear Mathematical Physics, vol. 15, Supplement, pp. 81–90, 2008.
- M. S. Bruzón and M. L. Gandarias, “Travelling wave solutions for a generalized Benjamin-Bona-Mahony-Burgers equation,” International Journal of Mathematical Models and Methods in Applied Sciences, vol. 2, no. 1, pp. 103–108, 2008.
- M. S. Bruzón, M. L. Gandarias, and J. C. Camacho, “Symmetry analysis and solutions for a generalization of a family of BBM equations,” Journal of Nonlinear Mathematical Physics, vol. 15, supplement 3, pp. 81–90, 2008.
- H. Tari and D. D. Ganji, “Approximate explicit solutions of nonlinear BBMB equations by Hes methods and comparison with the exact solution,” Physics Letters A, vol. 367, pp. 95–101, 2007.
- S. A. El-Wakil, M. A. Abdou, and A. Hendi, “New periodic wave solutions via Exp-function method,” Physics Letters A, vol. 372, no. 6, pp. 830–840, 2008.
- M. Kazeminia, P. Tolou, J. Mahmoudi, I. Khatami, and N. Tolou, “Solitary and periodic solutions of BBMB equation via Exp-function method,” Advanced Studies in Theoretical Physics, vol. 3, no. 12, pp. 461–471, 2009.
- C. A. Gómez and A. H. Salas, “Exact solutions for the generalized BBM equation with variable coefficients,” Mathematical Problems in Engineering, vol. 2010, Article ID 498249, 10 pages, 2010.
- A. Fakhari, G. Domairry, and Ebrahimpour, “Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution,” Physics Letters A, vol. 368, no. 1-2, pp. 64–68, 2007.
- M. Alqruan and K. Al-Khaled, “Sinc and solitary wave solutions to the generalized Benjamin-Bona-Mahony-Burgers equations,” Physica Scripta, vol. 83, no. 6, 2011.
- K. Omrani and M. Ayadi, “Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 1, pp. 239–248, 2008.
- F. Z. Gao, J. X. Qiu, and Q. Zhang, “Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation,” Journal of Scientific Computing, vol. 41, no. 3, pp. 436–460, 2009.
- W. S. Don and D. Gottlieb, “The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points,” SIAM Journal on Numerical Analysis, vol. 31, no. 6, pp. 1519–1534, 1994.
- H. Y. Li, H. Wu, and H. P. Ma, “The Legendre Galerkin-Chebyshev collocation method for Burgers-like equations,” IMA Journal of Numerical Analysis, vol. 23, no. 1, pp. 109–124, 2003.
- H. Wu, H. P. Ma, and H. Y. Li, “Optimal error estimates of the Chebyshev-Legendre spectral method for solving the generalized Burgers equation,” SIAM Journal on Numerical Analysis, vol. 41, no. 2, pp. 659–672, 2003.
- J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer, Berlin, Germany, 1976.
- J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, China, 2006.
- C. Bernardi and Y. Maday, “Spectral methods,” in Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., Techniques of Scientific Computing (Part 2), pp. 209–486, North-Holland, Amsterdam, The Netherlands, 1997.
- J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, Mineola, NY, USA, 2nd edition, 2000.
- B. Y. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998.
- L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, Pa, USA, 2000.
- J. Shen, “Efficient Chebyshev-Legendre Galerkin methods for elliptic problems,” in Proceedings of the 3rd International Conference on Spectral and High Order Methods (ICOSAHOM ’95), pp. 233–239, Houston Journal of Mathematics, June 1995.
- H. P. Ma, “Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws,” SIAM Journal on Numerical Analysis, vol. 35, no. 3, pp. 869–892, 1998.
- H. P. Ma, “Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws,” SIAM Journal on Numerical Analysis, vol. 35, no. 3, pp. 893–908, 1998.
- H. P. Ma and W. W. Sun, “A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations,” SIAM Journal on Numerical Analysis, vol. 38, no. 5, pp. 1425–1438, 2000.
- T. G. Zhao, C. Li, Z. L. Zang, and Y. J. Wu, “Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 1046–1056, 2012.
- T. G. Zhao, Y. J. Wu, and H. P. Ma, “Error analysis of Chebyshev-Legendre pseudospectral method for a class of nonclassical parabolic equation,” Journal of Scientific Computing. In press.
- W. Zhang and H. P. Ma, “The Chebyshev-Legendre collocation method for a class of optimal control problems,” International Journal of Computer Mathematics, vol. 85, no. 2, pp. 225–240, 2008.
- J. Shen, “Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials,” SIAM Journal on Scientific Computing, vol. 15, no. 6, pp. 1489–1505, 1994.
- B. K. Alpert and V. Rokhlin, “A fast algorithm for the evaluation of Legendre expansions,” SIAM Journal on Scientific and Statistical Computing, vol. 12, no. 1, pp. 158–179, 1991.