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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 527467, 15 pages
http://dx.doi.org/10.1155/2012/527467
Research Article

A Galerkin Solution for Burgers' Equation Using Cubic B-Spline Finite Elements

Department of Mathematics, Faculty of Education, Suez Canal University, Al-Arish 45111, Egypt

Received 13 May 2012; Accepted 24 July 2012

Academic Editor: Benchawan Wiwatanapataphee

Copyright © 2012 A. A. Soliman. 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.

Linked References

  1. J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics,” Quarterly of Applied Mathematics, vol. 9, pp. 225–236, 1951.
  2. J. M. Burgers, “A mathematical model illustrating the theory of turbulence,” in Advances in Applied Mechanics, pp. 171–199, Academic Press, New York, NY, USA, 1948.
  3. E. Varo\=glu and W. D. L. Finn, “Space-time finite elements incorporating characteristics for the Burgers equation,” International Journal for Numerical Methods in Engineering, vol. 16, pp. 171–184, 1980. View at Publisher · View at Google Scholar
  4. E. Hopf, “The partial differential equation ut+uuxx=vuxx,” Communications on Pure and Applied Mathematics, vol. 3, pp. 201–230, 1950.
  5. E. R. Benton and G. W. Platzman, “A table of solutions of the one-dimensional Burgers equation,” Quarterly of Applied Mathematics, vol. 30, pp. 195–212, 1972.
  6. D. U. Rosenberg, Methods for Solution of Partial Differntial Equations, vol. 113, American Elsevier, New York, NY, USA, 1969.
  7. J. Caldwell, P. Wanless, and A. E. Cook, “A finite element approach to Burgers' equation,” Applied Mathematical Modelling, vol. 5, no. 3, pp. 189–193, 1981. View at Publisher · View at Google Scholar
  8. B. M. Herbst, S. W. Schoombie, and A. R. Mitchell, “A moving Petrov-Galerkin method for transport equations,” International Journal for Numerical Methods in Engineering, vol. 18, no. 9, pp. 1321–1336, 1982. View at Publisher · View at Google Scholar
  9. L. R. T. Gardner, G. A. Gardner, and A. H. A. Ali, “A method of lines solutions for Burgers’ equation,” in Computer Mech, Y. K. Cheung, et al., Ed., pp. 1555–1561, Balkema, Rotterdam, 1991.
  10. I. Christie, D. F. Griffiths, A. R. Mitchell, and J. M. Sanz-Serna, “Product approximation for nonlinear problems in the finite element method,” Journal of Numerical Analysis, vol. 1, no. 3, pp. 253–266, 1981. View at Publisher · View at Google Scholar
  11. S. G. Rubin and R. A. Graves, Jr., “Viscous flow solutions with a cubic spline approximation,” Computers & Fluids, vol. 3, pp. 1–36, 1975.
  12. S. G. Rubin and P. K. Khosla, “Higher-order numerical solutions using cubic splines,” American Institute of Aeronautics and Astronautics, vol. 14, no. 7, pp. 851–858, 1976.
  13. J. Caldwell, “Application of cubic splines to the nonlinear Burgers' equation,” in Numerical Methods for Nonlinear Problems, vol. 3, pp. 253–261, Pineridge Press, 1986.
  14. A. H. A. Ali, G. A. Gardner, and L. R. T. Gardner, “A collocation solution for Burgers' equation using cubic Cubic-spline finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 100, no. 3, pp. 325–337, 1992. View at Publisher · View at Google Scholar
  15. İ. Dağ, D. Irk, and B. Saka, “A numerical solution of the Burgers' equation using cubic B-splines,” Applied Mathematics and Computation, vol. 163, no. 1, pp. 199–211, 2005. View at Publisher · View at Google Scholar
  16. A. A. Soliman, “Numerical technique for Burgers’ equation based on similarity reductions,” in International Conference on Applied Computational Fluid Dynamics, pp. 559–566., Bejing, China, October 2000.
  17. A. A. Soliman, “The modified extended tanh-function method for solving Burgers-type equations,” Physica A, vol. 361, no. 2, pp. 394–404, 2006. View at Publisher · View at Google Scholar
  18. K. R. Raslan, “A collocation solution for Burgers equation using quadratic B-spline finite elements,” International Journal of Computer Mathematics, vol. 80, no. 7, pp. 931–938, 2003. View at Publisher · View at Google Scholar
  19. G. A. Gardner, A. H. A. Ali, and L. R. T. Gardner, “A finite element solution for the korteweg-de vries equation using cubic B-spline shape function,” in Proceedings of the Information Security Policy Made Easy (ISNME '89), R. Gruber, J. Periaux, and R. P. Shaw, Eds., vol. 2, pp. 565–570, Springer, 1989.
  20. L. R. T. Gardner and G. A. Gardner, “Solitary waves of the regularised long-wave equation,” Journal of Computational Physics, vol. 91, no. 2, pp. 441–459, 1990. View at Publisher · View at Google Scholar
  21. O. C. Zienkiewicz, The Finite Elment Method, McGraw Hill, London, UK, 3th edition, 1977.
  22. P. M. Prenter, Splines and Variational Methods, John Wiley & Sons, New York, NY, USA, 1989.
  23. H. Nguyen and J. Reynen, “A space-time finite element approach to Burgers' equation,” in Numerical Methods for Nonlinear Problems, E. Hinton, Ed., vol. 2, pp. 718–728, Pineridge Press, 1984.