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Mathematical Problems in Engineering
Volume 2009, Article ID 370765, 11 pages
http://dx.doi.org/10.1155/2009/370765
Research Article

Numerical Solutions of the Generalized Burgers-Huxley Equation by a Differential Quadrature Method

1Department of Mathematics, Faculty of Art and Science, Pamukkale University, 20070 Denizli, Turkey
2Department of Civil Engineering, Faculty of Engineering, Pamukkale University, 20070 Denizli, Turkey

Received 29 July 2008; Accepted 26 January 2009

Academic Editor: Francesco Pellicano

Copyright © 2009 Murat Sari and Gürhan Gürarslan. 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. X. Y. Wang, Z. S. Zhu, and Y. K. Lu, “Solitary wave solutions of the generalised Burgers-Huxley equation,” Journal of Physics A, vol. 23, no. 3, pp. 271–274, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. G. Estevez, “Non-classical symmetries and the singular manifold method: the Burgers and the Burgers-Huxley equations,” Journal of Physics A, vol. 27, no. 6, pp. 2113–2127, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. T. Darvishi, S. Kheybari, and F. Khani, “Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 10, pp. 2091–2103, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. Javidi, “A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 338–344, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Javidi and A. Golbabai, “A new domain decomposition algorithm for generalized Burger's-Huxley equation based on Chebyshev polynomials and preconditioning,” Chaos, Solitons & Fractals. In press. View at Publisher · View at Google Scholar
  6. I. Hashim, M. S. M. Noorani, and M. R. Said Al-Hadidi, “Solving the generalized Burgers-Huxley equation using the Adomian decomposition method,” Mathematical and Computer Modelling, vol. 43, no. 11-12, pp. 1404–1411, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. I. Hashim, M. S. M. Noorani, and B. Batiha, “A note on the Adomian decomposition method for the generalized Huxley equation,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1439–1445, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  8. H. N. A. Ismail, K. Raslan, and A. A. Abd Rabboh, “Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations,” Applied Mathematics and Computation, vol. 159, no. 1, pp. 291–301, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  9. A. Molabahramia and F. Khani, “The homotopy analysis method to solve the Burgers-Huxley equation,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 589–600, 2009. View at Publisher · View at Google Scholar
  10. A.-M. Wazwaz, “Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 754–761, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. B. Batiha, M. S. M. Noorani, and I. Hashim, “Application of variational iteration method to the generalized Burgers-Huxley equation,” Chaos, Solitons & Fractals, vol. 36, no. 3, pp. 660–663, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. B. Batiha, M. S. M. Noorani, and I. Hashim, “Numerical simulation of the generalized Huxley equation by He's variational iteration method,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1322–1325, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. O. Yu. Efimova and N. A. Kudryashov, “Exact solutions of the Burgers-Huxley equation,” Journal of Applied Mathematics and Mechanics, vol. 68, no. 3, pp. 413–420, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. Bellman, B. G. Kashef, and J. Casti, “Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations,” Journal of Computational Physics, vol. 10, pp. 40–52, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. W. Bert and M. Malik, “Differential quadrature method in computational mechanics: a review,” Applied Mechanics Review, vol. 49, no. 1, pp. 1–28, 1996. View at Google Scholar
  16. M. Sari, “Differential quadrature method for singularly perturbed two-point boundary value problems,” Journal of Applied Sciences, vol. 8, pp. 1091–1096, 2008. View at Google Scholar
  17. C. Shu, Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, Ph.D. thesis, University of Glasgow, Glasgow, UK, 1991.
  18. C. Shu and B.E. Richards, “Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations,” International Journal for Numerical Methods in Fluids, vol. 15, no. 7, pp. 791–798, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. U. Yücel, “Approximations of Sturm-Liouville eigenvalues using differential quadrature (DQ) method,” Journal of Computational and Applied Mathematics, vol. 192, no. 2, pp. 310–319, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. U. Yücel and M. Sari, “Differential quadrature method (DQM) for a class of singular two-point boundary value problems,” International Journal of Computer Mathematics, vol. 86, no. 3, pp. 465–475, 2009. View at Publisher · View at Google Scholar
  21. Z. Zong and K. Y. Lam, “A localized differential quadrature (LDQ) method and its application to the 2D wave equation,” Computational Mechanics, vol. 29, no. 4-5, pp. 382–391, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. C. Shu and Y. T. Chew, “Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems,” Communications in Numerical Methods in Engineering, vol. 13, no. 8, pp. 643–653, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. C. Shu and H. Xue, “Explicit computation of weighting coefficients in the harmonic differential quadrature,” Journal of Sound and Vibration, vol. 204, no. 3, pp. 549–555, 1997. View at Publisher · View at Google Scholar
  24. C. Shu, Differential Quadrature and Its Application in Engineering, Springer, London, UK, 2000. View at MathSciNet
  25. S. Gottlieb and C.-W. Shu, “Total variation diminishing Runge-Kutta schemes,” Mathematics of Computation, vol. 67, no. 221, pp. 73–85, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. Fitzhugh, “Mathematical models of excitation and propagation in nerve,” in Biological Engineering, H. P. Schwan, Ed., pp. 1–85, McGraw-Hill, New York, NY, USA, 1969. View at Google Scholar
  27. A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of Physiology, vol. 117, pp. 500–544, 1952. View at Google Scholar