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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 760542, 12 pages
http://dx.doi.org/10.1155/2013/760542
Research Article

A Jacobi Collocation Method for Solving Nonlinear Burgers-Type Equations

1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Eskiehir Yolu 29 Km, 06810 Ankara, Turkey
4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
5Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt

Received 24 July 2013; Accepted 15 August 2013

Academic Editor: Soheil Salahshour

Copyright © 2013 E. H. Doha 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.

Linked References

  1. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, New York, NY, USA, 2006. View at MathSciNet
  2. E. H. Doha and A. H. Bhrawy, “An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 558–571, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Nguyen and C. Delcarte, “A spectral collocation method to solve Helmholtz problems with boundary conditions involving mixed tangential and normal derivatives,” Journal of Computational Physics, vol. 200, no. 1, pp. 34–49, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. H. Bhrawy and A. S. Alofi, “A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 62–70, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. H. Bhrawy, E. Tohidi, and F. Soleymani, “A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals,” Applied Mathematics and Computation, vol. 219, no. 2, pp. 482–497, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. H. Bhrawy and M. Alshomrani, “A shifted legendre spectral method for fractional-ordermulti-point boundary value problems,” Advances in Difference Equations, vol. 2012, article 8, 2012.
  7. E. H. Doha, A. H. Bhrawy, D. Baleanu, and S. S. Ezz-Eldien, “On shifted Jacobi spectral approximations for solving fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 15, pp. 8042–8056, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. L. Zhu and Q. Fan, “Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2333–2341, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931–4943, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. Saadatmandi and M. Dehghan, “The use of sinc-collocation method for solving multi-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 593–601, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “On shifted Jacobi spectral method for high-order multi-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 10, pp. 3802–3810, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. Ierley, B. Spencer, and R. Worthing, “Spectral methods in time for a class of parabolic partial differential equations,” Journal of Computational Physics, vol. 102, no. 1, pp. 88–97, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. H. Tal-Ezer, “Spectral methods in time for hyperbolic equations,” SIAM Journal on Numerical Analysis, vol. 23, no. 1, pp. 11–26, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. Tal-Ezer, “Spectral methods in time for parabolic problems,” SIAM Journal on Numerical Analysis, vol. 26, no. 1, pp. 1–11, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. A. Coutsias, T. Hagstrom, J. S. Hesthaven, and D. Torres, “Integration preconditioners for differential operators in spectral tau-methods,” in Proceedings of the 3rd International Conference on Spectral and High Order Methods, A. V. Ilin and L. R. Scott, Eds., pp. 21–38, University of Houston, Houston, Tex, USA, 1996, Houston Journal of Mathematics.
  16. F.-y. Zhang, “Spectral and pseudospectral approximations in time for parabolic equations,” Journal of Computational Mathematics, vol. 16, no. 2, pp. 107–120, 1998. View at Zentralblatt MATH · View at MathSciNet
  17. J.-G. Tang and H.-P. Ma, “A Legendre spectral method in time for first-order hyperbolic equations,” Applied Numerical Mathematics, vol. 57, no. 1, pp. 1–11, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. H. Bhrawy, “A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients,” Applied Mathematics and Computation, vol. 222, pp. 255–264, 2013. View at Publisher · View at Google Scholar
  19. H. N. A. Ismail, K. Raslan, and A. A. A. 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
  20. H. Bateman, “Some recent researchers on the motion of fluids,” Monthly Weather Review, vol. 43, pp. 163–170, 1915. View at Publisher · View at Google Scholar
  21. 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. View at MathSciNet
  22. M. K. Kadalbajoo and A. Awasthi, “A numerical method based on Crank-Nicolson scheme for Burgers' equation,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1430–1442, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Gülsu, “A finite difference approach for solution of Burgers' equation,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1245–1255, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. P. Kim, “Invariantization of the Crank-Nicolson method for Burgers' equation,” Physica D, vol. 237, no. 2, pp. 243–254, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. H. Nguyen and J. Reynen, “A space-time least-square finite element scheme for advection-diffusion equations,” Computer Methods in Applied Mechanics and Engineering, vol. 42, no. 3, pp. 331–342, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. H. Nguyen and J. Reynen, “A space-time finite element approach to Burgers' equation,” in Numerical Methods for Non-Linear Problems, C. Taylor, E. Hinton, D. R. J. Owen, and E. Onate, Eds., vol. 2, pp. 718–728, 1984. View at Zentralblatt MATH · View at MathSciNet
  27. L. R. T. Gardner, G. A. Gardner, and A. Dogan, “A Petrov-Galerkin finite element scheme for Burgers' equation,” The Arabian Journal for Science and Engineering C, vol. 22, no. 2, pp. 99–109, 1997. View at Zentralblatt MATH · View at MathSciNet
  28. L. R. T. Gardner, G. A. Gardner, and A. Dogan, in A Least-Squares Finite Element Scheme For Burgers Equation, Mathematics, University of Wales, Bangor, UK, 1996.
  29. S. Kutluay, A. Esen, and I. Dag, “Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method,” Journal of Computational and Applied Mathematics, vol. 167, no. 1, pp. 21–33, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. R. C. Mittal and R. K. Jain, “Numerical solutions of nonlinear Burgers' equation with modified cubic B-splines collocation method,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7839–7855, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1224–1244, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. E. H. Doha and A. H. Bhrawy, “A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations,” Numerical Methods for Partial Differential Equations, vol. 25, no. 3, pp. 712–739, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. Kazem, “An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1126–1136, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  34. A. Ahmadian, M. Suleiman, S. Salahshour, and D. Baleanu, “A Jacobi operational matrix for solving a fuzzy linear fractional differential equation,” Advances in Difference Equations, vol. 2013, p. 104, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  35. E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1820–1832, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. A. Veksler and Y. Zarmi, “Wave interactions and the analysis of the perturbed Burgers equation,” Physica D, vol. 211, no. 1-2, pp. 57–73, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. A. Veksler and Y. Zarmi, “Freedom in the expansion and obstacles to integrability in multiple-soliton solutions of the perturbed KdV equation,” Physica D, vol. 217, no. 1, pp. 77–87, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. Z.-Y. Ma, X.-F. Wu, and J.-M. Zhu, “Multisoliton excitations for the Kadomtsev-Petviashvili equation and the coupled Burgers equation,” Chaos, Solitons and Fractals, vol. 31, no. 3, pp. 648–657, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. R. A. Fisher, “The wave of advance of advantageous genes,” Annals of Eugenics, vol. 7, pp. 335–369, 1937.
  40. A. J. Khattak, “A computational meshless method for the generalized Burger's-Huxley equation,” Applied Mathematical Modelling, vol. 33, no. 9, pp. 3718–3729, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology, Academic Press, London, UK, 1986. View at MathSciNet
  42. D. A. Frank, Diffusion and Heat Exchange in Chemical Kinetics, Princeton University Press, Princeton, NJ, USA, 1955.
  43. A.-M. Wazwaz, “The extended tanh method for abundant solitary wave solutions of nonlinear wave equations,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1131–1142, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. Y. Tan, H. Xu, and S.-J. Liao, “Explicit series solution of travelling waves with a front of Fisher equation,” Chaos, Solitons and Fractals, vol. 31, no. 2, pp. 462–472, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. J. Canosa, “Diffusion in nonlinear multiplicate media,” Journal of Mathematical Physics, vol. 31, pp. 1862–1869, 1969.
  47. A.-M. Wazwaz, “Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 639–656, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. E. Fan and Y. C. Hon, “Generalized tanh method extended to special types of nonlinear equations,” Zeitschrift fur Naturforschung A, vol. 57, no. 8, pp. 692–700, 2002. View at Scopus