About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 741278, 12 pages
http://dx.doi.org/10.1155/2013/741278
Research Article

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni Suef 62511, Egypt
3Department of Mathematics, Faculty of Science, Um-Al-Qurah University, Makkah 21955, Saudi Arabia

Received 20 January 2013; Revised 6 June 2013; Accepted 6 June 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 A. H. Bhrawy 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, Scientific Computation, Springer, Berlin, Germany, 2006. View at Zentralblatt MATH · View at MathSciNet
  2. C. I. Gheorghiu, Spectral Methods for Differential Problems, T. Popoviciu Institute of Numerical Analysis, Cluj-Napoca, Romaina, 2007.
  3. 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
  4. E. H. Doha, W. M. Abd-Elhameed, and A. H. Bhrawy, “New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials,” Collectanea Mathematica, 2013. View at Publisher · View at Google Scholar
  5. K. Zhang, J. Li, and H. Song, “Collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays,” Applied Mathematics and Computation, vol. 218, no. 22, pp. 10848–10860, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. S. K. Vanani and F. Soleymani, “Tau approximate solution of weakly singular Volterra integral equations,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 494–502, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. 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 MathSciNet
  8. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2364–2373, 2011. 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. D. Rostamy, K. Karimi, L. Gharacheh, and M. Khaksarfard, “Spectral method for fractional quadratic Riccati differential equation,” Journal of Applied Mathematics and Bioinformatics, vol. 2, pp. 85–97, 2012.
  11. 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
  12. A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals,” Boundary Value Problems, vol. 2012, article 62, 2012. View at Publisher · View at Google Scholar
  13. K. Maleknejad and M. Attary, “A Chebysheve collocation method for the solution of higher-order Fredholm-Volterra integro-differential equations system,” Scientific Bulletin, Universitatea Politehnica din Bucuresti Series A, vol. 74, no. 4, pp. 17–28, 2012. View at MathSciNet
  14. M. Maleki, I. Hashim, M. T. Kajani, and S. Abbasbandy, “An adaptive pseudospectral method for fractional order boundary value problems,” Abstract and Applied Analysis, vol. 2012, Article ID 381708, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. G. Szegő, Orthogonal Polynomials, Colloquium Publications. XXIII. American Mathematical Society, 1939. View at MathSciNet
  16. E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 947230, 21 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. A. H. Bhrawy, E. H. Doha, and R. M. Hafez, “A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 947230, 16 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. H. Kim and R. Sakthivel, “Travelling wave solutions for time-delayed nonlinear evolution equations,” Applied Mathematics Letters, vol. 23, no. 5, pp. 527–532, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Dehghan and R. Salehi, “Solution of a nonlinear time-delay model in biology via semi-analytical approaches,” Computer Physics Communications, vol. 181, no. 7, pp. 1255–1265, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Ghasemi and M. T. Kajani, “Numerical solution of time-varying delay systems by Chebyshev wavelets,” Applied Mathematical Modelling, vol. 35, no. 11, pp. 5235–5244, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. X. T. Wang, “Numerical solution of delay systems containing inverse time by hybrid functions,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 535–546, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. X. T. Wang, “Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 849–856, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. X. T. Wang, “Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions,” Journal of the Franklin Institute, vol. 344, no. 7, pp. 941–953, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. S. Sedaghat, Y. Ordokhani, and M. Dehghan, “Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4815–4830, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. E. Tohidi, A. H. Bhrawy, and Kh. Erfani, “A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation,” Applied Mathematical Modelling, vol. 37, pp. 4283–4294, 2013.
  26. I. Ali, H. Brunner, and T. Tang, “Spectral methods for pantograph-type differential and integral equations with multiple delays,” Frontiers of Mathematics in China, vol. 4, no. 1, pp. 49–61, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  27. D. Trif, “Direct operatorial tau method for pantograph-type equations,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 2194–2203, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  28. R. Sakthivel, C. Chun, and J. Lee, “New travelling wave solutions of burgers equation with finite transport memory,” Zeitschrift fur Naturforschung A, vol. 65, no. 8, pp. 633–640, 2010. View at Scopus
  29. R. Sakthivel, “Robust stabilization the Korteweg-de Vries-Burgers equation by boundary control,” Nonlinear Dynamics, vol. 58, no. 4, pp. 739–744, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. K. Pandey, L. Verma, and A. K. Verma, “Du Fort-Frankel finite difference scheme for Burgers equation,” Arabian Journal of Mathematics, vol. 2, no. 1, pp. 91–101, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  31. Z.-Z. Sun and X.-N. Wu, “A difference scheme for Burgers equation in an unbounded domain,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 285–304, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. R. C. Mittal and R. Jiwari, “A differential quadrature method for numerical solutions of Burgers-type equations,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22, no. 6-7, pp. 880–895, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  33. D. Rostamy and K. Karimi, “Hypercomplex mathematics and HPM for the time-delayed Burgers equation with convergence analysis,” Numerical Algorithms, vol. 58, no. 1, pp. 85–101, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. A. G. Kudryavtsev and O. A. Sapozhnikov, “Determination of the exact solutions to the inhomogeneous burgers equation with the use of the darboux transformation,” Acoustical Physics, vol. 57, no. 3, pp. 311–319, 2011. View at Publisher · View at Google Scholar · View at Scopus
  35. H. Caglar and M. F. Ucar, “Non-polynomial spline method for the solution of non-linear Burgers equation,” Chaos and Complex Systems, pp. 213–218, 2013.
  36. M. Wang and F. Zhao, “Haar Wavelet method for solving two-dimensional Burgers' equation,” in Proceedings of the 2nd International Congress on Computer Applications and Computational Science, Advances in Intelligent and Soft Computing, vol. 145, pp. 381–387, 2012.
  37. J. Zhang, P. Wei, and M. Wang, “The investigation into the exact solutions of the generalized time-delayed Burgers-Fisher equation with positive fractional power terms,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 36, no. 5, pp. 2192–2196, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. S. Rendine, A. Piazza, and L. L. Cavalli-Sforza, “Simulation and separation by principle components of multiple demic expansions in Europe,” The American Naturalist, vol. 128, pp. 681–706, 1986.
  39. E. S. Fahmy, H. A. Abdusalam, and K. R. Raslan, “On the solutions of the time-delayed Burgers equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 12, pp. 4775–4786, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. A. J. M. Jawad, M. D. Petković, and A. Biswas, “Soliton solutions of Burgers equations and perturbed Burgers equation,” Applied Mathematics and Computation, vol. 216, no. 11, pp. 3370–3377, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. H. Kim and R. Sakthivel, “Travelling wave solutions for time-delayed nonlinear evolution equations,” Applied Mathematics Letters, vol. 23, no. 5, pp. 527–532, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. 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
  43. E. H. Doha, W. M. Abd-Elhameed, and M. A. Bassuony, “New algorithms for solving high evenorder differential equations using third and fourth Chebyshev-Galerkin methods,” Journal of Computational Physics, vol. 236, pp. 563–579, 2013.
  44. 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
  45. 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
  46. R. Naz, I. Naeem, and F. M. Mahomed, “First integrals for two linearly coupled nonlinear Duffing oscillators,” Mathematical Problems in Engineering, vol. 2011, Article ID 831647, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. M. Asadzadeh, D. Rostamy, and F. Zabihi, “Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrodinger equations,” Numerical Methods for Partial Differential Equations, 2013. View at Publisher · View at Google Scholar
  48. R. A. Van Gorder and K. Vajravelu, “A general class of coupled nonlinear differential equations arising in self-similar solutions of convective heat transfer problems,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 460–465, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. M. Huang and Z. Zhou, “Standing wave solutions for the discrete coupled nonlinear Schrodinger equations with unbounded potentials,” Abstract and Applied Analysis, vol. 2013, Article ID 842594, 6 pages, 2013. View at Publisher · View at Google Scholar