Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 871724, 17 pages
http://dx.doi.org/10.1155/2012/871724
Research Article

Some New Traveling Wave Solutions of the Nonlinear Reaction Diffusion Equation by Using the Improved (𝐺/𝐺)-Expansion Method

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Received 5 December 2011; Revised 23 February 2012; Accepted 24 February 2012

Academic Editor: Jun-Juh Yan

Copyright © 2012 Hasibun Naher and Farah Aini Abdullah. 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. K. W. Chow, “A class of exact, periodic solutions of nonlinear envelope equations,” Journal of Mathematical Physics, vol. 36, no. 8, pp. 4125–4137, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. R. Hirota, “Exact solution of the korteweg-de vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971. View at Publisher · View at Google Scholar · View at Scopus
  4. M. Wang and X. Li, “Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1257–1268, 2005. View at Publisher · View at Google Scholar
  5. C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, vol. 161 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982.
  6. L. Jianming, D. Jie, and Y. Wenjun, “Bäcklund transformation and new exact solutions of the Sharma-Tasso-Olver equation,” Abstract and Applied Analysis, vol. 2011, Article ID 935710, 8 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. Z. Yan and H. Zhang, “New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water,” Physics Letters A, vol. 285, no. 5-6, pp. 355–362, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. C. A. Gómez, A. H. Salas, and B. Acevedo Frias, “Exact solutions to KdV6 equation by using a new approach of the projective Riccati equation method,” Mathematical Problems in Engineering, vol. 2010, Article ID 797084, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. X. Li and M. Wang, “A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms,” Physics Letters A, vol. 361, no. 1-2, pp. 115–118, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1-5, pp. 67–75, 1996. View at Google Scholar · View at Scopus
  11. E. M. E. Zayed, H. A. Zedan, and K. A. Gepreel, “On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV of equations,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 285–303, 2004. View at Publisher · View at Google Scholar
  12. A.-M. Wazwaz, “The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1467–1475, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. E. Yusufoğlu and A. Bekir, “Exact solutions of coupled nonlinear Klein-Gordon equations,” Mathematical and Computer Modelling, vol. 48, no. 11-12, pp. 1694–1700, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. A. Bekir and A. C. Cevikel, “Solitary wave solutions of two nonlinear physical models by tanh-coth method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1804–1809, 2009. View at Publisher · View at Google Scholar · View at Scopus
  15. F. Taşcan and A. Bekir, “Analytic solutions of the (2+1)-dimensional nonlinear evolution equations using the sine-cosine method,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 3134–3139, 2009. View at Publisher · View at Google Scholar
  16. F. Taşcan and A. Bekir, “Applications of the first integral method to nonlinear evolution equations,” Chinese Physics B, vol. 19, no. 8, Article ID 080201, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. A. H. Salas and C. A. Gómez S., “Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 194329, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Exp-function method for traveling wave solutions of modified zakharov-kuznetsov equation,” Journal of King Saud University, vol. 22, no. 4, pp. 213–216, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. A. Bekir, “The exp-function method for ostrovsky equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 6, pp. 735–739, 2009. View at Google Scholar · View at Scopus
  21. A. Yıldırım and Z. Pınar, “Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1873–1880, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. H. Naher, F. A. Abdullah, and M. A. Akbar, “New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method,” Journal of Applied Mathematics, vol. 2012, Article ID 575387, 14 pages, 2012. View at Publisher · View at Google Scholar
  23. H. Naher, F. A. Abdullah, and M. A. Akbar, “The exp-function method for new exact solutions of the nonlinear partial differential equations,” International Journal of the Physical Sciences, vol. 6, no. 29, pp. 6706–6716, 2011. View at Google Scholar
  24. W. Zhang, “The extended tanh method and the exp-function method to solve a kind of nonlinear heat equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 935873, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. İ. Aslan, “Application of the exp-function method to nonlinear lattice differential equations for multi-wave and rational solutions,” Mathematical Methods in the Applied Sciences, vol. 34, no. 14, pp. 1707–1710, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. A. M. Wazwaz, “A new (2+1)-dimensional Korteweg-de-Vries equation and its extension to a new (3+1)-dimensional Kadomtsev-Petviashvili equation,” Physica Scripta, vol. 84, no. 3, Article ID 035010, 2011. View at Publisher · View at Google Scholar
  27. X. Liu, L. Tian, and Y. Wu, “Exact solutions of the generalized Benjamin-Bona-Mahony equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 796398, 5 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. A. Biswas, C. Zony, and E. Zerrad, “Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 153–156, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. R. Sassaman and A. Biswas, “Soliton perturbation theory for phi-four model and nonlinear Klein-Gordon equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3239–3249, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. R. Sassaman and A. Biswas, “Topological and non-topological solitons of the generalized Klein-Gordon equations,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 212–220, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. R. Sassaman, A. Heidari, and A. Biswas, “Topological and non-topological solitons of nonlinear Klein-Gordon equations by He's semi-inverse variational principle,” Journal of the Franklin Institute, vol. 347, no. 7, pp. 1148–1157, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. R. Sassaman, A. Heidari, F. Majid, E. Zerrad, and A. Biswas, “Topological and non-topological solitons of the generalized Klein-Gordon equations in 1+2 dimensions,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 17, no. 2, pp. 275–286, 2010. View at Google Scholar · View at Zentralblatt MATH
  33. R. Sassaman and A. Biswas, “Topological and non-topological solitons of the Klein-Gordon equations in 1+2 dimensions,” Nonlinear Dynamics, vol. 61, no. 1-2, pp. 23–28, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. R. Sassaman, M. Edwards, F. Majid, and A. Biswas, “1-Soliton solution of the coupled nonlinear Klien-Gordon equations,” Studies in Mathematical Sciences, vol. 1, no. 1, pp. 30–37, 2010. View at Google Scholar
  35. R. Sassaman and A. Biswas, “Soliton solution of the generalized Klien-Gordon equation by semi-inverse variational principle,” Mathematics in Engineering, Science and Aerospace, vol. 2, no. 1, pp. 99–104, 2011. View at Google Scholar
  36. R. Sassaman and A. Biswas, “1-Soliton solution of the Perturbed Klien-Gordon equation,” Physics Express, vol. 1, no. 1, pp. 9–14, 2011. View at Google Scholar
  37. A. Kiliçman and H. Eltayeb, “On a new integral transform and differential equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 463579, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. M. A. Noor, K. I. Noor, E. Al-Said, and M. Waseem, “Some new iterative methods for nonlinear equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 198943, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interaction,” Mathematical Problems in Engineering, vol. 2010, Article ID 695208, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. M. A. Abdou, “The extended F-expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 95–104, 2007. View at Publisher · View at Google Scholar
  41. A. S. Deakin and M. Davison, “An analytic solution for a Vasicek interest rate convertible bond model,” Journal of Applied Mathematics, vol. 2010, Article ID 263451, 5 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. P. Sunthrayuth and P. Kumam, “A new general iterative method for solution of a new general system of variational inclusions for nonexpansive semigroups in Banach spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 187052, 29 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. B. I. Yun, “An iteration method generating analytical solutions for Blasius problem,” Journal of Applied Mathematics, vol. 2011, Article ID 925649, 8 pages, 2011. View at Publisher · View at Google Scholar
  44. M. Wang, X. Li, and J. Zhang, “The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters. A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher · View at Google Scholar
  45. A. Malik, F. Chand, and S. C. Mishra, “Exact travelling wave solutions of some nonlinear equations by (G′/G)-expansion method,” Applied Mathematics and Computation, vol. 216, no. 9, pp. 2596–2612, 2010. View at Publisher · View at Google Scholar
  46. A. Bekir, “Application of the (G′/G)-expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008. View at Publisher · View at Google Scholar
  47. E. M. E. Zayed, “New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized (G′/G)-expansion method,” Journal of Applied Mathematics & Informatics, vol. 28, no. 1-2, pp. 383–395, 2010. View at Google Scholar
  48. H. Naher, F. A. Abdullah, and M. A. Akbar, “The (G′/G)-expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation,” Mathematical Problems in Engineering, vol. 2011, Article ID 218216, 11 pages, 2011. View at Publisher · View at Google Scholar
  49. M. Hayek, “Constructing of exact solutions to the KdV and Burgers equations with power-law nonlinearity by the extended (G′/G)-expansion method,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 212–221, 2010. View at Publisher · View at Google Scholar
  50. S. Guo and Y. Zhou, “The extended (G′/G)-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3214–3221, 2010. View at Publisher · View at Google Scholar
  51. E. M. E. Zayed and S. Al-Joudi, “Applications of an extended (G′/G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics,” Mathematical Problems in Engineering, vol. 2010, Article ID 768573, 19 pages, 2010. View at Publisher · View at Google Scholar
  52. J. Zhang, F. Jiang, and X. Zhao, “An improved (G′/G)-expansion method for solving nonlinear evolution equations,” International Journal of Computer Mathematics, vol. 87, no. 8, pp. 1716–1725, 2010. View at Publisher · View at Google Scholar
  53. Y. S. Hamad, M. Sayed, S. K. Elagan, and E. R. El-Zahar, “The improved (G′/G)-expansion method for solving (3+1)-dimensional potential-YTSF equation,” Journal of Modern Methods in Numerical Mathematics, vol. 2, no. 1-2, pp. 32–38, 2011. View at Google Scholar
  54. T. A. Nofel, M. Sayed, Y. S. Hamad, and S. K. Elagan, “The improved (G′/G)-expansion method for solving the fifth-order KdV equation,” Annals of Fuzzy Mathematics and Informatics, vol. 3, no. 1, pp. 9–17, 2012. View at Google Scholar
  55. Y. M. Zhao, Y. J. Yang, and W. Li, “Application of the improved (G′/G)-expansion method for the variant Boussinesq equations,” Applied Mathematical Sciences, vol. 5, no. 58, pp. 2855–2861, 2011. View at Google Scholar
  56. S. Tao and T. Xia, “An improved (G′/G)- expansion method and its application to the (3+1)-dimensional kdv equation,” in International Conference on Information Science and Technology (ICIST '11), pp. 280–286, March 2011. View at Publisher · View at Google Scholar · View at Scopus
  57. E. M. E. Zayed and K. A. Gepreel, “The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics,” Journal of Mathematical Physics, vol. 50, no. 1, Article ID 013502, 12 pages, 2009. View at Publisher · View at Google Scholar