Table of Contents
International Journal of Engineering Mathematics
Volume 2015, Article ID 107978, 35 pages
http://dx.doi.org/10.1155/2015/107978
Research Article

New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

Department of Applied Mathematics, Faculty of Mathematics Science, University of Tabriz, Tabriz 51569 84687, Iran

Received 30 May 2015; Revised 10 September 2015; Accepted 10 September 2015

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2015 Jalil Manafian and Mehrdad Lakestani. 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. P. Gray and S. K. Scott, Chemical Oscillations and Instabilities—Non-Linear Chemical Kinetics, Oxford Science Publications, Oxford, UK, 1990.
  2. M. M. Kabir and A. Khajeh, “New explicit solutions for the Vakhnenko and a generalized form of the nonlinear heat conduction equations via exp-function method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 10, pp. 1307–1318, 2009. View at Google Scholar · View at Scopus
  3. M. Dehghan, J. Manafian, and A. Saadatmandi, “Solving nonlinear fractional partial differential equations using the homotopy analysis method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 448–479, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. M. G. Sakar and F. Erdogan, “The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian's decomposition method,” Applied Mathematical Modelling, vol. 37, no. 20-21, pp. 8876–8885, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. M. G. Sakar and H. Ergören, “Alternative variational iteration method for solving the time-fractional Fornberg-Whitham equation,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 14, pp. 3972–3979, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. H. Jafari, A. Kadem, and D. Baleanu, “Variational iteration method for a fractional-order Brusselator system,” Abstract and Applied Analysis, vol. 2014, Article ID 496323, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. M. Matinfar, S. Mirzanezhad, M. Ghasemi, and M. Salehi, “Solving the interaction of electromagnetic wave with electron by VIM,” Journal of King Saud University—Science, vol. 27, no. 1, pp. 63–70, 2015. View at Publisher · View at Google Scholar · View at Scopus
  8. M. Dehghan and J. Manafian, “The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method,” Zeitschrift für Naturforschung A, vol. 64, no. 7-8, pp. 420–430, 2009. View at Google Scholar · View at Scopus
  9. E. Cuce and P. M. Cuce, “A successful application of homotopy perturbation method for efficiency and effectiveness assessment of longitudinal porous fins,” Energy Conversion and Management, vol. 93, pp. 92–99, 2015. View at Publisher · View at Google Scholar · View at Scopus
  10. A.-M. Wazwaz, “Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 755–760, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. J. Manafian Heris and M. Lakestani, “Solitary wave and periodic wave solutions for variants of the KdV-Burger and the K(n, n)-Burger equations by the generalized tanh-coth method,” Communications in Numerical Analysis, vol. 2013, 18 pages, 2013. View at Publisher · View at Google Scholar
  12. J. Manafian Heris and M. Lakestani, “Exact solutions for the integrable sixth-order drinfeld-sokolov-satsuma-hirota system by the analytical methods,” International Scholarly Research Notices, vol. 2014, Article ID 840689, 8 pages, 2014. View at Publisher · View at Google Scholar
  13. J. Manafian and I. Zamanpour, “Exact travelling wave solutions of the symmetric regularized long wave (SRLW) using analytical methods,” Statistics, Optimization & Information Computing, vol. 2, no. 1, pp. 47–55, 2014. View at Google Scholar · View at MathSciNet
  14. M. A. Abdou and A. A. Soliman, “Modified extended tanh-function method and its application on nonlinear physical equations,” Physics Letters A, vol. 353, no. 6, pp. 487–492, 2006. View at Publisher · View at Google Scholar · View at Scopus
  15. S. A. El-Wakil and M. A. Abdou, “New exact travelling wave solutions using modified extended tanh-function method,” Chaos, Solitons & Fractals, vol. 31, no. 4, pp. 840–852, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. Manafian Heris and M. Bagheri, “Exact solutions for the modified KDV and the generalized KDV equations via Exp-function method,” Journal of Mathematical Extension, vol. 4, no. 2, pp. 77–98, 2010. View at Google Scholar · View at MathSciNet
  17. J. Manafian Heris and I. Zamanpour, “Analytical treatment of the coupled Higgs equation and the Maccari system via Exp-function method,” Acta Universitatis Apulensis, vol. 33, pp. 203–216, 2013. View at Google Scholar · View at MathSciNet
  18. M. G. Hafez, M. N. Alam, and M. A. Akbar, “Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system,” Journal of King Saud University—Science, vol. 27, no. 2, pp. 105–112, 2015. View at Publisher · View at Google Scholar
  19. H.-O. Roshid and M. Azizur Rahman, “The exp(-ϕη)-expansion method with application in the (1+1)-dimensional classical Boussinesq equations,” Results in Physics, vol. 4, pp. 150–155, 2014. View at Publisher · View at Google Scholar · View at Scopus
  20. M. Fazli Aghdaei and J. Manafianheris, “Exact solutions of the couple Boiti-Leon-Pempinelli system by the generalized (G'/G)-expansion method,” Journal of Mathematical Extension, vol. 5, no. 2, pp. 91–104, 2011. View at Google Scholar
  21. H. Naher and F. A. Abdullah, “New approach of (G'/G)-expansion method and new approach of generalized (G'/G)-expansion method for nonlinear evolution equation,” AIP Advances, vol. 3, Article ID 032116, 2013. View at Publisher · View at Google Scholar
  22. A. J. M. Jawad, M. D. Petkovic, and A. Biswas, “Modified simple equation method for nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 869–877, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. M. N. Alam, M. A. Akbar, and S. T. Mohyud-Din, “A novel (G′/G)-expansion method and its application to the Boussinesq equation,” Chinese Physics B, vol. 23, no. 2, Article ID 020203, 2014. View at Publisher · View at Google Scholar · View at Scopus
  24. Y. Chen and Q. Wang, “Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation,” Chaos, Solitons and Fractals, vol. 24, no. 3, pp. 745–757, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. X. Zhao, L. Wang, and W. Sun, “The repeated homogeneous balance method and its applications to nonlinear partial differential equations,” Chaos, Solitons & Fractals, vol. 28, no. 2, pp. 448–453, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. A. Hajipour and S. M. Mahmoudi, “Application of exp-function method to Fitzhugh-Nagumo equation,” World Applied Sciences Journal, vol. 9, no. 1, pp. 113–117, 2010. View at Google Scholar
  27. J. A. Whitehead and A. C. Newell, “Finite bandwidth, finite amplitude convection,” Journal of Fluid Mechanics, vol. 38, no. 2, pp. 279–303, 1969. View at Publisher · View at Google Scholar · View at Scopus
  28. 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 · View at MathSciNet · View at Scopus
  29. M. Dehghan, J. M. Heris, and A. Saadatmandi, “Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses,” Mathematical Methods in the Applied Sciences, vol. 33, no. 11, pp. 1384–1398, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical Journal, vol. 1, no. 6, pp. 445–466, 1961. View at Publisher · View at Google Scholar
  31. J. Nagumo, S. Arimoto, and S. Yoshizawa, “An active pulse transmission line simulating nerve axon,” Proceedings of the IRE, vol. 50, no. 10, pp. 2061–2070, 1962. View at Publisher · View at Google Scholar
  32. G. Hariharan and K. Kannan, “Haar wavelet method for solving FitzHugh-Nagumo equation,” World Academy of Science, Engineering and Technology, vol. 67, pp. 560–564, 2010. View at Google Scholar
  33. T. Kawahara and M. Tanaka, “Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation,” Physics Letters A, vol. 97, no. 8, pp. 311–314, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. M. C. Nucci and P. A. Clarkson, “The nonclassical method is more general than the direct method for symmetry reductions: an example of the FitzHugh-Nagumo equation,” Physics Letters A, vol. 164, no. 1, pp. 49–56, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. H. Li and Y. Guo, “New exact solutions to the FitzHugh-Nagumo equation,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 524–528, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. H. Triki and A.-M. Wazwaz, “On soliton solutions for the Fitzhugh-Nagumo equation with time-dependent coefficients,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 3821–3828, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. R. Jiwari, R. K. Gupta, and V. Kumar, “Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh-Nagumo equation with time-dependent coefficients,” Ain Shams Engineering Journal, vol. 5, no. 4, pp. 1343–1350, 2014. View at Publisher · View at Google Scholar · View at Scopus
  38. Q. Zheng and J. Shen, “Pattern formation in the FitzHugh—Nagumo model,” Computers & Mathematics with Applications, vol. 70, no. 5, pp. 1082–1097, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  39. R. A. Van Gorder, “Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function H(x, t) in the homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1233–1240, 2012. View at Publisher · View at Google Scholar · View at Scopus
  40. F. Liu, P. Zhuang, I. Turner, V. Anh, and K. Burrage, “A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain,” Journal of Computational Physics, vol. 293, pp. 252–263, 2015. View at Publisher · View at Google Scholar
  41. A. H. Abbasian, H. Fallah, and M. R. Razvan, “Symmetric bursting behaviors in the generalized FitzHugh-Nagumo model,” Biological Cybernetics, vol. 107, no. 4, pp. 465–476, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  42. S. S. Ray and S. Sahooz, “New exact solutions of fractional zakharov-kuznetsov and modified zakharov-kuznetsov equations using fractional sub-equation method,” Communications in Theoretical Physics, vol. 63, no. 1, pp. 25–30, 2015. View at Publisher · View at Google Scholar
  43. S. Sahoo and S. S. Ray, “Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV-Zakharov-Kuznetsov equations,” Computers & Mathematics with Applications, vol. 70, no. 2, pp. 158–166, 2015. View at Publisher · View at Google Scholar · View at MathSciNet