Anika Tashin Khan and Hasibun Naher. Solitons and Periodic Solutions of the Fisher Equation with Nonlinear Ordinary Differential Equation as Auxiliary Equation.
. 2018; 6(6):244-252. doi: 10.12691/AJAMS-6-6-5
New extension of the generalized and improved (G’/G)-expansion method, Nonlinear evolution equation (NLEE), Fisher Equation, Wave solutions, Nonlinear partial differential equation (NLPDE)
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
[1] | Hirota, R. (1973). Exact envelope-soliton solutions of a nonlinear wave equation. Journal of Mathematical Physics, 14(7), 805-809. |
|
[2] | Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654. |
|
[3] | Abdou, M. A. (2007). The extended tanh method and its applications for solving nonlinear physical models. Applied Mathematics and Computation, 190(1), 988-996. |
|
[4] | Naher, H., Abdullah, F. A., & Akbar, M. A. (2011). The exp-function method for new exact solutions of the nonlinear partial differential equations. International Journal of Physical Sciences, 6(29), 6706-6716. |
|
[5] | He, Y., Li, S., & Long, Y. (2012). Exact solutions of the Klein-Gordon equation by modified Exp-function method. In Int. Math. Forum (Vol. 7, No. 4, pp. 175-182). |
|
[6] | Zhou, Y., Wang, M., & Wang, Y. (2003). Periodic wave solutions to a coupled KdV equations with variable coefficients. Physics Letters A, 308(1), 31-36. |
|
[7] | Ali, A. T. (2011). New generalized Jacobi elliptic function rational expansion method. Journal of computational and applied mathematics, 235(14), 4117-4127. |
|
[8] | Wang, M. (1995). Solitary wave solutions for variant Boussinesq equations. Physics letters A, 199(3-4), 169-172. |
|
[9] | Mohyud-Din, S. T., & Noor, M. A. (2006). Homotopy perturbation method for solving fourth-order boundary value problems. Mathematical Problems in Engineering, 2007. |
|
[10] | Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654. |
|
[11] | Soliman, A. A., & Abdo, H. A. (2012). New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method. arXiv preprint arXiv:1207.5127. |
|
[12] | Rogers, C., & Shadwick, W. F. (1982). Bäcklund transformations and their applications. Academic press. |
|
[13] | Daga, A., & Pradhan, V. (2014). Variational homotopy perturbation method for the nonlinear generalized regularized long wave equation. American. Journal of Applied Mathematics and Statistics, 2, 231-234. |
|
[14] | Prasad, H. S., & Reddy, Y. N. (2015). A Fifth Order Compact Difference Method for Singularly Perturbed Singular Boundary Value Problems. American Journal of Applied Mathematics and Statistics, 3(2), 49-53. |
|
[15] | Aboiyar, T., Luga, T., & Iyorter, B. V. (2015). Derivation of continuous linear multistep methods using Hermite polynomials as basis functions. American Journal of Applied Mathematics and Statistics, 3(6), 220-225. |
|
[16] | Naher, H., & Begum, F. A. (2015). Application of Linear ODE as Auxiliary Equation to the Nonlinear Evolution Equation. American Journal of Applied Mathematics and Statistics, 3(1), 23-28. |
|
[17] | Wang, M., Li, X., & Zhang, J. (2008). The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423. |
|
[18] | Zayed, E. M. E., & Gepreel, K. A. (2009). The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. Journal of Mathematical Physics, 50(1), 013502. |
|
[19] | Naher, H., Abdullah, F. A., & Akbar, M. A. (2011). The (𝐺′/𝐺)-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation. Mathematical Problems in Engineering. |
|
[20] | Feng, J., Li, W., & Wan, Q. (2011). Using (G′/G-)expansion method to seek the traveling wave solution of Kolmogorov-Petrovskii–Piskunov equation. Applied Mathematics and Computation, 217(12), 5860-5865. |
|
[21] | Naher, H., & Abdullah, F. A. (2012). The Basic (G'/G)-expansion method for the fourth order Boussinesq equation. Applied Mathematics, 3(10), 1144. |
|
[22] | Zhang, J., Jiang, F., & Zhao, X. (2010). An improved (G′/G)-expansion method for solving nonlinear evolution equations. International Journal of Computer Mathematics, 87(8), 1716-1725. |
|
[23] | Hamed, Y. S., Sayed, M., Elagan, S. K., & El-Zahar, E. R. (2011). The improved (G'/G)-expansion method for solving (3+1)-dimensional potential-YTSF equation. Journal of Modern Methods in Numerical Mathematics, 2(1-2), 32-39. |
|
[24] | Naher, H., Abdullah, F. A., & Bekir, A. (2012). Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method. AIP Advances, 2(4), 042163. |
|
[25] | Naher, H., & Abdullah, F. A. (2012). Some New Traveling Wave Solutions of the Nonlinear Reaction Diffusion Equation by Using the Improved (𝐺′/𝐺)-Expansion Method. Mathematical Problems in Engineering, 2012. |
|
[26] | Naher, H., & Abdullah, F. A. (2012). Some new solutions of the combined KdV-MKdV equation by using the improved G/G-expansion method. World Applied Sciences Journal, 16(11), 1559-1570. |
|
[27] | Naher, H., & Abdullah, F. A. (2013). The Improved (G'/G)-Expansion Method to the (3 Dimensional Kadomstev-Petviashvili Equation. American Journal of Applied Mathematics and Statistics, 1(4), 64-70. |
|
[28] | Naher, H., Abdullah, F. A., & Rashid, A. (2014). The generalized riccati equation together with the (G'/G)-Expansion method for the (3+ 1)-dimensional modified KDV-zakharov-kuznetsov equation. |
|
[29] | Naher, H., & Abdullah, F. A.. (2014). The improved (G'/G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis & Applications, 16(2), 220-235. |
|
[30] | Naher, H., Abdullah, F. A., & Rashid, A. (2014). Some New Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation via the Improved (G'/G)-Expansion Method. Journal of Computational Analysis & Applications, 17(2) 287-296. |
|
[31] | Akbar, M. A., Ali, N. H., & Zayed, E. M. E. (2012). A generalized and improved-expansion method for nonlinear evolution equations, Math. Prob. Eng., Article ID 459879, 22. |
|
[32] | Naher, H., Abdullah, F. A., & Akbar, M. A. (2013). Generalized and improved (G′/G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. PloS one, 8(5), e64618. |
|
[33] | Naher, H., & Abdullah, F. A. (2013). New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation. AIP Advances, 3(3), 032116. |
|
[34] | Naher, H., & Abdullah, F. A. (2016). Further extension of the generalized and improved (G′/G)-expansion method for nonlinear evolution equation. Journal of the Association of Arab Universities for Basic and Applied Sciences, 19(1), 52-58. |
|
[35] | Kudryashov, N. A. (2005). Exact solitary waves of the Fisher equation. Physics Letters A, 342(1-2), 99-106. |
|
[36] | Wazwaz, A. M., & Gorguis, A. (2004). An analytic study of Fisher's equation by using Adomian decomposition method. Applied Mathematics and Computation, 154(3), 609-620. |
|
[37] | Öziş, T., & Köroğlu, C. (2008). A novel approach for solving the Fisher equation using Exp-function method. Physics Letters A, 372(21), 3836-3840. |
|
[38] | Tan, Y., Xu, H., & Liao, S. J. (2007). Explicit series solution of travelling waves with a front of Fisher equation. Chaos, Solitons & Fractals, 31(2), 462-472. |
|
[39] | Ablowitz, M. J., & Zeppetella, A. (1979). Explicit solutions of Fisher's equation for a special wave speed. Bulletin of Mathematical Biology, 41(6), 835-840. |
|