Exact solutions of nonlinear evolution equations (NLEEs) are very crucial to realize the obscurity of many physical phenomena in mathematical science. The modified simple equation (MSE) method is especially effective and highly proficient mathematical instrument to obtaining exact traveling wave solutions to NLEEs arising in science, engineering and mathematical physics. Though the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations for balance number less or equal 2 but the method did not applied to solve if the balance number is greater than 2. In this article, the MSE method is executed to construct exact traveling wave solutions of nonlinear evolution equations namely Kuramoto-Sivashinsky equation with balance number equal to 3. The obtained solutions are expressed in terms of exponential and trigonometric functions including solitary and periodic solutions. Moreover the procedure of this method reduces huge volume of calculations.
| Published in | International Journal of Applied Mathematics and Theoretical Physics (Volume 3, Issue 2) |
| DOI | 10.11648/j.ijamtp.20170302.11 |
| Page(s) | 20-25 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Modified Simple Equation (MSE) Method, Kuramoto-Sivashinsky Equation, Nonlinear Evolution Equations (NLEEs), Exact Traveling Wave Solutions
| [1] | L. Yang, J. Liu, K. Yang, Exact solutions of nonlinear PDE nonlinear transformation and reduction of nonlinear PDE to a quadrature, Phys. Lett. A, 278 (2001) 267-270. |
| [2] | A. C. Cevikel, A. Beker, M. Akar, S. San, A procedure to construct exact solution of nonlinear evolution equations, Pramana J. Phys., 79(3) (2012) 337-344. |
| [3] | E. M. E. Zayed, H. A. Zedan, K. A. Gepreel, On the solitary wave solutions for nonlinear Hirota-Sasuma coupled KDV equations, Chaos, Solitons Fract., 22 (2004) 285-303R. Hirota, Exact envelope soliton solutions of a nonlinear wave equation, J. Math. Phys., 14(1973) 805-810. |
| [4] | M. Wang, Solitary wave solutions for variant Boussinesq equations, Phy. Lett. A, 199 (1995) 169-172. |
| [5] | W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele, A. Meerpoel, Exact solitary wave solutions of nonlinear evolution and wave equations using a Direct algebraic method, J. Phys. A: Math. General 19 (5) (1986) 607–628. |
| [6] | X. Feng, Exploratory approach to explicit solution of nonlinear evolutions equations, Int. J. Theor. Phys. 39 (2000) 207–222. Liu S., Fu Z., Liu S. D. and Zhao Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear equations. Phys. Lett. A. 289 (2001) 69-74. |
| [7] | Liu S., Fu Z., Liu S. D. and Zhao Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear equations. Phys. Lett. A. 289 (2001) 69-74. |
| [8] | M. L. Wang, Y. B. Zhou, The periodic wave equations for the Klein Gordon Schordinger equations, Phys. Lett. A, 318 (2003) 84-92. |
| [9] | H. A. Nassar, M. A. Abdel-Razek, A. K. Seddeek, Expanding the tanh-function method for solving nonlinear equations, Appl. Math., 2 (2011) 1096-1104. |
| [10] | M. J. Ablowitz, P. A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, 1991. |
| [11] | A. Bekir, A. Boz, Exact solutions for nonlinear evolution equations using Exp-function method, Phy. Lett. A, 372 (2008) 1619-1625. |
| [12] | M. A. Akbar, N. H. M. Ali, Exp-function method for Duffing equation and new solutions of (2+1) dimensional dispersive long wave equations, Prog. Appl. Math., 1 (2) (2011). |
| [13] | H. Naher, A. F. Abdullah, M. A. Akbar, The Exp-function method for new exact solutions of the nonlinear partial differential equations, Int. J. Phys. Sci., 6 (29) (2011) 6706-6716. |
| [14] | A. M. Wazwaz, A Applied sine-cosine method for handle nonlinear wave equations, Math. Comput. Modeling, 40 (2004) 499-508. |
| [15] | N. Taghizadeh and M. Mirzazadeh, The first integral method to some complex nonlinear partial differential equations, J.Comput. Appl. Math., 235 (2011) 4871-4877. |
| [16] | Y. Khan, H. Va´ zquez-Leal, N. Faraz, An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations, Appl. Math. Model. 37 (2013) 2702–2708. |
| [17] | J. Weiss, M. Tabor and G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys., 24 (1982) 522-526. |
| [18] | G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Boston, MA, 1994, Periodic. |
| [19] | 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, 285 (5-6) (2001) 355-362. |
| [20] | A. L. Guo and J. Lin, “Exact solutions of (2+1)-dimensional HNLS equation”, Commun. Theor. Phys., 54 (2010), pp. 401-406. |
| [21] | Y. He, S. Li, Y. Long, Exact solutions of the Klein–Gordon equation by modified Exp-function method, Int. Math. Forum, 7(4) (2012) 175-182. |
| [22] | Biswas A, Zony C, Zerrad E. Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation. Appl Math Comput 2008; 203:153–6. |
| [23] | Khan K, Akbar MA. 2014 Exact traveling wave solutions of Kadomtsev–Petviashvili equation. J. Egypt. Math. Soc. 23, 278–281. (doi:10.1016/j.joems.2014.03.010). |
| [24] | K. Khan and M. A. Akbar, Application of exp(−Φ(η))-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation, World Appl. Sci. J., 24(10) 2013 1373-1377. |
| [25] | R. Islam, M. N. Alam, A. K. M. K. Sazzad Hossain, H. O. Roshid, M. A. Akbar, Traveling wave solutions of nonlinear evolution equations via Exp(−Φ(η))-expansion method, Global J. Sci. Frontier Res., 13(11) (2013) 63-71. |
| [26] | M. N. Alam, M. A. Akbar and H. O. Roshid, Study of nonlinear evolution equations to construct traveling wave solutions via the new approach of generalized (G′/G) -expansion method, Math. Stat., 1(3) (2013) 102-112. |
| [27] | M. A. Akbar, N. H. M. Ali, Exact Solutions to Some Nonlinear Partial Differential Equations in Mathematical Physics Via the (G´/G) -Expansion Method, Research J. of App. Sciences, Engineering and Technology 6(19): 3527-3535, 2013. |
| [28] | M. A. Akbar, N. H. M. Ali, E. M. E. Zayed, Abundant exact traveling wave solutions of the generalized Bretherton equation via (G´/G) -expansion method, Commun. Theor. Phys., 57 (2012) 173-178. |
| [29] | He JH. An elementary introduction to recently developed asymptotic methods and nano- mechanics in textile engineering. Int. J. Mod. Phys. B2008; 22 (21) 3487–578. |
| [30] | Zhang, J., F. Jiang and X. Zhao, An improved (G´/G)-expansion method for solving nonlinear evolution equations, Int. J. Com. Math., 87 (8) (2010) 1716-1725. |
| [31] | J. Akter, M. A. Akbar, Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method. Results in physics 5 (2015) 125-130. |
| [32] | Md. Ashrafuzzaman Khan, M. Ali Akbar, Fethi Bin Muhammad Belgacem. Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method. Mathematics Letters. 2 (1) (2016) 1-18. |
| [33] | A. J. M. Jawad, M. D. Petkovic and A. Biswas, “Modified simple equation method for nonlinear evolution equations”, Appl. Math. Comput., 217 (2010) 869-877. |
| [34] | K. Khan, M. A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method, Ain Shams Eng. J., 4 (4) (2013) 903-909. |
APA Style
A. K. M. Kazi Sazzad Hossain, M. Ali Akbar. (2017). Traveling Wave Solutions of Nonlinear Evolution Equations Via the Modified Simple Equation Method. International Journal of Applied Mathematics and Theoretical Physics, 3(2), 20-25. https://doi.org/10.11648/j.ijamtp.20170302.11
ACS Style
A. K. M. Kazi Sazzad Hossain; M. Ali Akbar. Traveling Wave Solutions of Nonlinear Evolution Equations Via the Modified Simple Equation Method. Int. J. Appl. Math. Theor. Phys. 2017, 3(2), 20-25. doi: 10.11648/j.ijamtp.20170302.11
AMA Style
A. K. M. Kazi Sazzad Hossain, M. Ali Akbar. Traveling Wave Solutions of Nonlinear Evolution Equations Via the Modified Simple Equation Method. Int J Appl Math Theor Phys. 2017;3(2):20-25. doi: 10.11648/j.ijamtp.20170302.11
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Download@article{10.11648/j.ijamtp.20170302.11,
author = {A. K. M. Kazi Sazzad Hossain and M. Ali Akbar},
title = {Traveling Wave Solutions of Nonlinear Evolution Equations Via the Modified Simple Equation Method},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {3},
number = {2},
pages = {20-25},
doi = {10.11648/j.ijamtp.20170302.11},
url = {https://doi.org/10.11648/j.ijamtp.20170302.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20170302.11},
abstract = {Exact solutions of nonlinear evolution equations (NLEEs) are very crucial to realize the obscurity of many physical phenomena in mathematical science. The modified simple equation (MSE) method is especially effective and highly proficient mathematical instrument to obtaining exact traveling wave solutions to NLEEs arising in science, engineering and mathematical physics. Though the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations for balance number less or equal 2 but the method did not applied to solve if the balance number is greater than 2. In this article, the MSE method is executed to construct exact traveling wave solutions of nonlinear evolution equations namely Kuramoto-Sivashinsky equation with balance number equal to 3. The obtained solutions are expressed in terms of exponential and trigonometric functions including solitary and periodic solutions. Moreover the procedure of this method reduces huge volume of calculations.},
year = {2017}
}
TY - JOUR T1 - Traveling Wave Solutions of Nonlinear Evolution Equations Via the Modified Simple Equation Method AU - A. K. M. Kazi Sazzad Hossain AU - M. Ali Akbar Y1 - 2017/03/14 PY - 2017 N1 - https://doi.org/10.11648/j.ijamtp.20170302.11 DO - 10.11648/j.ijamtp.20170302.11 T2 - International Journal of Applied Mathematics and Theoretical Physics JF - International Journal of Applied Mathematics and Theoretical Physics JO - International Journal of Applied Mathematics and Theoretical Physics SP - 20 EP - 25 PB - Science Publishing Group SN - 2575-5927 UR - https://doi.org/10.11648/j.ijamtp.20170302.11 AB - Exact solutions of nonlinear evolution equations (NLEEs) are very crucial to realize the obscurity of many physical phenomena in mathematical science. The modified simple equation (MSE) method is especially effective and highly proficient mathematical instrument to obtaining exact traveling wave solutions to NLEEs arising in science, engineering and mathematical physics. Though the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations for balance number less or equal 2 but the method did not applied to solve if the balance number is greater than 2. In this article, the MSE method is executed to construct exact traveling wave solutions of nonlinear evolution equations namely Kuramoto-Sivashinsky equation with balance number equal to 3. The obtained solutions are expressed in terms of exponential and trigonometric functions including solitary and periodic solutions. Moreover the procedure of this method reduces huge volume of calculations. VL - 3 IS - 2 ER -
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