Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method

M. Ashrafuzzaman Khan; M. Ali Akbar

doi:doi:10.11648/j.acm.20150403.14

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Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method

M. Ashrafuzzaman Khan, M. Ali Akbar

Received: 10 April 2015 Accepted: 21 April 2015 Published: 30 April 2015

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Abstract

Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.

DOI	10.11648/j.acm.20150403.14
Published in	Applied and Computational Mathematics (Volume 4, Issue 3, June 2015)
Page(s)	122-129
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

Keywords

MSE Method, Nonlinear Evolution Equations, Solitary Wave Solutions, Exact Solutions, Generalized Fifth-Order Kdv Equation

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M. Ashrafuzzaman Khan, M. Ali Akbar. (2015). Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method. Applied and Computational Mathematics, 4(3), 122-129. https://doi.org/10.11648/j.acm.20150403.14

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ACS Style

M. Ashrafuzzaman Khan; M. Ali Akbar. Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method. Appl. Comput. Math. 2015, 4(3), 122-129. doi: 10.11648/j.acm.20150403.14

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AMA Style

M. Ashrafuzzaman Khan, M. Ali Akbar. Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method. Appl Comput Math. 2015;4(3):122-129. doi: 10.11648/j.acm.20150403.14

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@article{10.11648/j.acm.20150403.14,
  author = {M. Ashrafuzzaman Khan and M. Ali Akbar},
  title = {Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3},
  pages = {122-129},
  doi = {10.11648/j.acm.20150403.14},
  url = {https://doi.org/10.11648/j.acm.20150403.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.14},
  abstract = {Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.},
 year = {2015}
}

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TY  - JOUR
T1  - Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method
AU  - M. Ashrafuzzaman Khan
AU  - M. Ali Akbar
Y1  - 2015/04/30
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.14
DO  - 10.11648/j.acm.20150403.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 122
EP  - 129
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150403.14
AB  - Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.
VL  - 4
IS  - 3
ER  -

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Author Information

M. Ashrafuzzaman Khan

Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh
M. Ali Akbar

Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

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APA Style

M. Ashrafuzzaman Khan, M. Ali Akbar. (2015). Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method. Applied and Computational Mathematics, 4(3), 122-129. https://doi.org/10.11648/j.acm.20150403.14

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ACS Style

M. Ashrafuzzaman Khan; M. Ali Akbar. Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method. Appl. Comput. Math. 2015, 4(3), 122-129. doi: 10.11648/j.acm.20150403.14

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AMA Style

M. Ashrafuzzaman Khan, M. Ali Akbar. Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method. Appl Comput Math. 2015;4(3):122-129. doi: 10.11648/j.acm.20150403.14

Copy | Download

@article{10.11648/j.acm.20150403.14,
  author = {M. Ashrafuzzaman Khan and M. Ali Akbar},
  title = {Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3},
  pages = {122-129},
  doi = {10.11648/j.acm.20150403.14},
  url = {https://doi.org/10.11648/j.acm.20150403.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.14},
  abstract = {Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method
AU  - M. Ashrafuzzaman Khan
AU  - M. Ali Akbar
Y1  - 2015/04/30
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.14
DO  - 10.11648/j.acm.20150403.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 122
EP  - 129
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150403.14
AB  - Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.
VL  - 4
IS  - 3
ER  -

Copy | Download