Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method

Md. Ashrafuzzaman Khan; M. Ali Akbar; Fethi Bin Muhammad Belgacem

doi:doi:10.11648/j.ml.20160201.11

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Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method

Md. Ashrafuzzaman Khan, M. Ali Akbar, Fethi Bin Muhammad Belgacem

Published in Mathematics Letters (Volume 2, Issue 1)

Received: 14 November 2015 Accepted: 30 March 2016 Published: 3 June 2016

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Abstract

Although the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations in the field of engineering and mathematical physics, it has some drawbacks. Particularly, if the balance number is greater than 1, the method cannot be expected to yield any solution. In this article, we present a process to implement the modified simple equation method to solve nonlinear evolution equations for balance number greater than 1, namely with balance number equal to 2. To validate our theory through applications, two equations have been chosen to undergo the proposed process, the Boussinesq and the Fisher equations, to which traveling wave are found and analyzed. For special parameters values, solitary wave solutions are originated 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.

Published in	Mathematics Letters (Volume 2, Issue 1)
DOI	10.11648/j.ml.20160201.11
Page(s)	1-18
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

Boussinesq Equation, Fisher Equation, Modified Simple Equation Method, Nonlinear Evolution Equations, Solitary Wave Solutions

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APA Style

Md. Ashrafuzzaman Khan, M. Ali Akbar, Fethi Bin Muhammad Belgacem. (2016). Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method. Mathematics Letters, 2(1), 1-18. https://doi.org/10.11648/j.ml.20160201.11

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ACS Style

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. Math. Lett. 2016, 2(1), 1-18. doi: 10.11648/j.ml.20160201.11

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AMA Style

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. Math Lett. 2016;2(1):1-18. doi: 10.11648/j.ml.20160201.11

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@article{10.11648/j.ml.20160201.11,
  author = {Md. Ashrafuzzaman Khan and M. Ali Akbar and Fethi Bin Muhammad Belgacem},
  title = {Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method},
  journal = {Mathematics Letters},
  volume = {2},
  number = {1},
  pages = {1-18},
  doi = {10.11648/j.ml.20160201.11},
  url = {https://doi.org/10.11648/j.ml.20160201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20160201.11},
  abstract = {Although the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations in the field of engineering and mathematical physics, it has some drawbacks. Particularly, if the balance number is greater than 1, the method cannot be expected to yield any solution. In this article, we present a process to implement the modified simple equation method to solve nonlinear evolution equations for balance number greater than 1, namely with balance number equal to 2. To validate our theory through applications, two equations have been chosen to undergo the proposed process, the Boussinesq and the Fisher equations, to which traveling wave are found and analyzed.  For special parameters values, solitary wave solutions are originated 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 = {2016}
}

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TY  - JOUR
T1  - Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method
AU  - Md. Ashrafuzzaman Khan
AU  - M. Ali Akbar
AU  - Fethi Bin Muhammad Belgacem
Y1  - 2016/06/03
PY  - 2016
N1  - https://doi.org/10.11648/j.ml.20160201.11
DO  - 10.11648/j.ml.20160201.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 1
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20160201.11
AB  - Although the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations in the field of engineering and mathematical physics, it has some drawbacks. Particularly, if the balance number is greater than 1, the method cannot be expected to yield any solution. In this article, we present a process to implement the modified simple equation method to solve nonlinear evolution equations for balance number greater than 1, namely with balance number equal to 2. To validate our theory through applications, two equations have been chosen to undergo the proposed process, the Boussinesq and the Fisher equations, to which traveling wave are found and analyzed.  For special parameters values, solitary wave solutions are originated 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  - 2
IS  - 1
ER  -

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Author Information

Md. Ashrafuzzaman Khan

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

Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh
Fethi Bin Muhammad Belgacem

Department of Mathematics, Faculty of Basic Education, PAAET, Al-Ardhyia, Kuwait

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Plain Text BibTeX RIS

APA Style

Md. Ashrafuzzaman Khan, M. Ali Akbar, Fethi Bin Muhammad Belgacem. (2016). Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method. Mathematics Letters, 2(1), 1-18. https://doi.org/10.11648/j.ml.20160201.11

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ACS Style

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. Math. Lett. 2016, 2(1), 1-18. doi: 10.11648/j.ml.20160201.11

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AMA Style

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. Math Lett. 2016;2(1):1-18. doi: 10.11648/j.ml.20160201.11

Copy | Download

@article{10.11648/j.ml.20160201.11,
  author = {Md. Ashrafuzzaman Khan and M. Ali Akbar and Fethi Bin Muhammad Belgacem},
  title = {Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method},
  journal = {Mathematics Letters},
  volume = {2},
  number = {1},
  pages = {1-18},
  doi = {10.11648/j.ml.20160201.11},
  url = {https://doi.org/10.11648/j.ml.20160201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20160201.11},
  abstract = {Although the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations in the field of engineering and mathematical physics, it has some drawbacks. Particularly, if the balance number is greater than 1, the method cannot be expected to yield any solution. In this article, we present a process to implement the modified simple equation method to solve nonlinear evolution equations for balance number greater than 1, namely with balance number equal to 2. To validate our theory through applications, two equations have been chosen to undergo the proposed process, the Boussinesq and the Fisher equations, to which traveling wave are found and analyzed.  For special parameters values, solitary wave solutions are originated 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 = {2016}
}

Copy | Download

TY  - JOUR
T1  - Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method
AU  - Md. Ashrafuzzaman Khan
AU  - M. Ali Akbar
AU  - Fethi Bin Muhammad Belgacem
Y1  - 2016/06/03
PY  - 2016
N1  - https://doi.org/10.11648/j.ml.20160201.11
DO  - 10.11648/j.ml.20160201.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 1
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20160201.11
AB  - Although the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations in the field of engineering and mathematical physics, it has some drawbacks. Particularly, if the balance number is greater than 1, the method cannot be expected to yield any solution. In this article, we present a process to implement the modified simple equation method to solve nonlinear evolution equations for balance number greater than 1, namely with balance number equal to 2. To validate our theory through applications, two equations have been chosen to undergo the proposed process, the Boussinesq and the Fisher equations, to which traveling wave are found and analyzed.  For special parameters values, solitary wave solutions are originated 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  - 2
IS  - 1
ER  -

Copy | Download