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

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.

Published in Applied and Computational Mathematics (Volume 4, Issue 3)
DOI 10.11648/j.acm.20150403.14
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), 2015. 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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Cite This Article
  • 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

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

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

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