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The G'/G2 - Expansion Method for Solving Fractional Burgers - Fisher and Burgers Equations

Received: 18 April 2020     Accepted: 12 May 2020     Published: 27 May 2020
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Abstract

In this paper, we apply G’/G2-Expansion method to discover a strategy for the approximate solution of the generalized fractional Burger-Fisher equation and fractional Burger equation. The given fractional Burger-Fisher and burger equation through substitution are converted into nonlinear ordinary differential equations, in the sense of the Jumarie’s modified Riemann-Liouville fractional derivative. The travelling wave solution is approximated by the G’/G2-Expansion method with unknown parameters that can be expressed by trigonometric functions, exponential functions, hyperbolic functions and rational functions. These results reveal that the proposed method is very effective and simple in performing a solution to the nonlinear fractional partial differential equation.

Published in Applied and Computational Mathematics (Volume 9, Issue 3)
DOI 10.11648/j.acm.20200903.12
Page(s) 56-63
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), 2020. Published by Science Publishing Group

Keywords

G’/G2-expansion Method, Burgers-Fisher Equation, Burgers Equation

References
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[2] M. Alaroud, M. Al-Smadi, R. R. Ahmad, and U. K. Salma Din, (2019). An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations. Symmetry, vol. 11 (2), pp. 205.
[3] Saleh Alshammari, Mohammed Al-Smadi, Mohammad Al Shammari, Ishak Hashim, and Mohd Almie Alias, (2019). Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions. International journal of differential equations, vol. 2019, pp. 1-11.
[4] S. Hasan, M. Al-Smadi, A. Freihet, and S. Momani, (2019). Two computational approaches for solving a fractional obstacle system in Hilbert space. Springer International publishing AG, vol. 2019 (1), pp. 55.
[5] Cheng Chen and Yao-Lin Jiang, (2018). Simplest equation method for some time-fractional partial differential equations with conformable derivative. Computers and Mathematics with Applications, vol. 75, Issue 8, pp. 2978-2988.
[6] E. A. Yousif, E. A-B. Abdel-Salam, M. A. El-Aasser, (2018). On the solution of the space-time fractional cubic nonlinear Schrödinger equation. Elsevier, Results in Physics, vol. 8, pp. 702-708.
[7] M. Alaroud, M. Al-Smadi, R. R. Ahmad, and U. K. Salma Din, (2018). Computational optimization of residual power series algorithm for certain classes of fuzzy fractional differential equations. International journal of differential Equations, vol. 2018, pp. 1-11.
[8] M. Ali Akbar, Norhashidah Hj. Mohd. Ali, Ripan Roy, (2018). Closed form solutions of two time fractional nonlinear wave equations. Elsevier, Results in Physics, vol. 9, pp. 1031-1039.
[9] Shumaila Javeeda, Summaya Saifa, Asif Waheedb and Dumitru Baleanuc, (2018) Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers. Elsevier, Results in Physics, vol. 9, pp. 1275-1281.
[10] Arzu Akbulut a and Melike Kaplan, (2018). Auxiliary equation method for time-fractional differential equations with conformable derivative. Computers and Mathematics with Applications, vol. 75, Issue 3, pp. 876-882.
[11] Dazhi Zhao. Maokang Luo (2017). General conformable fractional derivative and its physical interpretation. Springer-Verlag Italia, 54, pp. 903–917.
[12] K. Hosseini and R. Ansari,(2017). New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Waves in Random and Complex Media, vol. 27, pp. 628-636.
[13] Nematollah Kadkhoda (2017). Application of (G/G^2 )-expansion method for solving fractional differential equations. International journal of applied and computational mathematics, 3, pp. 1415–1424.
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  • APA Style

    Abaker A. Hassaballa. (2020). The G'/G2 - Expansion Method for Solving Fractional Burgers - Fisher and Burgers Equations. Applied and Computational Mathematics, 9(3), 56-63. https://doi.org/10.11648/j.acm.20200903.12

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    ACS Style

    Abaker A. Hassaballa. The G'/G2 - Expansion Method for Solving Fractional Burgers - Fisher and Burgers Equations. Appl. Comput. Math. 2020, 9(3), 56-63. doi: 10.11648/j.acm.20200903.12

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    AMA Style

    Abaker A. Hassaballa. The G'/G2 - Expansion Method for Solving Fractional Burgers - Fisher and Burgers Equations. Appl Comput Math. 2020;9(3):56-63. doi: 10.11648/j.acm.20200903.12

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  • @article{10.11648/j.acm.20200903.12,
      author = {Abaker A. Hassaballa},
      title = {The G'/G2 - Expansion Method for Solving Fractional Burgers - Fisher and Burgers Equations},
      journal = {Applied and Computational Mathematics},
      volume = {9},
      number = {3},
      pages = {56-63},
      doi = {10.11648/j.acm.20200903.12},
      url = {https://doi.org/10.11648/j.acm.20200903.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200903.12},
      abstract = {In this paper, we apply G’/G2-Expansion method to discover a strategy for the approximate solution of the generalized fractional Burger-Fisher equation and fractional Burger equation. The given fractional Burger-Fisher and burger equation through substitution are converted into nonlinear ordinary differential equations, in the sense of the Jumarie’s modified Riemann-Liouville fractional derivative. The travelling wave solution is approximated by the G’/G2-Expansion method with unknown parameters that can be expressed by trigonometric functions, exponential functions, hyperbolic functions and rational functions. These results reveal that the proposed method is very effective and simple in performing a solution to the nonlinear fractional partial differential equation.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - The G'/G2 - Expansion Method for Solving Fractional Burgers - Fisher and Burgers Equations
    AU  - Abaker A. Hassaballa
    Y1  - 2020/05/27
    PY  - 2020
    N1  - https://doi.org/10.11648/j.acm.20200903.12
    DO  - 10.11648/j.acm.20200903.12
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20200903.12
    AB  - In this paper, we apply G’/G2-Expansion method to discover a strategy for the approximate solution of the generalized fractional Burger-Fisher equation and fractional Burger equation. The given fractional Burger-Fisher and burger equation through substitution are converted into nonlinear ordinary differential equations, in the sense of the Jumarie’s modified Riemann-Liouville fractional derivative. The travelling wave solution is approximated by the G’/G2-Expansion method with unknown parameters that can be expressed by trigonometric functions, exponential functions, hyperbolic functions and rational functions. These results reveal that the proposed method is very effective and simple in performing a solution to the nonlinear fractional partial differential equation.
    VL  - 9
    IS  - 3
    ER  - 

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Author Information
  • Department of Mathematics, College of Applied & Industrial Sciences, Bahri University, Khartoum, Sudan

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