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(G'/G)-EXPANSION METHOD TO SEEK TRAVELING WAVE SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PDES ARISING IN NATURAL SCIENCES

Year 2023, Volume: 7 Issue: 2, 303 - 318, 23.07.2023
https://doi.org/10.31197/atnaa.1125691

Abstract

The (G'/G)-expansion method with the aid of symbolic computational system can be used to obtain the traveling wave solutions (hyperbolic, trigonometric and rational solutions) for nonlinear time-fractional evolution equations arising in mathematical physics and biology. In this work, we will process the analytical solutions of the time-fractional classical Boussinesq equation, the time-fractional Murray equation, and the space-time fractional Phi-four equation. With the fact that the method which we will propose in this paper is also a standard, direct and computerized method, the exact solutions for these equations are obtained.

References

  • [1] 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, (2015), http://dx.doi.org/10.1016/j.rinp.2015.01.008.
  • [2] L. Bangqing, X. Meiping, M. Yulan, , New exact solutions of (2+1)-dimensional generalization of shallow water wave equation by (G'/G)-expansion method, Applied Mechanics and Materials Vols 20-23 (2010) pp 1516-1521.
  • [3] R. M. Cherniha , New Ansatze and Exact Solutions for Nonlinear Reaction-Di ¨ ffusion Equations Arising in Mathematical Biology, Symmetry in Nonlinear Mathematical Physics, 1 (1997) 138146.
  • [4] R. M. Cherniha , New Exact Solutions Of One Nonlinear Equation In Mathematical Biology And Their Proprieties , Ukrainian Mathematical Journal, 53 (2001) (10) 1712-727.
  • [5] M. Djilali , A. Hakem and A. Benali , Exact Solutions of Kupershmidt Equation, Approximate Solutions for Time-Fractional Kupershmidt Equation: A Comparison Study. International Journal of Analysis and Applications, 18 (3) (2020) pp. 493-512.
  • [6] M. Djilali, A. Hakem , Solving Some Important Nonlinear Time-Fractional Evolution Equations By Using The (G/G)-Expansion Method, Journal of Science and Arts, 53 (4) (2020) pp. 815-832.
  • [7] M. Djilali, A. Hakem and A. Benali ,A Comparison Between Analytical and Numerical Solutions for Time-Fractional Coupled Dispersive Long-Wave Equations, Fundamentals of Contemporary Mathematical Sciences 2 (1) (2021) pp. 8-29.
  • [8] Y. Guo, S. Lai , New exact solutions for an (N +1)-dimensional generalized Boussinesq equation, Nonlinear Analysis, 72 (2010) 2863-2873.
  • [9] O. V. Kaptsov , Construction Of Exact Solutions Of The Boussinesq Equation , Journal of Applied Mechanics and Technical Physics, 39 (3) (1998) 389-392.
  • [10] F. Mahmud, M. Samsuzzoha and M. A. Akbar , The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fisher equation, Results in Physics, 7 (2017) 4296-430.
  • [11] A. Malik, F. Chand, H. Kumar and S. C. Mishra , Exact solutions of some physical models using the (G'/G)- expansion method . PRAMANA -journal of physics. Vol. 78, No. 4 (2012) pp. 513-529.
  • [12] D. Wang , W. Sun, C. Kong, H. Zhang, New extended rational expansion method and exact solutions of Boussinesq equation and JimboMiwa equations, Applied Mathematics and Computation, 189 (2007) 878886.
  • [13] W. Hereman , A. Nuseir , Symbolic methods to construct exact solutions of nonlinear partial differential equations, Mathematics and Computers in Simulation, 43 (1997) 13-27.
  • [14] A.M. Wazwaz, Analytic study on nonlinear variants of the RLW and the PHI-four equations, Communications in Nonlinear Science and Numerical Simulation, 12 (2007) 314-327. [15] ?. Akagi, T. Aydemir , Comparison between the (G'/G)- expansion method and the modified extended tanh method. Open Phys. ; 14 (2016) :88-94.
  • [16] P. A. Clarkson (1989), New similarity solutions for the modified Boussinesq equation, J. Phys. A : Math. Gen., 22 2355-2367.
  • [17] D. Levi and P. Winternitz ,Non-classical symmetry reduction: example of the Boussinesq equation , J. Phys. A: Math. Gen, 22 (1989 ) 2915-2924.
  • [18] X. Deng, M. Zhao , X. Li , Travelling wave solutions for a nonlinear variant of the PHI-four equation, Mathematical and Computer Modelling, 49 (2009) 617-622.
  • [19] A. Khajeh, M.M. Kabir, A. Y. Koma ,New Exact and Explicit Travelling Wave Solutions for the Coupled Higgs Equation and a Nonlinear Variant of the PHI-four Equation, International Journal of Nonlinear Sciences & Numerical Simulation, 11 (9) (2010) 725-741.
  • [20] M. Alquran, H. M. Jaradat, M. I. Syam , Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method, Nonlinear Dyn, 90 (2017) 2525-2529.
Year 2023, Volume: 7 Issue: 2, 303 - 318, 23.07.2023
https://doi.org/10.31197/atnaa.1125691

Abstract

References

  • [1] 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, (2015), http://dx.doi.org/10.1016/j.rinp.2015.01.008.
  • [2] L. Bangqing, X. Meiping, M. Yulan, , New exact solutions of (2+1)-dimensional generalization of shallow water wave equation by (G'/G)-expansion method, Applied Mechanics and Materials Vols 20-23 (2010) pp 1516-1521.
  • [3] R. M. Cherniha , New Ansatze and Exact Solutions for Nonlinear Reaction-Di ¨ ffusion Equations Arising in Mathematical Biology, Symmetry in Nonlinear Mathematical Physics, 1 (1997) 138146.
  • [4] R. M. Cherniha , New Exact Solutions Of One Nonlinear Equation In Mathematical Biology And Their Proprieties , Ukrainian Mathematical Journal, 53 (2001) (10) 1712-727.
  • [5] M. Djilali , A. Hakem and A. Benali , Exact Solutions of Kupershmidt Equation, Approximate Solutions for Time-Fractional Kupershmidt Equation: A Comparison Study. International Journal of Analysis and Applications, 18 (3) (2020) pp. 493-512.
  • [6] M. Djilali, A. Hakem , Solving Some Important Nonlinear Time-Fractional Evolution Equations By Using The (G/G)-Expansion Method, Journal of Science and Arts, 53 (4) (2020) pp. 815-832.
  • [7] M. Djilali, A. Hakem and A. Benali ,A Comparison Between Analytical and Numerical Solutions for Time-Fractional Coupled Dispersive Long-Wave Equations, Fundamentals of Contemporary Mathematical Sciences 2 (1) (2021) pp. 8-29.
  • [8] Y. Guo, S. Lai , New exact solutions for an (N +1)-dimensional generalized Boussinesq equation, Nonlinear Analysis, 72 (2010) 2863-2873.
  • [9] O. V. Kaptsov , Construction Of Exact Solutions Of The Boussinesq Equation , Journal of Applied Mechanics and Technical Physics, 39 (3) (1998) 389-392.
  • [10] F. Mahmud, M. Samsuzzoha and M. A. Akbar , The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fisher equation, Results in Physics, 7 (2017) 4296-430.
  • [11] A. Malik, F. Chand, H. Kumar and S. C. Mishra , Exact solutions of some physical models using the (G'/G)- expansion method . PRAMANA -journal of physics. Vol. 78, No. 4 (2012) pp. 513-529.
  • [12] D. Wang , W. Sun, C. Kong, H. Zhang, New extended rational expansion method and exact solutions of Boussinesq equation and JimboMiwa equations, Applied Mathematics and Computation, 189 (2007) 878886.
  • [13] W. Hereman , A. Nuseir , Symbolic methods to construct exact solutions of nonlinear partial differential equations, Mathematics and Computers in Simulation, 43 (1997) 13-27.
  • [14] A.M. Wazwaz, Analytic study on nonlinear variants of the RLW and the PHI-four equations, Communications in Nonlinear Science and Numerical Simulation, 12 (2007) 314-327. [15] ?. Akagi, T. Aydemir , Comparison between the (G'/G)- expansion method and the modified extended tanh method. Open Phys. ; 14 (2016) :88-94.
  • [16] P. A. Clarkson (1989), New similarity solutions for the modified Boussinesq equation, J. Phys. A : Math. Gen., 22 2355-2367.
  • [17] D. Levi and P. Winternitz ,Non-classical symmetry reduction: example of the Boussinesq equation , J. Phys. A: Math. Gen, 22 (1989 ) 2915-2924.
  • [18] X. Deng, M. Zhao , X. Li , Travelling wave solutions for a nonlinear variant of the PHI-four equation, Mathematical and Computer Modelling, 49 (2009) 617-622.
  • [19] A. Khajeh, M.M. Kabir, A. Y. Koma ,New Exact and Explicit Travelling Wave Solutions for the Coupled Higgs Equation and a Nonlinear Variant of the PHI-four Equation, International Journal of Nonlinear Sciences & Numerical Simulation, 11 (9) (2010) 725-741.
  • [20] M. Alquran, H. M. Jaradat, M. I. Syam , Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method, Nonlinear Dyn, 90 (2017) 2525-2529.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Medjahed Djilali 0000-0002-4390-9404

Hakem Alı 0000-0001-6145-4514

Early Pub Date August 3, 2023
Publication Date July 23, 2023
Published in Issue Year 2023 Volume: 7 Issue: 2

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