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Dergiler / Advances in the Theory of Nonlinear Analysis and its Application / 2023 / Cilt: 7 Sayı: 2

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

Medjahed Djilali(Djilali Liabes university)0000-0002-4390-9404
Hakem Alı(Djilali Liabes university)0000-0001-6145-4514
Sayfa
303–318
Yayın Tarihi
2023-07-23
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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.

📌 3 Maddede Bu Makale

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Amaç & Hipotez: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.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı: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.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Exact solutions · Traveling wave solutions · Boussinesq equation

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.

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Bu Makaleye Atıf Yap

Djilali, M., Alı, H. (2023). (G'/G)-EXPANSION METHOD TO SEEK TRAVELING WAVE SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PDES ARISING IN NATURAL SCIENCES. Advances in the Theory of Nonlinear Analysis and its Application, Cilt: 7 Sayı: 2, 303–318. https://doi.org/10.31197/atnaa.1125691

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Kaynakça (38)

  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.
  15. P. A. Clarkson (1989), New similarity solutions for the modified Boussinesq equation, J. Phys. A: Math. Gen., 22 2355-2367.
  16. D. Levi and P. Winternitz,Non-classical symmetry reduction: example of the Boussinesq equation, J. Phys. A: Math. Gen, 22 (1989 ) 2915-2924.
  17. 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.
  18. 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.
  19. 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.
  20. 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.
  21. 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.
  22. 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.
  23. R. M. Cherniha, New Exact Solutions Of One Nonlinear Equation In Mathematical Biology And Their Proprieties, Ukrainian Mathematical Journal, 53 (2001) (10) 1712-727.
  24. 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.
  25. 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.
  26. 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.
  27. Y. Guo, S. Lai, New exact solutions for an (N +1)-dimensional generalized Boussinesq equation, Nonlinear Analysis, 72 (2010) 2863-2873.
  28. O. V. Kaptsov, Construction Of Exact Solutions Of The Boussinesq Equation, Journal of Applied Mechanics and Technical Physics, 39 (3) (1998) 389-392.
  29. 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.
  30. 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.
  31. 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.
  32. W. Hereman, A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Mathematics and Computers in Simulation, 43 (1997) 13-27.
  33. 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.
  34. P. A. Clarkson (1989), New similarity solutions for the modified Boussinesq equation, J. Phys. A: Math. Gen., 22 2355-2367.
  35. D. Levi and P. Winternitz,Non-classical symmetry reduction: example of the Boussinesq equation, J. Phys. A: Math. Gen, 22 (1989 ) 2915-2924.
  36. 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.
  37. 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.
  38. 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.