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A novel technique to construct exact solutions for nonlinear partial differential equations

  • Behzad Ghanbari
  • Dumitru BaleanuEmail author
Regular Article
  • 28 Downloads
Part of the following topical collections:
  1. Focus Point on Fractional Differential Equations in Physics: Recent Advantages and Future Direction

Abstract.

The aim of the manuscript is to present a new exact solver of nonlinear partial differential equations. The proposed technique is developed by extending the \( \phi^{6}\)-model expansion method as a known method. The corresponding exact solutions are given in terms of Jacobi elliptic functions. Some new optical solutions of the resonant nonlinear Schrödinger equation are constructed within this newly proposed method. For some specific choices of the modulus of Jacobi elliptic functions, various solutions of the equation are introduced. Some numerical simulations are also included to emphasize that all parameters have major influences for the solitary waves behaviours. The proposed technique is very simple and straightforward, and can be employed to solve other non-linear partial differential equations.

Notes

References

  1. 1.
    A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optik 158, 399 (2018)ADSCrossRefGoogle Scholar
  2. 2.
    M.S. Osman, H.I. Abdel-Gawad, M.A. El Mahdy, Results Phys. 8, 1054 (2018)ADSCrossRefGoogle Scholar
  3. 3.
    M.S. Osman, Optik 156, 169 (2018)ADSCrossRefGoogle Scholar
  4. 4.
    F. Gao, X.-J. Yang, H.M. Srivastava, Therm. Sci. 21, 2307 (2017)CrossRefGoogle Scholar
  5. 5.
    X.-J. Yang, J.A.T. Machado, D. Baleanu, C. Cattani, Chaos 26, 084312 (2016)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    X.-J. Yang, F. Gao, H.M. Srivastava, Comput. Math. Appl. 73, 203 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    X.-J. Yang, J.T. Machado, D. Baleanu, Fractals 25, 1740006 (2017)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    X.-J. Yang, F. Gao, H.M. Srivastava, J. Comput. Appl. Math. 339, 285 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    X.-J. Yang, F. Gao, Therm. Sci. 21, 133 (2017)CrossRefGoogle Scholar
  10. 10.
    X.-J. Yang, Therm. Sci. 21, S79 (2017)CrossRefGoogle Scholar
  11. 11.
    X.-J. Yang, Appl. Math. Lett. 64, 193 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    X.-J. Yang, F. Gao, Therm. Sci. 21, 133 (2017)CrossRefGoogle Scholar
  13. 13.
    Y. Yugui, Y. Xiaojun, Z. Mingzheng, C. Cattani, Therm. Sci. 21, S129 (2017)CrossRefGoogle Scholar
  14. 14.
    A.I. Aliyu, M. Inc, A. Yusuf, D. Baleanu, Mod. Phys. Lett. B 32, 1850373 (2018)ADSCrossRefGoogle Scholar
  15. 15.
    A.I. Aliyu, M. Inc, A. Yusuf, D. Baleanu, Commun. Theor. Phys. 70, 511 (2018)ADSCrossRefGoogle Scholar
  16. 16.
    A.I. Aliyu, M. Inc, Symmetry 10, 341 (2018)CrossRefGoogle Scholar
  17. 17.
    M. Inc, A.I. Aliyu, A. Yusuf, D. Baleanu, Mod. Phys. Lett. B 32, 1850202 (2018)ADSCrossRefGoogle Scholar
  18. 18.
    A. Yusuf, A.I. Aliyu, M.S. Hashemi, Eur. Phys. J. Plus 133, 168 (2018)CrossRefGoogle Scholar
  19. 19.
    D. Baleanu, M. Inc, A. Yusuf, A.I. Aliyu, Open Phys. 16, 302 (2018)CrossRefGoogle Scholar
  20. 20.
    Q. Zhou, X. Xiong, Q. Zhu, Y. Liu, H. Yu, P. Yao, A. Biswas, M. Belic, J. Optoelectron. Adv. Mater. 17, 82 (2015)Google Scholar
  21. 21.
    E.M.E. Zayed, Abdul-Ghani Al-Nowehy, Optik 143, 84 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    E.M.E. Zayed, Abdul-Ghani Al-Nowehy, Eur. Phys. J. Plus 132, 475 (2017)CrossRefGoogle Scholar
  23. 23.
    E.M.E. Zayed, Abdul-Ghani Al-Nowehy, Opt. Quantum Electron. 50, 164 (2018)CrossRefGoogle Scholar
  24. 24.
    B. Ghanbari, M. Inc, Eur. Phys. J. Plus 133, 142 (2018)CrossRefGoogle Scholar
  25. 25.
    M.S. Osman, B. Ghanbari, Optik 175, 328 (2018)ADSCrossRefGoogle Scholar
  26. 26.
    B. Ghanbari, N. Raza, Mod. Phys. Lett. B 33, 1950018 (2019)ADSCrossRefGoogle Scholar
  27. 27.
    M.S. Osman, Behzad Ghanbari, J.A.T. Machado, Eur. Phys. J. Plus 134, 20 (2019)CrossRefGoogle Scholar
  28. 28.
    B. Ghanbari, A. Yusuf, M. Inc, D. Baleanu, Adv. Differ. Equ. 2019, 49 (2019)CrossRefGoogle Scholar
  29. 29.
    B. Ghanbari, Mod. Phys. Lett. B 33, 1950106 (2019)ADSCrossRefGoogle Scholar
  30. 30.
    B. Ghanbari, A. Yusuf, M. Inc, D. Baleanu, Adv. Differ. Equ. 2019, 49 (2019)CrossRefGoogle Scholar
  31. 31.
    B. Ghanbari, M.S. Osman, D. Baleanu, Mod. Phys. Lett. A 34, 1950155 (2019)ADSCrossRefGoogle Scholar
  32. 32.
    M.S. Osman, B. Ghanbari, J.A.T. Machado, Eur. Phys. J. Plus 134, 20 (2019)CrossRefGoogle Scholar
  33. 33.
    B. Ghanbari, D. Baleanu, M.A. Qurashi, Symmetry 11, 1 (2019)Google Scholar
  34. 34.
    B. Ghanbari, M. Inc, L. Rada, J. Appl. Anal. Comput. 9, 1 (2019)MathSciNetGoogle Scholar
  35. 35.
    J.G. Liu, Y. He, Nonlinear Dyn. 92, 1103 (2018)CrossRefGoogle Scholar
  36. 36.
    J.G. Liu, J.Q. Du, Z.F. Zeng, B. Nie, Nonlinear Dyn. 88, 655 (2017)CrossRefGoogle Scholar
  37. 37.
    J.G. Liu, J.Q. Du, Z.F. Zeng, G.P. Ai, Chaos 26, 103114 (2016)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    H.M. Baskonus, Nonlinear Dyn. 86, 177 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    H.M. Baskonus, AIP Conf. Proc. 1798, 020018 (2017)CrossRefGoogle Scholar
  40. 40.
    G. Yel, H.M. Baskonus, H. Bulut, Opt. Quantum Electron. 49, 1 (2017)CrossRefGoogle Scholar
  41. 41.
    A. Ciancio, H.M. Baskonus, T.A. Sulaiman, H. Bulut, Indian J. Phys. 92, 1281 (2018)ADSCrossRefGoogle Scholar
  42. 42.
    C. Cattani, T.A. Sulaiman, H.M. Baskonus, H. Bulut, Opt. Quantum Electron. 50, 138 (2018)CrossRefGoogle Scholar
  43. 43.
    C. Cattani, T.A. Sulaiman, H.M. Baskonus, H. Bulut, Eur. Phys. J. Plus 133, 288 (2018)CrossRefGoogle Scholar
  44. 44.
    Haci Mehmet Baskonus, Axioms 8, 18 (2019)CrossRefGoogle Scholar
  45. 45.
    S.M. El-Shaboury, M.K. Ammar, W.M. Yousef, Appl. Math. Nonlinear Sci. 2, 403 (2017)MathSciNetCrossRefGoogle Scholar
  46. 46.
    P.K. Pandey, Appl. Math. Nonlinear Sci. 3, 649 (2018)MathSciNetCrossRefGoogle Scholar
  47. 47.
    E.I. Eskitascoglu, M.B. Akta, H.M. Baskonus, Appl. Math. Nonlinear Sci. 4, 93 (2019)CrossRefGoogle Scholar
  48. 48.
    H.M. Baskonus, H. Bulut, T.A. Sulaiman, Appl. Math. Nonlinear Sci. 4, 129 (2019)CrossRefGoogle Scholar
  49. 49.
    M. Dewasurendra, K. Vajravelu, Appl. Math. Nonlinear Sci. 3, 114 (2018)Google Scholar
  50. 50.
    T. Caraballo, M. Herrera-Cobos, P. Marin-Rubio, Appl. Math. Nonlinear Sci. 2, 73 (2017)MathSciNetCrossRefGoogle Scholar
  51. 51.
    E.M.E. Zayed, K.A.E. Alurrfi, Appl. Math. Comput. 289, 111 (2016)MathSciNetGoogle Scholar
  52. 52.
    E.M.E. Zayed, Abdul-Ghani Al-Nowehy, Optik 130, 1295 (2017)ADSCrossRefGoogle Scholar
  53. 53.
    E.M.E. Zayed, R.M.A. Shohib, Optik 158, 970 (2018)ADSCrossRefGoogle Scholar
  54. 54.
    E.M.E. Zayed, A.-G. Al-Nowehy, M.E.M. Elshater, Eur. Phys. J. Plus 133, 417 (2018)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Engineering ScienceKermanshah University of TechnologyKermanshahIran
  2. 2.Department of Mathematics, Faculty of Engineering and Natural SciencesBahcesehir UniversityIstanbulTurkey
  3. 3.Department of Mathematics, Faculty of Arts and SciencesCankaya UniversityAnkaraTurkey
  4. 4.Institute of Space SciencesMagurele-BucharestMG-23Romania

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