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Comparison between the generalized tanh–coth and the (G′/G)-expansion methods for solving NPDEs and NODEs

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Abstract

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G ′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.

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References

  1. T Achouri and K Omrani, Commun. Nonlinear Sci. Numer. Simul. 14, 2025 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  2. M J Ablowitz and P A Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, 1991)

  3. R Hirota, The direct method in soliton theory (Cambridge University Press, 2004)

  4. M Dehghan, J Manafian and A Saadatmandi, Numer. Meth. Partial Differential Eq. J. 26, 448 (2010)

    MathSciNet  Google Scholar 

  5. M Dehghan, J Manafian and A Saadatmandi, Z. Naturforsch. A 65, 935 (2010)

    Article  ADS  Google Scholar 

  6. S Saha Ray and S Sahoo, Math. Meth. Appl. Sci. 38, 2840 (2015)

    Article  MathSciNet  Google Scholar 

  7. J H He, Int. J. Nonlinear Mech. 34, 699 (1999)

    Article  ADS  Google Scholar 

  8. M Dehghan, J Manafian and A Saadatmandi, Math. Meth. Appl. Sci. 33, 1384 (2010)

    MathSciNet  Google Scholar 

  9. M Dehghan and J Manafian, Z. Naturforsch. A 64, 420 (2009)

    Article  ADS  Google Scholar 

  10. S Saha Ray and S Sahoo, Appl. Math. Comput. 253, 72 (2015)

    MathSciNet  Google Scholar 

  11. A M Wazwaz, Appl. Math. Comput. 177, 755 (2006)

    MathSciNet  Google Scholar 

  12. S Sahoo and S Saha Ray, Phys. A: Stat. Mech. Appl. 434, 240 (2015)

    Article  MathSciNet  Google Scholar 

  13. J Manafian Heris and M Lakestani, Commun. Numer. Anal. 2013, 1 (2013)

    Article  MathSciNet  Google Scholar 

  14. S Saha Ray and S Sahoo, Rep. Math. Phys. 75, 63 (2015)

    Article  MathSciNet  Google Scholar 

  15. S Saha Ray and S Sahoo, Commun. Theor. Phys. 25, 63 (2015)

    Google Scholar 

  16. S Sahoo and S Saha Ray, Comput. Math. Appl. 70, 158 (2015)

    Article  MathSciNet  Google Scholar 

  17. X H Menga, W J Liua, H W Zhua, C Y Zhang and B Tian, Physica A 387, 97 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  18. M Fazli Aghdaei and J Manafianheris, J. Math. Ext. 5, 91 (2011)

    MathSciNet  Google Scholar 

  19. M Dehghan, J Manafian Heris and A Saadatmandi, Int. J. Numer. Methods for Heat Fluid Flow 21, 736 (2011)

    Article  MathSciNet  Google Scholar 

  20. M Dehghan, J Manafian Heris and A Saadatmandi, Int. J. Mod. Phys. B 25, 2965 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  21. J Manafian Heris and M Bagheri, J. Math. Ext. 4, 77 (2010)

    MathSciNet  Google Scholar 

  22. A J M Jawad, M D Petkovic and A Biswas, Appl. Math. Comput. 217, 869 (2010)

    MathSciNet  Google Scholar 

  23. W Malfliet, Am. J. Phys. 60, 650 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  24. W Malfliet and W Hereman, Phys. Scr. 54, 569 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  25. H Naher and F A Abdullah, AIP Adv. 3, 032116 (2013)

    Article  ADS  Google Scholar 

  26. J Manafian Heris and M Lakestani, Int. Sch. Res. Not. 2014, 840689 (2014)

    Google Scholar 

  27. F Chand and A K Malik, Int. J. Nonlinear Sci. 14, 416 (2012)

    MathSciNet  Google Scholar 

  28. M Wang, X Li and J Zhang, Phys. Lett. A 372, 417 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  29. J Zhang, X Wei and Y Lu, Phys. Lett. A 372, 3653 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  30. E Fan, Phys. Lett. A 277, 212 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  31. A M Wazwaz, Chaos, Solitons and Fractals 188, 1930 (2007)

    MathSciNet  Google Scholar 

  32. A M Wazwaz, Appl. Math. Comput. 184, 1002 (2007)

    MathSciNet  Google Scholar 

  33. J W Cahn and J E Hilliard, J. Chem. Phys. 28, 258 (2005)

    Article  ADS  Google Scholar 

  34. S M Allen and J W Cahn, Acta Metall. Mater. 27, 1085 (1979)

    Article  Google Scholar 

  35. Q Du, C Liu and X Wang, J. Comput. Phys. 212, 757 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  36. P Yue, J J Feng, C Liu and J Shen, J. Fluid Mech. 515, 293 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  37. X Yang, J J Feng, C Liu and J Shen, J. Comput. Phys. 218, 417 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  38. Y Ugurlu and D Kaya, Comput. Math. Appl. 56, 3038 (2008)

    Article  MathSciNet  Google Scholar 

  39. F Tascan and A Bekir, Appl. Math. Comput. 207, 279 (2009)

    MathSciNet  Google Scholar 

  40. C M Elliott and D A French, IMA J. Appl. Math. 38, 97 (1987)

    Article  MathSciNet  Google Scholar 

  41. K Parand and J A Rad, J. King Saud Univ-Sci. 24, 1 (2012)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Science, University of Tabriz, Tabriz, Iran

    JALIL MANAFIAN & MEHRDAD LAKESTANI

  2. Department of Mathematics-Computer, Eskişehir Osmangazi University, 26480, Eskişehir, Turkey

    AHMET BEKIR

Authors
  1. JALIL MANAFIAN
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  2. MEHRDAD LAKESTANI
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  3. AHMET BEKIR
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Correspondence to AHMET BEKIR.

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MANAFIAN, J., LAKESTANI, M. & BEKIR, A. Comparison between the generalized tanh–coth and the (G′/G)-expansion methods for solving NPDEs and NODEs. Pramana - J Phys 87, 95 (2016). https://doi.org/10.1007/s12043-016-1292-9

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  • DOI: https://doi.org/10.1007/s12043-016-1292-9

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