Analytic study of sixth-order thin-film equation by tan(ϕ/2)-expansion method

  • Jalil Manafian
  • Mehdi Fazli Aghdaei
  • Manouchehr Zadahmad


A improvement of the expansion methods, namely, the improved \(\tan (\phi (\xi )/2)\)-expansion method for solving the sixth-order thin-film equation is proposed. As a result, many new and more general exact traveling wave solutions are obtained including singular kink-type solutions. We obtained the further solutions comparing with other methods as Flitton and King (Eur J Appl Math 15:713–754, 2004) and Taha et al. (J King Saud Univ Sci 26:75–78, 2014). Recently this method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including kink and rational solutions have been found. These solutions might play important role in engineering and physics fields. Also the results demonstrate that the introduced method is powerful tools for solving the nonlinear partial differential equations.


Improved \(\tan (\phi (\xi )/2)\)-expansion method Sixth-order thin-film equation Traveling wave Singular kink-type solution 

Mathematics Subject Classification

65D19 65H10 35A20 35A24 35C08 35G50 



The authors are very grateful to the referees for their help in the improvement of this paper.


  1. Ansini, L., Giacomelli, L.: Doubly nonlinear thin-film equations in one space dimension. Arch. Ration. Mech. Anal. 173, 89–131 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. Armitage, J.P., Garduno, F.S., Maini, P.K., Satnoianu, R.A.: Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete Contin. Dyn. Syst. Ser. B 1, 339–362 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. Aronson, D.G.: Some problems in nonlinear difffusion. In: Fasano, A., Primiceno, M. (eds.) Lecture Notes in Mathematics. Springer, New York (1986)Google Scholar
  4. Baskonus, H.M., Bulut, H.: Exponential prototype structures for (2 + 1)-dimensional Boiti–Leon–Pempinelli systems in mathematical physics. Waves Random Complex Media 26, 201–208 (2016)MathSciNetGoogle Scholar
  5. Chen, Y., Wang, Q.: Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1 + 1)-dimensional dispersive long wave equation. Chaos Solitons Fract. 24, 745–757 (2005)ADSCrossRefMATHGoogle Scholar
  6. Crank, J., Gupta, R.S.: A moving boundary problem arising from the diffusion of oxygen in absorbing tissue. J. Inst. Math. Appl. 10, 19–33 (1972)MathSciNetCrossRefMATHGoogle Scholar
  7. Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Z. Naturforsch. 64a, 420–430 (2009)ADSGoogle Scholar
  8. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. J. 26, 448–479 (2010a)MathSciNetMATHGoogle Scholar
  9. Dehghan, M., Manafian, J., Saadatmandi, A.: Application of semi-analytic methods for the Fitzhugh–Nagumo equation, which models the transmission of nerve impulses. Math. Methods Appl. Sci. 33, 1384–1398 (2010b)MathSciNetMATHGoogle Scholar
  10. Dehghan, M., Heris, J.M., Saadatmandi, A.: Application of the exp-function method for solving a partial differential equation arising in biology and population genetics. Int. J. Numer. Methods Heat Fluid Flow 21, 736–753 (2011)MathSciNetCrossRefGoogle Scholar
  11. Flitton, J.C., King, J.R.: Moving boundary and fixed domain problems for a sixth-order thin film equation. Eur. J. Appl. Math. 15, 713–754 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. Hafez, M.G., Alam, M.N., Akbar, M.A.: Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system. J. King Saud Univ.-Sci. 27, 105–112 (2015)CrossRefGoogle Scholar
  13. Heris, J.M., Lakestani, M.: Solitary wave and periodic wave solutions for variants of the KdV-Burger and the K(n, n)-Burger equations by the generalized tanh–coth method. Commun. Numer. Anal. 2013, 1–18 (2013)MathSciNetCrossRefGoogle Scholar
  14. Howison, S.D., Mayers, D.F., Smith, W.R.: Numerical and asymptotic solution of a sixth order nonlinear diffusion equation and related coupled systems. IMA J. Appl. Math. 57, 79–98 (1996)MathSciNetCrossRefMATHGoogle Scholar
  15. Huang, R., Suo, Z.: Wrinkling of a compressed elastic film on a viscous layer. J. Appl. Phys. 91, 1135–1142 (2002)ADSCrossRefMATHGoogle Scholar
  16. Hulshof, J.: Some aspects of the thin film equation. In: Proceeding of the European Congress in Mathematics, Vol. 2, 202. Birkhauser-Verlag, pp. 291–301 (2001)Google Scholar
  17. Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869–877 (2010)MathSciNetMATHGoogle Scholar
  18. Khana, K., Akbarb, A., Rashidi, M.M., Zamanpour, I.: Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method. Waves Random Complex Media (2015). doi: 10.1080/17455030.2015.1068964 MathSciNetGoogle Scholar
  19. Lee, J., Sakthivel, R.: Exact travelling wave solutions for some important nonlinear physical models. Pramana J. Phys. 80, 757–769 (2013)ADSCrossRefGoogle Scholar
  20. Liang, B., Wang, M., Cao, Y., Shen, H.: A thin film equation with a singular diffusion. Appl. Math. Comput. 227, 1–10 (2014)MathSciNetGoogle Scholar
  21. Manafian, J.: On the complex structures of the Biswas–Milovic equation for power, parabolic and dual parabolic law nonlinearities. Eur. Phys. J. Plus 130, 1–20 (2015)CrossRefGoogle Scholar
  22. Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the \(\tan (\phi /2)\)-expansion method. Optik 127, 4222–4245 (2016)ADSCrossRefGoogle Scholar
  23. Manafian, J., Lakestani, M.: New improvement of the expansion methods for solving the generalized Fitzhugh–Nagumo equation with time-dependent coefficients. Int. J. Eng. Math. 2015, 1079 (2015a). doi: 10.1155/2015/107978 Article ID78, 35 pages
  24. Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015b)CrossRefGoogle Scholar
  25. Manafian, J., Lakestani, M.: Solitary wave and periodic wave solutions for Burgers, Fisher, Huxley and combined forms of these equations by the (G\(^{\prime }\)/G)-expansion method. Pramana J. Phys. 130, 31–52 (2015c)ADSCrossRefGoogle Scholar
  26. Manafian, J., Lakestani, M.: Application of \(\tan (\phi /2)\)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity. Optik 127, 2040–2054 (2016a)ADSCrossRefGoogle Scholar
  27. Manafian, J., Lakestani, M.: Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Opt. Quantum Electron. 48, 1–32 (2016b)CrossRefGoogle Scholar
  28. Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via \(tan(\phi /2)\)-expansion method. Optik 127, 5543–5551 (2016c)ADSCrossRefGoogle Scholar
  29. Manafian, J., Lakestani, M., Bekir, A.: Study of the analytical treatment of the (2 + 1)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach. Int. J. Appl. Comput. Math. 2, 243–268 (2016)MathSciNetCrossRefGoogle Scholar
  30. Myers, T.G.: Surface Tension Driven Thin Film Flows. Mechanics of Thin Film Coatings. Wiley, London (1996)Google Scholar
  31. Myers, T.G.: Thin films with high surface tension. SIAM Rev. 40, 441–462 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. Priya, N.V., Senthilvelan, M.: Generalized Darboux transformation and N-th order rogue wave solution of a general coupled nonlinear Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 20, 401–420 (2015). doi: 10.1016/j.cnsns.2014.06.001 ADSMathSciNetCrossRefMATHGoogle Scholar
  33. Rashidi, M.M., Hayat, T., Keimanesh, T., Yousefian, H.: A study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method. Heat Transf.-Asian Res. 42, 31–45 (2013)CrossRefGoogle Scholar
  34. Saba, F., Jabeen, S., Akbar, H., Mohyud-Din, S.T.: Modified alternative (G\(^{\prime }\)/G)-expansion method to general Sawada–Kotera equation of fifth-order. J. Egypt. Math. Soc. 23, 416–423 (2015)MathSciNetCrossRefMATHGoogle Scholar
  35. Taha, W.M., Noorani, M.S.M., Hashim, I.: New exact solutions of sixth-order thin-film equation. J. King Saud Univ. Sci. 26, 75–78 (2014)CrossRefGoogle Scholar
  36. Wazwaz, A.M.: Travelling wave solutions for combined and double combined sine–cosine-Gordon equations by the variable separated ODE method. Appl. Math. Comput. 177, 755–760 (2006)MathSciNetMATHGoogle Scholar
  37. Yildirim, A., Pinar, Z.: Application of the exp-function method for solving nonlinear reaction–diffusion equations arising in mathematical biology. Comput. Math. Appl 60, 1873–1880 (2010)MathSciNetCrossRefMATHGoogle Scholar
  38. Zhang, X., Zhao, J., Liu, J., Tang, B.: Homotopy perturbation method for two dimensional time-fractional wave equation. Appl. Math. Model. 38, 5545–5552 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Jalil Manafian
    • 1
  • Mehdi Fazli Aghdaei
    • 2
  • Manouchehr Zadahmad
    • 3
  1. 1.Young Researchers and Elite Club, Ilkhchi BranchIslamic Azad UniversityIlkhchiIran
  2. 2.Department of MathematicsPayame Noor UniversityTehranIran
  3. 3.Department of Computer Engineering, Ilkhchi BranchIslamic Azad UniversityIlkhchiIran

Personalised recommendations