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Nonlinear Dynamics

, Volume 95, Issue 1, pp 669–684 | Cite as

Application of the ITEM for solving three nonlinear evolution equations arising in fluid mechanics

  • Cevat Teymuri Sendi
  • Jalil ManafianEmail author
  • Hasan Mobasseri
  • Mohammad Mirzazadeh
  • Qin Zhou
  • Ahmet Bekir
Original Paper
  • 74 Downloads

Abstract

In this paper, by introducing new approach, the improved \(\tan \left( \phi (\xi )/2\right) \)-expansion method (ITEM) is further extended into the Vakhnenko–Parkes (VP) equation, the generalized regularized-long-wave (GRLW) equation and the symmetric regularized-long-wave (SRLW) equation in fluid mechanic. We extended the ITEM proposed by Manafian et al. (Int J Appl Comput Math 2:243–268, 2016) to construct new types of soliton wave solutions of nonlinear partial differential equations (NPDEs). The merit of the presented method is finding the further solutions of the considering problems including soliton, periodic, kink and kink-singular wave solutions. Comparing our new results with other results shows that our results give the further solutions. The results of applying this procedure (Figs. 1, 2, 3, 4, 5, 6) to the studied cases show the high efficiency of the new technique. Finally, these solutions might play an important role in engineering, physics and applied mathematics fields.

Keywords

Improved \(\tan (\phi (\xi )/2)\)-expansion method Vakhnenko–Parkes equation Generalized regularized-long-wave equation Symmetric regularized-long-wave equation 

Mathematics Subject Classification

35Q79 35Q51 35Q35 

References

  1. 1.
    Abazari, R.: Application of \(G^{\prime }/G\)-expansion method to travelling wave solutions of three nonlinear evolution equation. Comput. Fluids 39, 1957–1963 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aghdaei, M.F., Manafian, J.: Optical soliton wave solutions to the resonant Davey–Stewartson system. Opt. Quantum Electron. 48, 1–33 (2016)CrossRefGoogle Scholar
  3. 3.
    Alam, M.N., Akbar, M.A., Mohyud-Din, S.T.: A novel (G’/G)-expansion method and its application to the Boussinesq equation. Chin. Phys. B 23, 020203 (2014)CrossRefGoogle Scholar
  4. 4.
    Ali, S., Rizvi, S.T.R., Younis, M.: Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients. Nonlinear Dyn. 82, 1755–1762 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cheema, N., Younis, M.: New and more general traveling wave solutions for nonlinear Schrödinger equation. Waves Random Complex Media 26, 84–91 (2016)CrossRefGoogle Scholar
  6. 6.
    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 Fractals 24, 745–757 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Z. Naturforschung A 64a, 420–430 (2009)Google Scholar
  8. 8.
    Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Diff. Equ. J. 26, 448–479 (2010)MathSciNetzbMATHGoogle Scholar
  9. 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 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dehghan, M., Manafian, J., 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. 11.
    Guo, P.F., Zhang, L.W., Liew, K.M.: Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method. Appl. Math. Comput. 240, 91–101 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Islam, W., Younis, M., Rizvi, S.T.R.: Optical solitons with time fractional nonlinear Schrödinger equation and competing weakly nonlocal nonlinearity. Optik 130, 562–567 (2017)CrossRefGoogle Scholar
  13. 13.
    Jafari, H., Kadkhoda, N., Khalique, C.M.: Travelling wave solutions of nonlinear evolution equations using the simplest equation method. Comput. Math. Appl. 64, 2084–2088 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jawad, A.J.M., Petković, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869–877 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Liu, L., Zheng, L., Liu, F., Zhang, X.: Exact solution and invariant for fractional Cattaneo anomalous diffusion of cells in two-dimensional comb framework. Nonlinear Dyn. 89, 213–224 (2017)zbMATHCrossRefGoogle Scholar
  16. 16.
    Luo, X.-G., Wub, Q.-B., Zhang, B.-Q.: Revisit on partial solutions in the Adomian decomposition method: solving heat and wave equations. J. Math. Anal. Appl. 321, 353–363 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Ma, W.X.: Complexiton solutions to the Korteweg–de Vires equation. Phys. Lett. A 301, 35–44 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ma, W.X., Fuchssteiner, B.: Explicit and exact solutions of Kolmogorov–PetrovskII–Piskunov equation. Int. J. Nonlinear Mech. 31, 329–338 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ma, W.X., Maruno, K.: Complexiton solutions of the Toda lattice equation. Phys. A 343, 219–237 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ma, W.X., You, Y.: Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 357, 1753–1778 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Ma, W.X., You, Y.: Rational solutions of the Toda lattice equation in Casoratian form. Chaos Solitons Fractals 22, 395–406 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ma, W.X., Zhou, D.T.: Explicit exact solution of a generalized KdV equation. Acta Math. Scita. 17, 168–174 (1997)Google Scholar
  23. 23.
    Ma, W.X., Wu, H.Y., He, J.S.: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A 364, 29–32 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    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
  25. 25.
    Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolutionequations by the \(tan(\phi /2)\)-expansion method. Optik Int. J. Electron Opt. 127, 4222–4245 (2016)Google Scholar
  26. 26.
    Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015)CrossRefGoogle Scholar
  27. 27.
    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 (2015)Google Scholar
  28. 28.
    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, 1–35 (2015)zbMATHCrossRefGoogle Scholar
  29. 29.
    Manafian, J., Lakestani, M.: A new analytical approach to solve some the fractional-order partial differential equations. Indian J. Phys. 90, 1–16 (2016)CrossRefGoogle Scholar
  30. 30.
    Manafian, J., Lakestani, M.: Application of \(tan(\phi /2)\) -expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity. Optik Int. J. Electron Opt. 127, 2040–2054 (2016)Google Scholar
  31. 31.
    Manafian, J., Lakestani, M.: Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Opt. Quantum Electron. 48, 1–32 (2016)CrossRefGoogle Scholar
  32. 32.
    Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via \(tan(\phi /2)\)-expansion method. Optik Int. J. Electron Opt. 127, 5543–5551 (2016)Google Scholar
  33. 33.
    Manafian, J., Lakestani, M.: Optical soliton solutions for the Gerdjikov–Ivanov model via \(tan(\phi /2)\)-expansion method. Optik Int. J. Electron Opt. 127, 9603–9620 (2016)Google Scholar
  34. 34.
    Manafian, J., Zamanpour, I.: Exact travelling wave solutions of the symmetric regularized long wave (SRLW) using analytical methods. Stat. Opt. Inf. Comput. 2, 47–55 (2014)MathSciNetGoogle Scholar
  35. 35.
    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
  36. 36.
    Manafian, J., Lakestani, M., Bekir, A.: Comparison between the generalized tanh–coth and the \(G^{\prime }/G\)-expansion methods for solving NPDE’s and NODE’s. Pramana J. Phys. 87, 1–14 (2016)Google Scholar
  37. 37.
    Manafian, J., Aghdaei, M.F., Zadahmad, M.: Analytic study of sixth-order thin-film equation by \(tan(\phi /2)\)-expansion method. Opt. Quantum Electron. 48, 1–16 (2016)Google Scholar
  38. 38.
    Mirzazadeh, M., Eslami, M.: Exact multisoliton solutions of nonlinear Klein–Gordon equation in \(1+2\) dimensions. Eur. Phys. J. Plus 128, 1–9 (2015)Google Scholar
  39. 39.
    Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Some relatively new techniques for nonlinear problems. Math. Probl. Eng. 2009, 234849 (2009)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Traveling wave solutions of seventh-order generalized KdV equations using He’s polynomials. Int. J. Nonlinear Sci. Numer. Simul. 10, 223–229 (2009)zbMATHCrossRefGoogle Scholar
  41. 41.
    Mohyud-Din, S.T., Noor, M.A., Noor, K.I., Hosseini, M.M.: Variational iteration method for re-formulated partial differential equations. Int. J. Nonlinear Sci. Numer. Simul. 11, 87–92 (2010)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Mohyud-Din, S.T., Noor, M.A., Waheed, A.: Exp-function method for generalized traveling solutions of Calogero–Degasperis–Fokas equation. Z. Naturforschung A J. Phys. Sci. 65a, 78–84 (2010)Google Scholar
  43. 43.
    Mohyud-Din, S.T., Yildirim, A., Sariaydin, S.: Numerical soliton solution of the Kaup–Kupershmidt equation. Int. J. Numer. Methods Heat Fluid Flow 21, 272–281 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Mohyud-Din, S.T., Yildirim, A., Sezer, A.S.: Numerical soliton solutions of the improved Boussinesq equation. Int. J. Numer. Methods Heat Fluid Flow 21, 822–827 (2011)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Mohyud-Din, S.T., Negahdary, E., Usman, M.: A meshless numerical solution of the family of generalized fifth-order Korteweg–de Vries equations. Int. J. Numer. Methods Heat Fluid Flow 22, 641–658 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Mohyud-Din, S.T., Khan, Y., Naeem, F., Yildirim, A.: Exp-function method for solitary and periodic solutions of Fitzhugh Nagumo equations. Int. J. Numer. Methods Heat Fluid Flow 22, 335–341 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Noor, M.A., Mohyud-Din, S.T., Waheed, A.: Exp-function method for generalized traveling solutions of master partial differential equations. Acta Appl. Math. 104, 131–137 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Noor, M.A., Mohyud-Din, S.T., Waheed, A., Al-Said, E.A.: Exp-function method for traveling wave solutions of nonlinear evolution equations. Appl. Math. Comput. 216, 477–483 (2010)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Rashidi, M.M., Erfani, E.: A new analytical study of MHD stagnation-point flow in porous media with heat transfer. Comput. Fluids 40, 172–178 (2011)zbMATHCrossRefGoogle Scholar
  50. 50.
    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
  51. 51.
    Rashidi, M.M., Hayat, T., Keimanesh, M., Hendi, A.A.: New analytical method for the study of natural convection flow of a non-Newtonian fluid. Int. J. Numer. Methods Heat Fluid Flow 23, 436–450 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Sardar, A., Husnine, S.M., Rizvi, S.T.R., Younis, M., Ali, K.: Multiple travelling wave solutions for electrical transmission line model. Nonlinear Dyn. 82, 1317–1324 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Seyedi, S.H., Saray, B.N., Nobari, M.R.H.: Using interpolation scaling functions based on Galerkin method for solving non-Newtonian fluid flow between two vertical flat plates. Appl. Math. Comput. 269, 488–496 (2015)MathSciNetGoogle Scholar
  54. 54.
    Seyedi, S.H., Saray, B.N., Ramazani, A.: On the multiscale simulation of squeezing nanofluid flow by a high precision scheme. Powder Technol. 340, 264–273 (2018)CrossRefGoogle Scholar
  55. 55.
    Tang, G., Wang, S., Wang, G.: Solitons and complexitons solutions of an integrable model of (2+1)-dimensional Heisenberg ferromagnetic spin chain. Nonlinear Dyn. 88, 2319–2327 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Vakhnenko, V.O., Parkes, E.J.: The two loop soliton solution of the Vakhnenko equation. Nonlinearity 11, 1457–1464 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Vakhnenko, V.O., Parkes, E.J.: The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos Solitons Fractals 13, 1819–1826 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Vakhnenko, V.O., Parkes, E.J., Michtchenko, A.V.: The Vakhnenko equation from the viewpoint of the inverse scattering method for the KdV equation. Int. J. Diff. Equ. Appl. 1, 429–449 (2000)Google Scholar
  59. 59.
    Vakhnenko, V.O., Parkes, E.J., Michtchenko, A.V.: The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method. Chaos Solitons Fractals 45, 846–852 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Xu, F.: Application of exp-function method to symmetric regularized long wave (SRLW) equation. Phys. Lett. A. 372, 252–257 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Ye, Y., Song, J., Shen, S., Di, Y.: New coherent structures of the Vakhnenko–Parkes equation. Results Phys. 2, 170–174 (2012)CrossRefGoogle Scholar
  62. 62.
    Younis, M., Rizvi, S.T.R.: Dispersive dark optical soliton in (2+1)-dimensions by G’/G-expansion with dual-power law nonlinearity. Optik 126, 5812–5814 (2015)CrossRefGoogle Scholar
  63. 63.
    Younis, M., Rizvi, S.T.R.: Optical soliton like pulses in ring cavity fibers of carbon nanotubes. J. Nanoelectron. Optoelectron. 11, 276–279 (2015)CrossRefGoogle Scholar
  64. 64.
    Younis, M., Cheemaa, N., Mahmood, S.A., Rizvi, S.T.R.: On optical solitons: the chiral nonlinear Schrödinger equation with perturbation and Bohm potential. Opt. Quantum Electron. 48, 542–556 (2016)CrossRefGoogle Scholar
  65. 65.
    Yusufoǧlu, E., Bekir, A.: A travelling wave solution to the Ostrovsky equation. Appl. Math. Comput. 186, 256–260 (2007)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Zhao, X., Wang, L., Sun, W.: The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos Solitons Fractals 28, 448–453 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Tabriz BranchIslamic Azad UniversityTabrizIran
  2. 2.Young Researchers and Elite Club, Ilkhchi BranchIslamic Azad UniversityIlkhchiIran
  3. 3.Department of Management, Science and Research BranchIslamic Azad UniversityTehranIran
  4. 4.Department of Engineering Sciences, Faculty of Technology and Engineering, East of GuilanUniversity of GuilanRudsar-VajargahIran
  5. 5.School of Electronics and Information EngineeringWuhan Donghu UniversityWuhanPeople’s Republic of China
  6. 6.Neighbourhood of AkcaglanEskisehirTurkey

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