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On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering

  • T A Sulaiman
  • H Bulut
  • A Yokus
  • H M Baskonus
Original Paper
  • 80 Downloads

Abstract

The studies of the dynamic behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. In this study, we investigate the coupled Boussinesq equation which arises in the shallow water waves for two-layered fluid flow. The modified exp \((-\varphi (\zeta ))\)-expansion function method is utilized in reaching the solutions to this equation such as the topological kink-type soliton and singular soliton solutions. The interesting 2D and 3D graphics of the obtained analytical solutions in this study are presented. Via one of the reported analytical solutions, the finite forward difference method is used in obtaining the approximate numerical and exact solutions to this equation. The Fourier–Von Neumann analysis is used in checking the stability of the used numerical method with the studied model. The \(L_{2}\) and \(L_{\infty }\) error norms are computed. We finally present a comprehensive conclusion to this study.

Keywords

Shallow water waves Coupled Boussinesq equation MEFM FDM Exact and numerical approximations 

PACS Nos.

02.60.Lj 02.70.Wz 03.65.-w 

References

  1. [1]
    C T Sindi and J Manafian Eur. Phys. J. Plus 132 67 (2017)CrossRefGoogle Scholar
  2. [2]
    H M Baskonus, T A Sulaiman and H Bulut, Optik 131 1036 (2017)ADSCrossRefGoogle Scholar
  3. [3]
    H Bulut, T A Sulaiman and H M Baskonus Opt. Quant. Electron 48 564 (2016)CrossRefGoogle Scholar
  4. [4]
    H M Baskonus Nonlinear Dyn 86(1) 177 (2016)MathSciNetCrossRefGoogle Scholar
  5. [5]
    H Bulut, T A Sulaiman and B Demirdag Nonlinear Dyn 91 1985 (2018)CrossRefGoogle Scholar
  6. [6]
    T A Sulaiman, T Akturk, H Bulut and H M Baskonus Journal of Electromagnetic Waves and Applications 32(9) 1093 (2018)CrossRefGoogle Scholar
  7. [7]
    A Yokus, T A Sulaiman and H Bulut Opt. Quant. Electron 50 31 (2018)CrossRefGoogle Scholar
  8. [8]
    H M Baskonus, H Bulut and T A Sulaiman Eur. Phys. J. Plus 132 482 (2017)CrossRefGoogle Scholar
  9. [9]
    R Pal, H Kaur, T S Raju and C N Kumar Nonlinear Dyn 89(1) 617 (2017)CrossRefGoogle Scholar
  10. [10]
    E Fan and H Zhang Physics Letters A 246(5) 403(1998)ADSCrossRefGoogle Scholar
  11. [11]
    E Fan and J Zhang Physics Letters A 305(6) 383 (2002)ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    Z Zhang Rom Journ Phys 60 1384 (2015)Google Scholar
  13. [13]
    C Wang Nonlinear Dyn 85(2) 1119-1126 (2016)MathSciNetCrossRefGoogle Scholar
  14. [14]
    X J Yang, F Gao and H M Srivastava Computers and Mathematics with Applications 73(2) 203 (2017)MathSciNetCrossRefGoogle Scholar
  15. [15]
    H M Baskonus and H Bulut Waves in Random and Complex Media 26(2) 201 (2016)CrossRefGoogle Scholar
  16. [16]
    W Tan and Z Dai Nonlinear Dyn 89(4) 2723 (2017)CrossRefGoogle Scholar
  17. [17]
    M T Darvishi, M Najafi and A M Wazwaz Ocean Engineering 130 228 (2017)CrossRefGoogle Scholar
  18. [18]
    M Eslami, M A Mirzazadeh and A Neirameh Pramana 84(1) 3 (2015)ADSCrossRefGoogle Scholar
  19. [19]
    E V Krishnan, S Kumar and A Biswas Nonlinear Dyn 70(2) 1213 (2012)CrossRefGoogle Scholar
  20. [20]
    H Zeybek and S B G Karakoc SpringerPlus 5 199 (2016)CrossRefGoogle Scholar
  21. [21]
    H Borluk and G M Muslu Numerical Methods for Partial Differential Equations 31 995 (2014)CrossRefGoogle Scholar
  22. [22]
    W M K A W Zaimi, A Ishak and I Pop Journal of King Saud University-Science 25 143 (2013)CrossRefGoogle Scholar
  23. [23]
    Y Dereli Int J Nonlinear Sci 13(1) 28(2012)MathSciNetGoogle Scholar
  24. [24]
    H Demiray J App Eng Math 1(1) 49 (2011)MathSciNetGoogle Scholar
  25. [25]
    C Yong and L Biao Chinese Physics 13(3) 302 (2004)CrossRefGoogle Scholar
  26. [26]
    A R Seadawy and D Lu Results in Physics 6 590 (2016)ADSCrossRefGoogle Scholar
  27. [27]
    A R Seadawy and D Lu Results in Physics 7 43 (2017)ADSCrossRefGoogle Scholar
  28. [28]
    A R Seadawy Mathematical Methods and Applied Sciences 40(5) 1598 (2017)ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    A R Seadawy The European Physical Journal Plus 132 29 (2017)ADSCrossRefGoogle Scholar
  30. [30]
    A R Seadawy Optik 139 31 (2017)ADSCrossRefGoogle Scholar
  31. [31]
    A R Seadawy Journal of Electromagnetic Waves and Applications 1353 (14) 1353 (2017)MathSciNetCrossRefGoogle Scholar
  32. [32]
    A R Seadawy The Pramana-Journal of Physics 89(3) 49 (2017)ADSCrossRefGoogle Scholar
  33. [33]
    A R Seadawy Applied Mathematical Sciences 6(82) 4081 (2012)MathSciNetGoogle Scholar
  34. [34]
    Z Lu and H Zhang Chaos, Solitons and Fractals 19 527 (2004)ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    C Dai and Y Wang Chaos, Solitons and Fractals 39 350 (2009)ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    H Zhang Communications in Nonlinear Science and Numerical Simulation 12(5) 627 (2007)ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    H Bulut, H A Isik and T A Sulaiman, ITM Web of Conferences 3 01019 (2017)CrossRefGoogle Scholar
  38. [38]
    O A Ilhan, T A Sulaiman, H Bulut and H M Baskonus, Eur. Phys. J. Plus 133 27 (2018)CrossRefGoogle Scholar
  39. [39]
    R I Nuruddeen, L Muhammad, A M Nass and T A Sulaiman, Palestine Journal of Mathematics 7(1) 262 (2018)MathSciNetGoogle Scholar
  40. [40]
    H Bulut, T A Sulaiman and H M Baskonus, Optik 163 49 (2018)ADSCrossRefGoogle Scholar
  41. [41]
    J Zhang, F Jiang and X Zhao International Journal of Computer Mathematics 87(8) 1716 (2010)MathSciNetCrossRefGoogle Scholar
  42. [42]
    H M Baskonus, H Bulut and F B M Belgacem Journal of Computational and Applied Mathematics 312 257 (2017)MathSciNetCrossRefGoogle Scholar
  43. [43]
    E Aranda and P Pedregal Journal of the Franklin Institute 351(1) 3865 (2014)CrossRefGoogle Scholar
  44. [44]
    A Ashyralyev and Y Ozdemir Journal of the Franklin Institute 351(2) 602 (2014)MathSciNetCrossRefGoogle Scholar
  45. [45]
    N S Akbar, S Nadeem, R U Haq and Z H Khan Indian Journal of Physics 87(11) 1121 (2013)ADSCrossRefGoogle Scholar
  46. [46]
    H Bulut, T A Sulaiman, H M Baskonus and T Akturk Opt Quant Electron 50 134 (2018)CrossRefGoogle Scholar
  47. [47]
    A Yokus, H M Baskonus, T A Sulaiman and H Bulut Numerical Methods of Partial Differential Equations 34(1) 211 (2018)CrossRefGoogle Scholar
  48. [48]
    A R Seadawy Physics of Plasmas 21 052107 (2014)ADSCrossRefGoogle Scholar
  49. [49]
    A R Seadawy Computers and Mathematics with Applications 70(4) 345 (2015)MathSciNetCrossRefGoogle Scholar
  50. [50]
    A R Seadawy The European Physical Journal Plus 130 182 (2015)CrossRefGoogle Scholar
  51. [51]
    A R Seadawy Physica A 439 124 (2015)ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    A R Seadawy Appl. Math. Inf. Sci. 10(1) 209 (2016)MathSciNetCrossRefGoogle Scholar
  53. [53]
    A R Seadawy Computers and Mathematics with Applications 71(1) 201 (2016)MathSciNetCrossRefGoogle Scholar
  54. [54]
    A R Seadawy Physica A 455 44 (2016)ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    H M Baskonus, H Bulut and A Atangana Smart Materials and Structures 25(3) 035022 (2016)CrossRefGoogle Scholar
  56. [56]
    A Ciancio, H M Baskonus, T A Sulaiman and H Bulut Indian J Phys 92(10) 1281 (2018)ADSCrossRefGoogle Scholar
  57. [57]
    O A Ilhan, H Bulut, T A Sulaiman and H M Baskonus Indian J Phys 92(8) 999 (2018)ADSCrossRefGoogle Scholar
  58. [58]
    S Duran, M Askin and T A Sulaiman IJOCTA 7(3) 240 (2017)MathSciNetGoogle Scholar
  59. [59]
    A J M Jawad, M D Petkovic, P Laketa and A Biswas Scientia Iranica B 20(1) 179 (2013)Google Scholar
  60. [60]
    A Wazwaz Journal of Computational and Applied Mathematics 207(1) 18 (2007)ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    D H Peregrine J Fluid Mech 27 815 (1967)ADSCrossRefGoogle Scholar
  62. [62]
    A Yokus and D Kaya Journal of Nonlinear Science and Applications 10 3419 (2017)CrossRefGoogle Scholar

Copyright information

© Indian Association for the Cultivation of Science 2018

Authors and Affiliations

  1. 1.Department of MathematicsFirat UniversityElazigTurkey
  2. 2.Department of MathematicsFederal UniversityDutseNigeria
  3. 3.Department of Mathematics EducationFinal International UniversityKyreniaCyprus
  4. 4.Department of ActuaryFirat UniversityElazigTurkey
  5. 5.Department of Computer EngineeringMunzur UniversityTunceliTurkey

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