Skip to main content

Advertisement

SpringerLink
Log in

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

  • Original Paper
  • Published:
Indian Journal of Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. C T Sindi and J Manafian Eur. Phys. J. Plus 132 67 (2017)

    Article  Google Scholar 

  2. H M Baskonus, T A Sulaiman and H Bulut, Optik 131 1036 (2017)

    Article  ADS  Google Scholar 

  3. H Bulut, T A Sulaiman and H M Baskonus Opt. Quant. Electron 48 564 (2016)

    Article  Google Scholar 

  4. H M Baskonus Nonlinear Dyn 86(1) 177 (2016)

    Article  MathSciNet  Google Scholar 

  5. H Bulut, T A Sulaiman and B Demirdag Nonlinear Dyn 91 1985 (2018)

    Article  Google Scholar 

  6. T A Sulaiman, T Akturk, H Bulut and H M Baskonus Journal of Electromagnetic Waves and Applications 32(9) 1093 (2018)

    Article  Google Scholar 

  7. A Yokus, T A Sulaiman and H Bulut Opt. Quant. Electron 50 31 (2018)

    Article  Google Scholar 

  8. H M Baskonus, H Bulut and T A Sulaiman Eur. Phys. J. Plus 132 482 (2017)

    Article  Google Scholar 

  9. R Pal, H Kaur, T S Raju and C N Kumar Nonlinear Dyn 89(1) 617 (2017)

    Article  Google Scholar 

  10. E Fan and H Zhang Physics Letters A 246(5) 403(1998)

    Article  ADS  Google Scholar 

  11. E Fan and J Zhang Physics Letters A 305(6) 383 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  12. Z Zhang Rom Journ Phys 60 1384 (2015)

    Google Scholar 

  13. C Wang Nonlinear Dyn 85(2) 1119-1126 (2016)

  14. X J Yang, F Gao and H M Srivastava Computers and Mathematics with Applications 73(2) 203 (2017)

    Article  MathSciNet  Google Scholar 

  15. H M Baskonus and H Bulut Waves in Random and Complex Media 26(2) 201 (2016)

    Article  Google Scholar 

  16. W Tan and Z Dai Nonlinear Dyn 89(4) 2723 (2017)

    Article  Google Scholar 

  17. M T Darvishi, M Najafi and A M Wazwaz Ocean Engineering 130 228 (2017)

    Article  Google Scholar 

  18. M Eslami, M A Mirzazadeh and A Neirameh Pramana 84(1) 3 (2015)

    Article  ADS  Google Scholar 

  19. E V Krishnan, S Kumar and A Biswas Nonlinear Dyn 70(2) 1213 (2012)

    Article  Google Scholar 

  20. H Zeybek and S B G Karakoc SpringerPlus 5 199 (2016)

    Article  Google Scholar 

  21. H Borluk and G M Muslu Numerical Methods for Partial Differential Equations 31 995 (2014)

    Article  Google Scholar 

  22. W M K A W Zaimi, A Ishak and I Pop Journal of King Saud University-Science 25 143 (2013)

    Article  Google Scholar 

  23. Y Dereli Int J Nonlinear Sci 13(1) 28(2012)

    MathSciNet  Google Scholar 

  24. H Demiray J App Eng Math 1(1) 49 (2011)

    MathSciNet  Google Scholar 

  25. C Yong and L Biao Chinese Physics 13(3) 302 (2004)

    Article  Google Scholar 

  26. A R Seadawy and D Lu Results in Physics 6 590 (2016)

    Article  ADS  Google Scholar 

  27. A R Seadawy and D Lu Results in Physics 7 43 (2017)

    Article  ADS  Google Scholar 

  28. A R Seadawy Mathematical Methods and Applied Sciences 40(5) 1598 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  29. A R Seadawy The European Physical Journal Plus 132 29 (2017)

    Article  ADS  Google Scholar 

  30. A R Seadawy Optik 139 31 (2017)

    Article  ADS  Google Scholar 

  31. A R Seadawy Journal of Electromagnetic Waves and Applications 1353 (14) 1353 (2017)

    Article  MathSciNet  Google Scholar 

  32. A R Seadawy The Pramana-Journal of Physics 89(3) 49 (2017)

    Article  ADS  Google Scholar 

  33. A R Seadawy Applied Mathematical Sciences 6(82) 4081 (2012)

    MathSciNet  Google Scholar 

  34. Z Lu and H Zhang Chaos, Solitons and Fractals 19 527 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  35. C Dai and Y Wang Chaos, Solitons and Fractals 39 350 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  36. H Zhang Communications in Nonlinear Science and Numerical Simulation 12(5) 627 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  37. H Bulut, H A Isik and T A Sulaiman, ITM Web of Conferences 3 01019 (2017)

  38. O A Ilhan, T A Sulaiman, H Bulut and H M Baskonus, Eur. Phys. J. Plus 133 27 (2018)

    Article  Google Scholar 

  39. R I Nuruddeen, L Muhammad, A M Nass and T A Sulaiman, Palestine Journal of Mathematics 7(1) 262 (2018)

    MathSciNet  Google Scholar 

  40. H Bulut, T A Sulaiman and H M Baskonus, Optik 163 49 (2018)

    Article  ADS  Google Scholar 

  41. J Zhang, F Jiang and X Zhao International Journal of Computer Mathematics 87(8) 1716 (2010)

    Article  MathSciNet  Google Scholar 

  42. H M Baskonus, H Bulut and F B M Belgacem Journal of Computational and Applied Mathematics 312 257 (2017)

    Article  MathSciNet  Google Scholar 

  43. E Aranda and P Pedregal Journal of the Franklin Institute 351(1) 3865 (2014)

    Article  Google Scholar 

  44. A Ashyralyev and Y Ozdemir Journal of the Franklin Institute 351(2) 602 (2014)

    Article  MathSciNet  Google Scholar 

  45. N S Akbar, S Nadeem, R U Haq and Z H Khan Indian Journal of Physics 87(11) 1121 (2013)

    Article  ADS  Google Scholar 

  46. H Bulut, T A Sulaiman, H M Baskonus and T Akturk Opt Quant Electron 50 134 (2018)

    Article  Google Scholar 

  47. A Yokus, H M Baskonus, T A Sulaiman and H Bulut Numerical Methods of Partial Differential Equations 34(1) 211 (2018)

    Article  Google Scholar 

  48. A R Seadawy Physics of Plasmas 21 052107 (2014)

    Article  ADS  Google Scholar 

  49. A R Seadawy Computers and Mathematics with Applications 70(4) 345 (2015)

    Article  MathSciNet  Google Scholar 

  50. A R Seadawy The European Physical Journal Plus 130 182 (2015)

    Article  Google Scholar 

  51. A R Seadawy Physica A 439 124 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  52. A R Seadawy Appl. Math. Inf. Sci. 10(1) 209 (2016)

    Article  MathSciNet  Google Scholar 

  53. A R Seadawy Computers and Mathematics with Applications 71(1) 201 (2016)

    Article  MathSciNet  Google Scholar 

  54. A R Seadawy Physica A 455 44 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  55. H M Baskonus, H Bulut and A Atangana Smart Materials and Structures 25(3) 035022 (2016)

    Article  Google Scholar 

  56. A Ciancio, H M Baskonus, T A Sulaiman and H Bulut Indian J Phys 92(10) 1281 (2018)

  57. O A Ilhan, H Bulut, T A Sulaiman and H M Baskonus Indian J Phys 92(8) 999 (2018)

  58. S Duran, M Askin and T A Sulaiman IJOCTA 7(3) 240 (2017)

    Article  MathSciNet  Google Scholar 

  59. A J M Jawad, M D Petkovic, P Laketa and A Biswas Scientia Iranica B 20(1) 179 (2013)

    Google Scholar 

  60. A Wazwaz Journal of Computational and Applied Mathematics 207(1) 18 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  61. D H Peregrine J Fluid Mech 27 815 (1967)

    Article  ADS  Google Scholar 

  62. A Yokus and D Kaya Journal of Nonlinear Science and Applications 10 3419 (2017)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Firat University, Elazig, Turkey

    T A Sulaiman

  2. Department of Mathematics, Federal University, Dutse, Jigawa, Nigeria

    T A Sulaiman & H Bulut

  3. Department of Mathematics Education, Final International University, Kyrenia, Cyprus

    H Bulut

  4. Department of Actuary, Firat University, Elazig, Turkey

    A Yokus

  5. Department of Computer Engineering, Munzur University, Tunceli, Turkey

    H M Baskonus

Authors
  1. T A Sulaiman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H Bulut
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A Yokus
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. H M Baskonus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H M Baskonus.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sulaiman, T.A., Bulut, H., Yokus, A. et al. On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering. Indian J Phys 93, 647–656 (2019). https://doi.org/10.1007/s12648-018-1322-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12648-018-1322-1

Keywords

PACS Nos.

Search

Navigation