Skip to main content
SpringerLink
Log in

Numerical Investigations of Physical Processes for Regularized Long Wave Equation

  • Conference paper
  • First Online:
Progress in Intelligent Decision Science (IDS 2020)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1301))

Included in the following conference series:

Abstract

In this study, a numerical approach is used to investigate the solution of the regularized long wave (RLW) equation. A newly proposed quartic trigonometric-tension (QTT) B–spline based collocation scheme is constructed for spatial discretization. Then, the equation is transformed into a time-dependent system of differential equations, which is discretized by the Crank-Nicolson scheme. Approximate solutions of the RLW equation have successfully attained by the fully discretized system. The motions of the conservation laws of the RLW equation have also been computed numerically. Solitary wave propagation, interaction of two solitary waves, wave undulation and wave generations are simulated and results are compared to the existing literature.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Peregrine, D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25(2), 321–330 (1966)

    Article  MathSciNet  Google Scholar 

  2. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 272(1220), 47–78 (1972)

    MathSciNet  MATH  Google Scholar 

  3. Olver, P.J.: Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Phil. Soc. 85(143–160), 4 (1979)

    Google Scholar 

  4. Dogan, A.: Numerical solution of RLW equation using linear finite elements within Galerkin’s method. Appl. Math. Model. 26(7), 771–783 (2002)

    Article  Google Scholar 

  5. Gardner, L.R.T., Gardner, G.A., Dag, I.: A B-spline finite element method for the RLW equation. Commun. Numer. Methods Eng. 11, 59–98 (1995)

    Article  Google Scholar 

  6. Dogan, A.: Numerical solution of regularized long wave equation using Petrov-Galerkin method. Commun. Numer. Methods Eng. 17(7), 485–494 (2001)

    Article  MathSciNet  Google Scholar 

  7. Mohammadi, R.: Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation. Chin. Phys. B 24(5), 050206 (2015)

    Article  Google Scholar 

  8. Dag, I., Saka, B., Irk, D.: Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput. 159(2), 373–389 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Mei, L., Cehn, Y.: Numerical solutions of RLW equation using Galerkin method with extrapolation techniques. Comput. Phys. Commun. 183(8), 1609–1616 (2012)

    Article  MathSciNet  Google Scholar 

  10. Saka, B., Dag, I.: A numerical solution of the RLW equation by Galerkin method using quartic B-splines. Commun. Numer. Methods Eng. 24, 1339–1361 (2008)

    Article  MathSciNet  Google Scholar 

  11. Saka, B., Dag, I., Dogan, A.: Galerkin method for the numerical solution of the RLW equation using quadratic B-splines. Int. J. Comput. Math. 81(6), 727–739 (2004)

    Article  MathSciNet  Google Scholar 

  12. Gorgulu, M.Z., Dag, I., Irk, D.: Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method. Chin. Phys. B 26(8), 080202 (2017)

    Article  Google Scholar 

  13. Irk, D., Yildiz, P.K., Gorgulu, M.Z.: Quartic trigonometric B-spline algorithm for numerical solution of the regularized long wave equation. Turk. J. Math. 43(1), 112–125 (2019)

    Article  MathSciNet  Google Scholar 

  14. Zaki, S.I.: Solitary waves of the splitted RLW equation. Comput. Phys. Commun. 138(1), 80–91 (2001)

    Article  Google Scholar 

  15. Esen, A., Kutluay, S.: Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput. 174(2), 833–845 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Dag, I., Saka, B., Irk, D.: Galerkin method for the numerical solution of the RLW equation using quintic B-splines. J. Comput. Appl. Math. 190(1–2), 532–547 (2006)

    Article  MathSciNet  Google Scholar 

  17. Lu, C., Huang, W., Qiu, J.: An adaptive moving mesh finite element solution of the regularized long wave equation. J. Sci. Comput. 74(1), 122–144 (2018)

    Article  MathSciNet  Google Scholar 

  18. Jain, P.C., Shankar, R., Singh, T.V.: Numerical solution of regularized long-wave equation. Commun. Numer. Methods Eng. 9(7), 579–586 (1993)

    Article  MathSciNet  Google Scholar 

  19. Bhardwaj, D., Shankar, R.: A computational method for regularized long wave equation. Comput. Math. Appl. 40(12), 1397–1404 (2000)

    Article  MathSciNet  Google Scholar 

  20. Kutluay, S., Esen, A.: A finite difference solution of the regularized long-wave equation. Math. Probl. Eng. 2006, 1–14 (2006)

    Google Scholar 

  21. Inan, B., Bahadır, A.R.: A fully implicit finite difference scheme for the regularized long wave equation. Gen. Math. Notes 33(2), 40 (2016)

    Google Scholar 

  22. Oruc, O., Bulut, F., Esen, A.: Numerical solutions of regularized long wave equation by Haar wavelet method. Mediterr. J. Math. 13(5), 3235–3253 (2016)

    Article  MathSciNet  Google Scholar 

  23. Dağ, İ., Korkmaz, A., Saka, B.: Cosine expansion-based differential quadrature algorithm for numerical solution of the RLW equation. Numer. Methods Partial Diff. Equat. Int. J. 26(3), 544–560 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Korkmaz, A., Dağ, İ.: Numerical simulations of boundary-forced RLW equation with cubic B-spline-based differential quadrature methods. Arab. J. Sci. Eng. 38, 1151–1160 (2013)

    Article  MathSciNet  Google Scholar 

  25. Gardner, L.R.T., Gardner, G.A., Dogan, A.: A least-squares finite element scheme for the RLW equation. Commun. Numer. Methods Eng. 12(11), 795–804 (1996)

    Article  MathSciNet  Google Scholar 

  26. Dağ, İ.: Least-squares quadratic B-spline finite element method for the regularised long wave equation. Comput. Methods Appl. Mech. Eng. 182(1–2), 205–215 (2000)

    Article  Google Scholar 

  27. Gu, H., Chen, N.: Least-squares mixed finite element methods for the RLW equations. Numer. Methods Partial Diff. Equat. Int. J. 24(3), 749–758 (2008)

    Article  MathSciNet  Google Scholar 

  28. Dag, I., Ozer, N.: Approximation of the RLW equation by the least square cubic B spline finite element method. Appl. Math. Model. 25(3), 221–231 (2001)

    Article  Google Scholar 

  29. Guo, B.Y., Cao, W.M.: The Fourier pseudospectral method with a restrain operator for the RLW equation. J. Comput. Phys. 74(1), 110–126 (1988)

    Article  MathSciNet  Google Scholar 

  30. Djidjeli, K., Price, W.G., Twizell, E.H., Cao, Q.: A linearized implicit pseudo-spectral method for some model equations: the regularized long wave equations. Commun. Numer. Methods Eng. 19(11), 847–863 (2003)

    Article  MathSciNet  Google Scholar 

  31. Nuruddeen, R.I., Aboodh, K.S., Ali, K.K.: Investigating the tangent dispersive solitary wave solutions to the equal width and regularized long wave equations. J. King Saud Univ.-Sci. 32(1), 677–681 (2020)

    Article  Google Scholar 

  32. Alinia, N., Zarebnia, M.: A numerical algorithm based on a new kind of tension B-spline function for solving Burgers-Huxley equation. Numer. Algorithms 82, 1121–1142 (2019)

    Article  MathSciNet  Google Scholar 

  33. Hong, Q., Wang, Y., Gong, Y.: Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation. Numer. Methods Partial Diff. Equat. 36(2), 394–417 (2020)

    Article  MathSciNet  Google Scholar 

  34. Maharana, N., Nayak, A.K., Jena, P.: A comparative study of regularized long wave equations (RLW) using collocation method with cubic B-spline. In: Decision Science in Action, pp. 203–216. Springer, Singapore (2019)

    Google Scholar 

  35. Mittal, R.C., Rohila, R.: A fourth order cubic B-spline collocation method for the numerical study of the RLW and MRLW equations. Wave Motion 80, 47–68 (2018)

    Article  MathSciNet  Google Scholar 

  36. Wang, G., Fang, M.: Unified and extended form of three types of splines. J. Comput. Appl. Math. 216(2), 498–508 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science and Letters, Eskisehir Osmangazi University, Eskisehir, Turkey

    Ozlem Ersoy Hepson

  2. Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

    Gülsemay Yiğit

Authors
  1. Ozlem Ersoy Hepson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gülsemay Yiğit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ozlem Ersoy Hepson .

Editor information

Editors and Affiliations

  1. Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

    Tofigh Allahviranloo

  2. Faculty of Engineering and Natural Sciences, Bahçeşehir University, Istanbul, Turkey

    Soheil Salahshour

  3. Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

    Nafiz Arica

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Hepson, O.E., Yiğit, G. (2021). Numerical Investigations of Physical Processes for Regularized Long Wave Equation. In: Allahviranloo, T., Salahshour, S., Arica, N. (eds) Progress in Intelligent Decision Science. IDS 2020. Advances in Intelligent Systems and Computing, vol 1301. Springer, Cham. https://doi.org/10.1007/978-3-030-66501-2_58

Download citation

Publish with us

Policies and ethics

Search

Navigation