Skip to main content
SpringerLink
Log in

A Strang Splitting Approach Combined with Chebyshev Wavelets to Solve the Regularized Long-Wave Equation Numerically

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to obtain the numerical solutions of regularized long-wave (RLW) equation with various initial and boundary conditions. The performance of the proposed method measured with three different test problems. To measure the accuracy of the method, \(L_{2}\) and \(L_{\infty }\) error norms and the \(I_{1},I_{2,}\) \(I_{3}\) invariants are computed. The results of the computations are compared with the existing numerical and exact solutions in the literature.

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

Similar content being viewed by others

References

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

    Google Scholar 

  2. Lin, J., Xie, Z., Zhou, J.: High-order compact difference scheme for the regularized long wave equation. Commun. Numer. Methods Eng. 23, 135–156 (2007)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Shokri, A., Dehghan, M.: A meshless method the using radial basis functions for numerical solution of the regularized long wave equation. Numer. Methods Partial Differ. Equ. 26, 807–825 (2010)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Dag, İ.: Least square quadratic B-spline finite element method for the regularized long wave equation. Comput. Methods Appl. Mech. Eng. 182, 205–215 (2000)

    MATH  Google Scholar 

  10. Dag, I., Özer, M.N.: Approximation of RLW equation by least square cubic B-spline finite element method. Appl. Math. Model. 25, 221–231 (2001)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Dogan, A.: Numerical solution of RLW equation using linear finite elements within Galerkins method. Appl. Math. Model. 26, 771–783 (2002)

    MATH  Google Scholar 

  13. Jain, P.C., Shankar, R., Singh, T.V.: Numerical solutions of RLW equation. Commun. Numer. Methods Eng. 9, 587–594 (1993)

    Google Scholar 

  14. Raslan, K.R.: A computational method for the regularized long wave (RLW) equation. Appl. Math. Comput. 167(2), 1101–1118 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Dağ, İ., Korkmaz, A., Saka, B.: Cosine expansion based differential quadrature algorithm for numerical solution of the RLW equation. Numer. Methods Part D E 26(3), 544–560 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Saka, B., Dağ, İ.: Quartic B-spline collocation algorithms for numerical solution of the RLW equation. Numer. Methods Part D E 23(3), 731–751 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Saka, B., Dağ, İ., Irk, D.: Quintic B-spline collocation method for numerical solutions of the RLW equation. Anziam J. 49(3), 389–410 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  19. Saka, B., ahin, A., Dağ, İ.: B-spline collocation algorithms for numerical solution of the RLW equation. Numer. Methods Part D E 27, 581–607 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Siraj-ul-Islam, Sirajul Haq, Ali, Arshed: A meshfree method for the numerical solution of the RLW equation. J. Comput. Appl. Math. 223, 997–1012 (2009)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Görgülü, 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)

    Google Scholar 

  23. Yagmurlu, N.M., Ucar, Y., Celikkaya, I.: Operator splitting for numerical solutions of the RLW equation. J. Appl. Anal. Comput. 8(5), 1494–1510 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Dehghan, M., Salehi, R.: The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas. Comput. Phys. Commun. 182, 2540–2549 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math. 286, 211–231 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Oruç, Ö.: A computational method based on Hermite wavelets for two-dimensional Sobolev and regularized long wave equations in fluids. Numer. Methods Partial Differ. Equ. 34(5), 1693–1715 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Dehghan, M., Shafieeabyaneh, N.: Local radial basis function-finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher-Kolmogorov equations. Eng. Comput. (2019). https://doi.org/10.1007/s00366-019-00877-z. (In press)

    Article  Google Scholar 

  28. Abbaszadeh, M., Dehghan, M.: The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau regularized long wave (RRLW) equation with error analysis. Appl. Anal. 97, 1129–1153 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions. Comput. Math. Appl. 68, 212–237 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Assari, P., Dehghan, M.: Application of dual-Chebyshev wavelets for the numerical solution of boundary integral equations with logarithmic singular kernels. Eng. Comput. 35, 175–190 (2019)

    Google Scholar 

  31. Heydari, M.H., Hooshmandasl, M.R., Cattani, C.: A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation. Proc. Math. Sci. 128(2), 26 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Micula, S., Cattani, C.: On a numerical method based on wavelets for Fredholm–Hammerstein integral equations of the second kind. Math. Methods Appl. Sci. 41(18), 9103–9115 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Oruç, Ö., Bulut, F., Esen, A.: Chebyshev wavelet method for numerical solutions of coupled Burgers’ equation. Hacet. J. Math. Stat. 48(1), 1–16 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Oruç, Ö., Esen, A., Bulut, F.: A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers’ equation. Discrete Contin. Dyn. Syst. S 12(3), 533–542 (2019). https://doi.org/10.3934/dcdss.2019035

    Article  MathSciNet  MATH  Google Scholar 

  35. Oruç, Ö.: An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation. Eng. Comput. (2019). https://doi.org/10.1007/s00366-019-00734-z

    Article  Google Scholar 

  36. Cattani, C.: Haar wavelet-based technique for sharp jumps classification. Math. Comput. Model. 39(2–3), 255–278 (2004)

    MathSciNet  MATH  Google Scholar 

  37. Oruç, Ö., Bulut, F., Esen, A.: A numerical treatment based on Haar wavelets for coupled KdV equation. Int. J. Optim. Control Theor. Appl. (IJOCTA) 7(2), 195–204 (2017)

    MathSciNet  Google Scholar 

  38. Oruç, Ö.: A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations. Comput. Math. Appl. 77(7), 1799–1820 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Cattani, C., Rushchitskii, Y.Y.: Cubically nonlinear elastic waves: wave equations and methods of analysis. Int. Appl. Mech. 39(10), 1115–1145 (2003)

    MathSciNet  Google Scholar 

  40. Cattani, C.: On the existence of wavelet symmetries in archaea DNA. Comput. Math. Methods Med. 2012, 673934 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Daubechies, I.: Ten Lectures on Wavelet. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

  42. Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M.: A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type. Appl. Math. Model. 38, 1597–1606 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)

    MathSciNet  MATH  Google Scholar 

  44. Rubin, S.G., Graves, R.A.: Cubic spline approximation for problems in fluid mechanics, NASA TR R-436., Washington DC (1975)

  45. Eaton, J.W., Bateman, D., Hauberg, S., Wehbring, R.: GNU Octave version 5.1.0 manual: a high-level interactive language for numerical computations (2019). https://www.gnu.org/software/octave/doc/v5.1.0/. Accessed 8 Aug 2020

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

    MathSciNet  MATH  Google Scholar 

  47. Hunter, J.D.: Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9(3), 90–95 (2007)

    Google Scholar 

  48. Oruç, Ö., Bulut, F., Esen, A.: Numerical solutions of regularized long wave equation By Haar wavelet method. Mediterr. J. Math. 13(5), 3235–3253 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors received no funding for this work. Additionally, the authors would like to thank to the anonymous reviewers for their helpful comments and suggestions which improve to the paper.

Author information

Authors and Affiliations

  1. Eğil Vocational and Technical Anatolian High School, Diyarbakir, Turkey

    Ömer Oruç

  2. Department of Mathematics, İnönü University, 44280, Malatya, Turkey

    Alaattin Esen

  3. Department of Physics, İnönü University, 44280, Malatya, Turkey

    Fatih Bulut

Authors
  1. Ömer Oruç
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alaattin Esen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fatih Bulut
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ömer Oruç.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oruç, Ö., Esen, A. & Bulut, F. A Strang Splitting Approach Combined with Chebyshev Wavelets to Solve the Regularized Long-Wave Equation Numerically. Mediterr. J. Math. 17, 140 (2020). https://doi.org/10.1007/s00009-020-01572-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-020-01572-w

Keywords

Mathematics Subject Classification

Search

Navigation