Skip to main content
SpringerLink
Log in

A New Multi-Domain Spectral Method for Korteweg-de Vries Equation on The Whole Line

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Some composite generalized Laguerre-Legendre quasi-orthogonal approximation results on the whole line are established. A novel multi-domain composite generalized Laguerre-Legendre spectral scheme is provided for the Korteweg-de Vries equation on the whole line. The scheme features mobile common boundaries of the adjacent subdomains, which better fits the soliton wave governed by the Korteweg-de Vries equation. Convergence of the proposed scheme is proved. Numerical results show the efficiency of the scheme and coincide well with theoretical analysis.

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

Similar content being viewed by others

References

  1. Boyd, J.P.: The rate of convergence of Hermite function series. Math. Comp. 35, 1309–1316 (1980)

    Article  MathSciNet  Google Scholar 

  2. Christov, C.I.: A complete orthonormal system of functions in \(L^2(-\infty ,\infty )\) space. SIAM J. Appl. Math. 42, 1337–1344 (1982)

    Article  MathSciNet  Google Scholar 

  3. Coulaud, O., Funaro, D., Kavian, O.: Laguerre spectral approximation of elliptic problems in exterior domains. Comp. Mech. in Appl. Mech. and Engi. 80, 451–458 (1990)

  4. Johnson, Fox C., M., Guo B.-Y., Tang T.: Combined Hermite spectral-finite difference method for the Fokker-Planck equation. Math. Comp. 71, 1497–1528 (2001)

  5. Funaro, D.: Polynomial Approxiamtions of Differential Equations. Springer-Verlag, Berlin (1992)

    Book  Google Scholar 

  6. Funaro, D., Kavian, O.: Approximation of some diffusion evolution equation in unbounded domains by Hermite function. Math. Comp. 57, 597–619 (1999)

    Article  MathSciNet  Google Scholar 

  7. Initial-boundary value problem of the Korteweg-de Vries equation: Ton B.- A. J. Diff. Equa. 25, 288–309 (1977)

    Article  Google Scholar 

  8. Guo, B.-Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)

    Book  Google Scholar 

  9. Guo, B.-Y., Sun, T., Zhang, C.: Jacobi and Laguerre quasi-orthogonal approximations and related interpolations. Math. Comp. 82, 413–441 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Guo B.-Y.: Some developments in spectral methods for nonlinear partial differential equations in unbounded domains. In: Differential Geometry and Related Topics. Singapore: World Scientific. 68–90, (2002)

  11. Guo, B.-Y.: Some progress in spectral methods. Sci. China Math. 56, 2411–2438 (2013)

    Article  MathSciNet  Google Scholar 

  12. Guo, B.-Y.: Spectral and pseudospectral methods for unbounded domains. Sci. China Math. 45, 975–1024 (2015)

    MATH  Google Scholar 

  13. Guo, B.-Y., Ma, H.-P.: Composite Legendre-Laguerre approximation in unbounded domains. J. Comp. Math. 19, 101–112 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Ma, H.-P., Guo, B.-Y.: Composite Legendre-Laguerre pseudospectral approximation in unbounded domains. IMA J. Numer. Anal. 21, 587–602 (2001)

    Article  MathSciNet  Google Scholar 

  15. Guo, B.-Y., Shen, J.: Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numer. Math. 86, 635–654 (2000)

    Article  MathSciNet  Google Scholar 

  16. Chai, G., Wang, T.-J.: Mixed generalized Hermite-Fourier spectral method for Fokker-Planck equation of periodic field. Appl. Numer. Math. 133, 25–40 (2018)

    Article  MathSciNet  Google Scholar 

  17. Guo, B.-Y., Shen, J.: On spectral approximations using modified Legendre rational functions: Application to Korteweg-de Vries equation on the half line. Indina Math. J. 50, 181–204 (2001)

    Article  MathSciNet  Google Scholar 

  18. Zhang, C., Guo, B.-Y., Sun, T.: Laguerre Spectral Method for High Order Problems. Numer. Math. Theor. Meth. Appl. 6(3), 520–537 (2013)

    Article  MathSciNet  Google Scholar 

  19. Shen J., Tang T., Wang L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Vol. 41, Springer-Verlag, Berlin, Heidelberg(2011)

  20. Guo, B.-Y., Wang, T.-J.: Composite Generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel. Math. Comp. 78, 129–151 (2009)

    Article  MathSciNet  Google Scholar 

  21. Guo, B.-Y., Wang, T.-J.: Composite generalized Laguerre- Legendre spectral method for exterior problems. Adv. in Comp. Math. 32, 393–429 (2010)

    Article  Google Scholar 

  22. Guo, B.-Y., Zhang, X.-Y.: A new generalized Laguerre spectral approximation and its applications. J. Comput. Appl. Math. 181, 342–363 (2005)

    Article  MathSciNet  Google Scholar 

  23. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541 (1970)

    MATH  Google Scholar 

  24. Mastroianni, G., Occorsio, D.: Lagrange interpolation at Laguerre zeros in some weighted uniform spaces. Acta Math. Hungar. 91, 27–52 (2001)

    Article  MathSciNet  Google Scholar 

  25. Sun, T., Yi, L.-J.: A new Galerkin spectral element method for fourth-order boundary value problems. International Journal of Computer Mathematics 93(6), 915–928 (2016)

    Article  MathSciNet  Google Scholar 

  26. Shen, J., Wang, L.-L.: Some recent advances on spectral methods for unbounded domains. Commun. Comput. Phys. 5, 195–241 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Shen, J., Wang, L.-L.: Laguerre and composite Legendre-Laguerre dual-Petrov-Galerkin methods for third-order equations. Discr. Contin. Dyna. sys.-ser. B. 6(6), 1381–1402 (2006)

    MathSciNet  MATH  Google Scholar 

  28. Guo, B.-Y., Wang, T.-J.: Composite generalized Laguerre- Legendre spectral method for fourth-order exterior problems. J. Sci. Comput. 44, 255–285 (2010)

    Article  MathSciNet  Google Scholar 

  29. Wang, T.-J.: Composite generalized Laguerre spectral method for nonlinear Fokker-Planck equation on the whole line. Math. Meth. Appl. Sci. 40, 1462–1474 (2017)

    Article  MathSciNet  Google Scholar 

  30. Wang, T.-J., Guo, B.-Y.: Composite generalized Laguerre-Legendre pseudospectral method for Fokker-Planck equation in an infinite channel. Appl. Numer. Math. 58, 1448–1466 (2008)

    Article  MathSciNet  Google Scholar 

  31. Jiao, Y.-J., Wang, T.-J., Zhang, Q.: A Fully Discrete Spectral Method for Fisher’s Equation on The Whole Line. East Asian J. Appl. Math. 6(4), 400–415 (2016)

    Article  MathSciNet  Google Scholar 

  32. Yu, X.-H., Guo, B.-Y.: Spectral element method for mixed inhomogeneous boundary value problems of fourth order. J. Sci. Comput. 61, 673–701 (2014)

    Article  MathSciNet  Google Scholar 

  33. Sun, T., Wang, T.-J.: Multi-domain Decomposition Pseudospectral Method for Nonlinear Fokker-Planck Equations. Commun. Appl. Math. Comput. 1(2), 231–252 (2019)

    Article  MathSciNet  Google Scholar 

  34. Zhang, C., Guo, B.-Y.: Domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations on unbounded domains. J. Sci. Comput. 53, 451–480 (2012)

    Article  MathSciNet  Google Scholar 

  35. Zhang, Z.-Q., Ma, H.-P.: A rational spectral method for the KdV equation on the half line. J. Comput. Appl. Math. 230, 614–625 (2009)

    Article  MathSciNet  Google Scholar 

  36. Chen, J.-B., Pelinovsky, D.E.: Rogue periodic waves of the mKdV equation. London Math. Soci. Nonlinea. 31, 1955–1980 (2018)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees, Dr. Yue Zhu of Hangzhou Dianzi University and Dr. Lu-yu Wang of Zhejiang University for their help on polishing this work.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471003, China

    Tian-jun Wang

Authors
  1. Tian-jun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tian-jun Wang.

Ethics declarations

Competing Interests

The author confirm that this article has no competing interests with any institution or individual.

Additional information

Publisher's Note

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

Tian-jun Wang is supported in part by NSF of China [grant numbers 12171141, 11371123] and NSF of Henan Province [grant numbers 202300410156]

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Tj. A New Multi-Domain Spectral Method for Korteweg-de Vries Equation on The Whole Line. J Sci Comput 92, 32 (2022). https://doi.org/10.1007/s10915-022-01887-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01887-8

Keywords

Mathematics Subject Classification

Search

Navigation