Skip to main content
SpringerLink
Log in

A meshless method for numerical solution of the coupled Schrödinger-KdV equations

  • Published:
Computing Aims and scope Submit manuscript

Abstract

This paper presents meshless method using RBF collocation scheme for the coupled Schrödinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are studied. L 2 and L ∞ error norms between the results and exact solution is used as a performance measure. Moreover the results of numerical experiments are presented, and are compared with the findings of Finite Element method, finite difference Crank–Nicolson (CN) scheme and analytical solution to confirm the good accuracy of the presented scheme.

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.

Similar content being viewed by others

References

  1. Abdou MA, Soliman AA (2005) New applications of variational iteration method. Phys D Nonlinear Phenomena 211: 161–182

    MathSciNet  Google Scholar 

  2. Appert K, Vaclavik J (1977) Dynamics of coupled solitons. Phys Fluids 20: 1845–1849

    Article  MATH  Google Scholar 

  3. Bai D, Zhang L (2009) The finite element method for the coupled Schrödinger-KdV equations. Phys Lett A 373: 2237–2244

    Article  MathSciNet  Google Scholar 

  4. Fasshauer GE (2005) Meshless methods. In: Rieth M, Schommers W (eds) Handbook of theoretical and computational nanotechnology. American Scientific publishers, Stevenson Ranch

    Google Scholar 

  5. Franke C, Schaback R (1998) Solving partial differential equations by collocation using radial basis functions. Appl Math Comput 93: 73–82

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo B, Shen L (1982) The periodic initial value problem and the initial value problem for the system of KdV equation coupling with nonlinear Schrödinger equations. In: Proceedings of DD-3 Symposium, Chang Chun 417–435

  7. Hon YC, Cheung KF, Mao XZ, Kansa EJ (1999) A multiquadric solution for the shallow water equations. ASCE J Hydraul Eng 125: 524–533

    Article  Google Scholar 

  8. Hon YC, Mao XZ (1998) An efficient numerical scheme for Burgers equation. Appl Math Comput 95: 37–50

    Article  MathSciNet  MATH  Google Scholar 

  9. Hon YC, Schaback R (2001) On unsymmetric collocation by radial basis functions. Appl Math Comput 119: 177–186

    Article  MathSciNet  MATH  Google Scholar 

  10. Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics-I. Comput Math Appl 19: 127–145

    Article  MathSciNet  MATH  Google Scholar 

  11. Kansa EJ (1999) Motivation for using radial basis functions to solve PDEs, Tech. rep., Lawrence Livermore Laboratory

  12. Kaya D, El-Sayed M (2003) On the solution of the coupled Schrödinger-KdV equation by the decomposition method. Phys Lett A 313: 82–88

    Article  MathSciNet  MATH  Google Scholar 

  13. Kucukarslan S (2009) Homotopy perturbation method for coupled Schrödinger-KdV equation. Nonlinear Anal Real World Appl 10: 2264–2271

    Article  MathSciNet  Google Scholar 

  14. Micchelli CA (1986) Interpolation of scatterded data: distance matrix and conditionally positive definite functions. Constr Approx 2: 11–22

    Article  MathSciNet  MATH  Google Scholar 

  15. Power H, Barraco V (2002) A comparison analysis between unsymmetric and symmetric radial basis function numerical solution of the partial differential equations. Comput Math Appl 43: 551–583

    Article  MathSciNet  MATH  Google Scholar 

  16. Schaback R (2008) The meshless kernel-based method of lines for solving nonlinear evolution equations. Preprint, Göttingen

  17. Wendland H (1998) Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J Approx Theory 93: 258–272

    Article  MathSciNet  MATH  Google Scholar 

  18. Wendland H (2005) Scattered data approximation. Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  19. Werner H, Arndt H (1985) Gewöhnliche Differentialgleichungen, Hochschultexte. Springer, Berlin

    Google Scholar 

  20. Wloka JT (1987) Partial differential equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  21. Wu Z, Schaback R (1993) Local error estimates for radial basis function interpolation of scattered data. IMA J Numer Anal 13: 13–27

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang M (2007) Radial basis function interpolation in Sobolev space and its applications. J Comput Math 25: 201–210

    MathSciNet  Google Scholar 

  23. Zhang X, Song KZ, Lu MW, Liu X (2000) Meshless methods based on collocation with radial basis functions. Comput Mech 26: 333–343

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 1684613114, Iran

    A. Golbabai & A. Safdari-Vaighani

  2. Department of Mathematics and Statistics, Allameh Tabatabai University, Tehran, Iran

    A. Safdari-Vaighani

Authors
  1. A. Golbabai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Safdari-Vaighani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Golbabai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Golbabai, A., Safdari-Vaighani, A. A meshless method for numerical solution of the coupled Schrödinger-KdV equations. Computing 92, 225–242 (2011). https://doi.org/10.1007/s00607-010-0138-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-010-0138-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation