Skip to main content
Log in

An application of convergence acceleration techniques to a class of two-point boundary value problems on a semi-infinite domain

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

For boundary value problems posed on unbounded domains it is often appropriate to impose a boundary condition at “infinity”. For certain classes of boundary value problem obvious numerical difficulties can be avoided by truncating the unbounded domain and solving a sequence of finite domain problems instead. We introduce a novel technique which is straightforward to implement and which exploits information contained in this sequence in order to extrapolate to the unbounded case. The technique introduces a new and interesting application of a variety of convergence acceleration algorithms.

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

Access this article

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. R. Bellman,Stability Theory of Differential Equations (McGraw-Hill, 1953).

  2. E.S. Birger and N.B. Lyalikova, Discovery of the solutions of certain systems of differential equations with a given condition at infinity, I. Trans. Zh. v. M. 5, 6 (1965) 979.

    Google Scholar 

  3. C. Brezinski,Accéleration de la Convergence en Analyse Numérique, Lecture Notes in Mathematics, no. 584 (Springer, 1977).

  4. C. Brezinski,Algorithmes d'Accéleration de la Convergence, Etude Numérique (Technip, Paris, 1978).

    MATH  Google Scholar 

  5. C. Brezinski, A general extrapolation algorithm, Numer. Math. 35 (1980) 175.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. Brezinski, Algorithm 585 — A subroutine for general interpolation and extrapolation problems, ACM Trans. Math. Software 8 (1982) 290.

    Article  MATH  Google Scholar 

  7. L. Collatz,The Numerical Treatment of Differential Equations (Springer, 1951).

  8. J.P. Delahaye, Automatic selection of sequence transformations, Math. Comp. 37 (1981) 197.

    Article  MATH  MathSciNet  Google Scholar 

  9. J.P. Delahaye and B. Germaine-Bonne, The set of logarithmically convergent sequences cannot be accelerated, SIAM J. Numer. Anal. 19 (1982) 840.

    Article  MATH  MathSciNet  Google Scholar 

  10. R.A. Fisher, Ann. Eugen. 7 (1936) 355.

    Article  Google Scholar 

  11. B. Germaine-Bonne, Transformations de Suites, Rev. Française Automat. Informat. Recherche Oper. 7 (1973) 84.

    Google Scholar 

  12. F.R. de Hoog and R. Weiss, The numerical solution of boundary value problems with an essential singularity, SIAM J. Numer. Anal. 16 (1979) 637.

    Article  MATH  MathSciNet  Google Scholar 

  13. H.B. Keller,Numerical Solution of Two-Point Boundary Value Problems, no. 24, CBMS/NSF Regional Conference Series on Applied Mathematics (SIAM, Philadelphia, 1976).

    Book  Google Scholar 

  14. M. Lentini and H.B. Keller, Boundary value problems on semi-infinite intervals and their numerical solution, SIAM J. Numer. Anal. 17 (1980) 577.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. Levin, Development of non-linear transformations for improving convergence of sequences, Int. J. Comp. Math. 3 (1973) 371.

    Article  MATH  Google Scholar 

  16. P. Markowich, A theory for the approximation of solutions of boundary value problems on infinite intervals, SIAM J. Math. Anal. 13 (1982) 484.

    Article  MATH  MathSciNet  Google Scholar 

  17. R.M.M. Mattheij, On the computation of solutions of boundary value problems on infinite intervals, Math. Comp. 48 (1987) 533.

    Article  MATH  MathSciNet  Google Scholar 

  18. D.A. Smith and W.F. Ford, Acceleration of linear and logarithmic convergence, SIAM J. Numer. Anal. 2 (1979) 223.

    Article  MathSciNet  Google Scholar 

  19. D.A. Quinney, On computing travelling wave solutions in a model for the Belousov Zhabotinskii reaction, J. Inst. Math. Appl. 23 (1979) 193.

    Article  MATH  MathSciNet  Google Scholar 

  20. D.A. Quinney, The numerical computation of travelling wavespeeds in reaction diffusion equations,Proc. 3rd Int. Seminar on the Numerical Treatment of Differential Equations, Halle, Germany (1985) p. 127.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by C. Brezinski

Rights and permissions

Reprints and permissions

About this article

Cite this article

Croft, A. An application of convergence acceleration techniques to a class of two-point boundary value problems on a semi-infinite domain. Numer Algor 2, 307–320 (1992). https://doi.org/10.1007/BF02139471

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02139471

Keywords

Navigation