Skip to main content

Applications of nonstandard analysis to boundary value problems in singular perturbation theory

Part I: Theory of Singular Perturbations

Part of the Lecture Notes in Mathematics book series (LNM,volume 942)

Keywords

  • Singular Perturbation
  • Integral Curve
  • Integral Curf
  • Slow Manifold
  • Nonstandard Analysis

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BENOIT E., CALLOT J.L. DIENER F. et DIENER M., Chasse au canard. Collectanea Mathematica 31 (1980).

    Google Scholar 

  2. BOBO SEKE, Ombres des graphes de fonctions continues. Thèse Strasbourg (1981).

    Google Scholar 

  3. CALLOT, J.L. Bifurcation du portrait de phase pour des équations différentielles du second ordre. Thèse Strasbourg (1981).

    Google Scholar 

  4. CARRIER G.F., and PEARSON C.E., Ordinary Differential Equations. Ginn/Blaisdell, Waltham, Mass. (1968).

    MATH  Google Scholar 

  5. CARTIER P., Perturbations singulières des équations différentielles ordinaires et analyse non standard. Seminaire Bourbaki, No 580, Novembre 1981.

    Google Scholar 

  6. CODDINGTON E.A. and LEVINSON N., A Boundary Value Problem for a Nonlinear Differential Equation with a Small Parameter. Proc. Amer. Math. Soc. 3 (1952), 73–81.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. DIEKMANN D. and HILHORST D., How Many Jumps? Variationnal Characterisation of the Limit Solution of a Singular Perturbation Problem. Geometrical Approaches to Differential Equation, Lecture Notesin Math No 810, Springer Verlag (1980), 159–180.

    Google Scholar 

  8. DIENER F., Méthode du plan d'observabilité. Thèse Strasbourg 1981.

    Google Scholar 

  9. DIENER M., Etude générique des canards. Thèse Strasbourg (1981).

    Google Scholar 

  10. DORR F.W., PARTER S.V. and SHAMPINE L.F., Applications of the Maximum Principle to Singular Perturbation Problems. SIAM Review 15 (1973), 43–88.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. ECKHAUS W., Asymptotic Analysis of Singular Perturbations, North-Holland (1979).

    Google Scholar 

  12. FIFE P.C., Transition Layers in Singular Perturbation Problems. Jour. Diff. Eqns. 15 (1974), 77–105.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. FIFE P.C., Two Point Boundary Value Problems Admitting Interior Transition Layers (unpublished).

    Google Scholar 

  14. FRAENKEL L.E., On the Method of Matched Asymptotic Expansions I, II, III. Proc. Camb. Phil. Soc. 65 (1969), 209–284.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. GOZE M., Perturbations de Structures Géométriques. Thèse Mulhouse (1982).

    Google Scholar 

  16. HARTHONG J., Vision macroscopique de phénomènes périodiques. Thèse Strasbourg (1981).

    Google Scholar 

  17. HOWES F.A., Boundary-Interior Layers Interactions in Nonlinear Singular Perturbation Theory. Mem. Amer. Math. Soc. 15 (1989), No 203.

    Google Scholar 

  18. HOWES F.A. and PARTER S.V., A Model Nonlinear Problem Having a Continuous Locus of Singular Points. Studies Appl. Math. 58 (1978), 249–262.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. KEDEM G., PARTER S.V. and STEUERWALT M., The Solutions of a Model Nonlinear Singular Perturbation Problem Having a Continuous Locus of Singular Points. Studies Appl. Math. 63 (1980), 119–146.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. KEVORKIAN J. and COLE J.D., Perturbation Methods in Applied Mathematics. Springer Verlag, New-York (1981).

    CrossRef  MATH  Google Scholar 

  21. KOPELL N. and PARTER S.V., A Complete Analysis of a Model Nonlinear Singular Perturbation Problem Having a Continuous Locus of Singular Points. Advances Appl. Math. 2 (1981), 212–238.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. LUTZ R. and GOZE M., Nonstandard Analysis—A Practical Guide with Applications. Lecture Notes in Math. No 881, Springer Verlag (1981).

    Google Scholar 

  23. LUTZ R. et SARI T., Sur le comportement asymptotique des solutions dans un problème aux limites non linéaire, C.R. Acad. Sc. Paris 292 (1981), 925–928.

    MathSciNet  MATH  Google Scholar 

  24. NAYFEH A.H., Perturbation Methods. Wiley Intersciences (1973).

    Google Scholar 

  25. NELSON E., Internal Set Theory: A New Approach, to Nonstandard Analysis. Bull. Amer. Math. Soc. 83 (1977), 1165–1198.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. O'MALLEY R.E., Jr., Introduction to Singular Perturbations. Academic Press (1974).

    Google Scholar 

  27. O'MALLEY R.E., Jr., Phase Plane Solutions to some Singular Perturbation Problems. Journ. Math. Anal. Appl. 54 (1976), 449–466.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. REEB G., Séance-débat sur l'Analyse non Standard. Gazette des Mathématiciens, 8 (1977), 8–14.

    MATH  Google Scholar 

  29. REEB G., La mathématique non standard vieille de soixante ans? Publication IRMA-Strasbourg (1979).

    Google Scholar 

  30. REEB G., Mathématique non standard (Essai de Vulgarisation). Bulletin APMEP 328 (1981), 259–273.

    Google Scholar 

  31. REEB G., TROESCH A. et URLACHER E., Analyse non Standard. Séminaire LOI-Publication IRMA-Strasbourg (1978).

    Google Scholar 

  32. ROBINSON A., Nonstandard Analysis, North Holland, Amsterdam (1966).

    Google Scholar 

  33. SARI T., Sur le comportement asymptotique des solutions dans un problème aux limites semi-linéaire. C.R. Acad. Sc. Paris 292 (1981) 867–870.

    MathSciNet  MATH  Google Scholar 

  34. TROESCH A., Etude qualitative de systèmes différentiels: une approche basée sur l'analyse non standard. Thèse Strasbourg (1981).

    Google Scholar 

  35. URLACHER E., Oscillations de relaxation et analyse non standard. Thèse Strasbourg (1981).

    Google Scholar 

  36. VASIL'EVA A.B., Asymptotic Behaviour of Solutions to Certain Problems Involving Nonlinear Differential Equations Containing a Small Parameter Multiplying the Highest Derivatives. Russian Math. Surveys 18 (1963), 13–84.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. WASOW W.R., Asymptotic Expansions for Ordinary Differential Equations. Intersciences, New-York (1965).

    MATH  Google Scholar 

  38. WASOW W.R., The Capriciousness of Singular Perturbations. Nieuv. Arch. Wisk. 18 (1970), 190–210.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1982 Springer-Verlag

About this paper

Cite this paper

Lutz, R., Sari, T. (1982). Applications of nonstandard analysis to boundary value problems in singular perturbation theory. In: Eckhaus, W., de Jager, E.M. (eds) Theory and Applications of Singular Perturbations. Lecture Notes in Mathematics, vol 942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094743

Download citation

  • DOI: https://doi.org/10.1007/BFb0094743

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11584-7

  • Online ISBN: 978-3-540-39332-0

  • eBook Packages: Springer Book Archive