Skip to main content
SpringerLink
Log in

A Simple Multi-scale Procedure for Both Oscillatory and Boundary Layer Problems

  • Chapter
  • First Online:
Historical Developments in Singular Perturbations

  • 1104 Accesses

Abstract

In this final chapter, we shall develop a simple multiscale method by applying it to a varied sequence of examples.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

References

  1. M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons (Cambridge University Press, Cambridge, 2011)

    Book  Google Scholar 

  2. T. Bakri, Y.A. Kutnetsov, F. Verhulst, E. Doedel, Multiple solutions of a generalized singularly perturbed Bratu problem. Int. J. Bifurcation Chaos 22(1250095), (2012)

    Google Scholar 

  3. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978)

    MATH  Google Scholar 

  4. D.J. Benney, A.C. Newell, Sequential time closures for interacting random waves. J. Math. and Phys. 46, 363–393 (1967)

    MATH  MathSciNet  Google Scholar 

  5. F.J. Bourland, R. Haberman, The modulated phase shift for strongly nonlinear, slowly-varying, and weakly-damped oscillators. SIAM J. Appl. Math. 48, 737–748 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Chen, R.E. O’Malley, Jr., Multiple solutions of a singularly perturbed boundary value problem arising in chemical reactor theory. Lecture Notes in Math. 280, 314–319 (1972)

    Article  MathSciNet  Google Scholar 

  7. L.-Y. Chen, N. Goldenfeld, Y. Oono, Renormalization group and singular perturbations: Multiple-scales, boundary layers, and reductive perturbation theory. Phys. Rev. E 54(1), 376–394 (1996)

    Article  Google Scholar 

  8. H. Cheng, The renormalized two-scale method. Stud. Appl. Math 113, 381–387 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan Press, Boston, 2007)

    Google Scholar 

  10. H. Cheng, T.-T. Wu, An aging spring. Stud. Appl. Math. 49, 183–185 (1970)

    MATH  MathSciNet  Google Scholar 

  11. H. Chiba, C 1 approximation of vector fields based on the renormalization group method. SIAM J. Appl. Dyn. Syst. 7(3), 895–932 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. W. Eckhaus, On modulation equations of Ginzburg-Landau type, in ICIAM 91, ed. by R.E. O’Malley, Jr. (SIAM, Philadelphia, 1992), pp. 83–98

    Google Scholar 

  13. R. Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th edn. (Pearson Prentice-Hall, Upper Saddle River, NJ, 2012)

    Google Scholar 

  14. P.W. Hemker, A Numerical Study of Stiff Two-Point Boundary Value Problems (Mathematical Centre, Amsterdam, 1977)

    Google Scholar 

  15. F.C. Hoppensteadt, Singular perturbations on the infinite interval. Trans. Am. Math. Soc. 123, 521–535 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  16. C.J. Howls, Exponential asymptotics and boundary value problems: keeping both sides happy at all orders. Proc. R. Soc. Lond. A 466, 2771–2794 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. R.S. Johnson, Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering (Springer, New York, 2005)

    Google Scholar 

  18. J. Kevorkian, J.D. Cole, Multiple Scale and Singular Perturbation Methods (Springer, New York, 1996)

    Book  MATH  Google Scholar 

  19. E. Kirkinis, R.E. O’Malley, Jr., Renormalization group and multiple scales for the Swift-Hohenberg and Kuramoto-Sivashinski equations. (to appear)

    Google Scholar 

  20. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984)

    Book  MATH  Google Scholar 

  21. G.E. Latta, Singular perturbation problems. PhD thesis, California Institute of Technology, Pasadena, CA, 1951

    Google Scholar 

  22. V.P. Maslov, The Complex WKB Method for Nonlinear Equations I: Linear Theory (Birkhäuser, Basel, 1994)

    Book  MATH  Google Scholar 

  23. J.B. McLeod, S. Sadhu, Existence of solutions and asymptotic analysis of a class of singularly perturbed boundary value problems. Adv. Differ. Equ. 18(9–10), 825–848 (2013)

    MATH  MathSciNet  Google Scholar 

  24. P.D. Miller, Applied Asymptotic Analysis (Amer. Math. Soc., Providence, RI, 2006)

    MATH  Google Scholar 

  25. J.V. Moloney, A.C. Newell, Nonlinear Optics (Westview, Boulder, CO, 2004)

    MATH  Google Scholar 

  26. J.A. Morrison, Comparison of the modified method of averaging and the two-variable expansion procedure. SIAM Rev. 8, 66–85 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  27. B. Mudavanhu, R.E. O’Malley, Jr., A renormalization group method for nonlinear oscillators. Stud. Appl. Math. 107, 63–79 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. J.A. Murdock, Perturbations: Theory and Methods (Wiley-Interscience, New York, 1991)

    MATH  Google Scholar 

  29. A.H. Nayfeh, Perturbation Methods (Wiley, New York, 1973)

    MATH  Google Scholar 

  30. B. Noble, M.A. Hussain, Multiple scaling and a related expansion method with applications, in Lasers, Molecules, and Methods, ed. by J.O. Hirschfelder, R.L. Wyatt, R.D. Coalson (Wiley, New York, 1989), pp. 83–136

    Google Scholar 

  31. R.E. O’Malley, Jr., Boundary value problems for linear systems of ordinary differential equations involving many small parameters. J. Math. Mech. 18, 835–855 (1969)

    MATH  MathSciNet  Google Scholar 

  32. R.E. O’Malley, Jr., E. Kirkinis, Examples illustrating the use of renormalization techniques for singularly perturbed differential equations. Stud. Appl. Math. 122, 105–122 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. R.E. O’Malley, Jr., E. Kirkinis, A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383–410 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. R.E. O’Malley, Jr., E. Kirkinis, Two-timing and matched asymptotic expansions for singular perturbation problems. Eur. J. Appl. Math. 22, 613–629 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. R.E. O’Malley, Jr., E. Kirkinis, Variation of parameters and the renormalization group method. Stud. Appl. Math. 133, (2014)

    Google Scholar 

  36. R.E. O’Malley, Jr., D.B. Williams, Deriving amplitude equations for weakly-nonlinear oscillators and their generalizations. J. Comp. Appl. Math. 190, 3–21 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. L.A. Ostrovsky, A.S. Potapov, Modulated Waves: Theory and Applications (Johns Hopkins, Baltimore, 1999)

    MATH  Google Scholar 

  38. A.J. Roberts, Modelling Emergent Dynamics in Complex Systems (to appear)

    Google Scholar 

  39. L.A. Rubenfeld, On a derivative-expansion technique and some comments on multiple scaling in the asymptotic approximation of solutions of certain differential equations. SIAM Rev. 20, 79–105 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  40. P.J. Schmid, D.S. Henningson, Stability and Transition in Shear Flows, vol. 142 of Applied Mathematical Sciences (Springer, New York, 2001)

    Google Scholar 

  41. F. Verhulst, Profits and pitfalls of timescales in asymptotics. SIAM Rev. 56, (2014)

    Google Scholar 

  42. W. Wasow, On the asymptotic solution of boundary value problems for ordinary differential equations containing a parameter. J. Math. and Phys. 23, 173–183 (1944)

    MathSciNet  Google Scholar 

  43. E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd edn. (Wiley, New York, 1989)

    MATH  Google Scholar 

  44. W. Zhang, H.L. Ho, Y.H. Qian, F.B. Gao, A refined asymptotic perturbation method for nonlinear dynamical systems. Arch. Appl. Mech. 84, 591–606 (2014)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Seattle, WA, USA

    Robert E. O’Malley

Authors
  1. Robert E. O’Malley
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

O’Malley, R.E. (2014). A Simple Multi-scale Procedure for Both Oscillatory and Boundary Layer Problems. In: Historical Developments in Singular Perturbations. Springer, Cham. https://doi.org/10.1007/978-3-319-11924-3_6

Download citation

Publish with us

Policies and ethics

Search

Navigation