Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-08-18T20:25:42.388Z Has data issue: false hasContentIssue false
  • English
  • Français

A Survey in Mathematics for Industry: Two-timing and matched asymptotic expansions for singular perturbation problems

Published online by Cambridge University Press:  12 August 2011

R. E. O'MALLEY Jr.  and
E. KIRKINIS
R. E. O'MALLEY Jr.
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA email: omalley@amath.washington.edu, kirkinis@amath.washington.edu
E. KIRKINIS
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA email: omalley@amath.washington.edu, kirkinis@amath.washington.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383–410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations.

Keywords

Asymptotic methodsPerturbation theoryBoundary layersMultiple scalesAmplitude equations

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 6 , December 2011 , pp. 613 - 629
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Bender, C. M. & Orszag, S. A. (1978) Advanced Mathematical Methods for Scientists and Engineers. International Series in Pure and Applied Mathematics. McGraw-Hill, New York.Google Scholar
[2]Chen, L.-Y., Goldenfeld, N. & Oono, Y. (1994) Renormalization group theory for global asymptotic analysis. Phys. Rev. Lett. 73, 1311–1315.CrossRefGoogle ScholarPubMed
[3]Chen, L.-Y., Goldenfeld, N. & Oono, Y. (1996) Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. E 54 (1), 376–394.Google Scholar
[4]Cochran, J. A. (1962) Problems in Singular Perturbation Theory, PhD thesis, Stanford University, Palo Alto, CA.Google Scholar
[5]Cole, J. D. (1968) Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, MA.Google Scholar
[6]Dahlquist, G., Edsberg, L., Skollermo, G. & Soderlind, G. (1982) Are the numerical methods and software satisfactory for Chemical-Kinetics? Lect. Notes Math. 968, 149–164.Google Scholar
[7]deJager, E. M. Jager, E. M. & Jiang, F. (1996) The Theory of Singular Perturbations. North-Holland Series in Applied Mathematics and Mechanics, Vol. 42. North-Holland, Amsterdam.Google Scholar
[8]Du, L.-L., Wu, X.-H. & Chen, S. (2010) A novel mathematical modelling of multiple scales for a class of two-dimensional singular perturbed problems. Presented during the International Conference on Singular Perturbation Theory and Application, June 2010, Shanghai.Google Scholar
[9]Franz, S. & Roos, H.-G. (2011) The capriciousness of numerical methods for singular perturbations. SIAM Rev. 53, 157–173.CrossRefGoogle Scholar
[10]Goldenfeld, N., Martin, O. & Oono, Y. (1989) Intermediate asymptotics and renormalization group theory. J. Sci. Comput. 4, 355–372.Google Scholar
[11]Il'in, A. M. (1992) Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Translations of Mathematical Monographs, Vol. 102. American Mathematical Society, Providence, RI. Translated from the Russian by V. Minachin [V. V. Minakhin].Google Scholar
[12]Johnson, R. S. (2005) Singular Perturbation Theory. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York.Google Scholar
[13]Kevorkian, J. (1961) The Uniformly Valid Asymptotic Approximations to the Solutions of Certain Nonlinear Ordinary Differential Equations, PhD thesis, California Institute of Technology, Pasadena, CA.Google Scholar
[14]Kevorkian, J. & Cole, J. D. (1996) Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, Vol. 114. Springer-Verlag, New York.CrossRefGoogle Scholar
[15]Kirkinis, E. (2008a) Reduction of amplitude equations by the renormalization group approach. Phys. Rev. E 77 (1), 011105.CrossRefGoogle ScholarPubMed
[16]Kirkinis, E. (2008b) Secular series and renormalization group for amplitude equations. Phys. Rev. E 78 (1), 032104.CrossRefGoogle Scholar
[17]Kirkinis, E. (2010) Amplitude equations and asymptotic expansions for multi-scale problems. Asymptotic Anal. 67 (1-2), 1–16.CrossRefGoogle Scholar
[18]Kuzmak, G. E. (1959) Asymptotic solutions of nonlinear second order differential equations with variable coefficients. J. Appl. Math. Mech. 23, 730–744.CrossRefGoogle Scholar
[19]Lagerstrom, P. A. (1988) Matched Asymptotic Expansions. Applied Mathematical Sciences, Vol. 76. Springer-Verlag, New York. Ideas and Techniques.Google Scholar
[20]Levinson, N. (1950) The first boundary value problem for ϵΔu + A(x, y)u x + B(x, y)u y + C(x, y)u = D(x, y) for small ϵ. Ann. Math. 51 (2), 428–445.Google Scholar
[21]Lomov, S. A. (1992) Introduction to the General Theory of Singular Perturbations. Translations of Mathematical Monographs, Vol. 112. American Mathematical Society, Providence, RI.Google Scholar
[22]Mahony, J. J. (1961/1962) An expansion method for singular perturbation problems. J. Austral. Math. Soc. 2, 440–463.CrossRefGoogle Scholar
[23]Nayfeh, A. H. (1973) Perturbation Methods, John Wiley & Sons, New York-London-Sydney. Pure and Applied Mathematics.Google Scholar
[24]Olver, F. W. J. (1974) Asymptotics and Special Functions, Academic Press, New York-London. Computer Science and Applied Mathematics.Google Scholar
[25]OøMalley, R. E. Jr., (1991) Singular Perturbation Methods for Ordinary Differential Equations. Applied Mathematical Sciences, Vol. 89. Springer-Verlag, New York.Google Scholar
[26]O'Malley, R. E. Jr. & Kirkinis, E. (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124 (4), 383–410.CrossRefGoogle Scholar
[27]Smith, D. R. (1985) Singular-Perturbation Theory. An Introduction with Applications, Cambridge University Press, Cambridge, England, UK.Google Scholar
[28]Van Dyke, M. (1964) Perturbation Methods in Fluid Mechanics. Applied Mathematics and Mechanics, Vol. 8, Academic Press, New York.Google Scholar
[29]Vasil'eva, A. B., Butuzov, V. F. & Kalachev, L. V. (1995) The Boundary Function Method for Singular Perturbation Problems. SIAM Studies in Applied Mathematics, Vol. 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. With a foreword by Robert E. O'Malley, Jr.Google Scholar
[30]Wasow, W. (1944) On the asymptotic solution of boundary value problems for ordinary differential equations containing a parameter. J. Math. Phys. Mass. Inst. Tech. 23, 173–183.Google Scholar
[31]Wechselberger, M. (2005) Existence and bifurcation of canards in 3 in the case of a folded node. SIAM J. Appl. Dyn. Syst. 4 (1), 101–139 (electronic).CrossRefGoogle Scholar
[32]Wong, R. (2009) A panoramic view of asymptotics. In: Foundations of Computational Mathematics Hong Kong 2008. London Mathematical Society Lecture Note Series, Vol. 363. Cambridge University Press, England, UK, pp. 190–235.CrossRefGoogle Scholar
[33]Zauderer, E. (1989) Partial Differential Equations of Applied Mathematics, Pure and Applied Mathematics, 2nd ed., John Wiley & Sons, New York.Google Scholar