Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Necessary conditions for applicability of Poincaré-Lighthill perturbation theory

Author: Peter D. Usher
Journal: Quart. Appl. Math. 28 (1971), 463-471
MSC: Primary 34.53
DOI: https://doi.org/10.1090/qam/273146
MathSciNet review: 273146
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish necessary conditions for the applicability of Poincaré--Lighthill (or coordinate stretching) perturbation theory to ordinary differential equations. The criteria are simple consequences of a unique modification of the classical theory of coordinate stretching. The usefulness of the new approach and the role of the criteria of applicability are illustrated by means of simple examples.

References [Enhancements On Off] (What's this?)

  • [1] H. Poincaré, Les méthodes nouvelles de la mecanique celeste, vol. 1, Gauthier--Villars, Paris, 1892; reprint, Dover, New York, 1957
  • [2] M. J. Lighthill, A technique for rendering approximate solutions to physical problems uniformly valid, Philos. Mag. (7) 40 (1949), 1179–1201. MR 0033941
  • [3] Peter D. Usher, Coordinate stretching and interface location. II. A new 𝑃𝐿 expansion, J. Computational Phys. 3 (1968/1969), 29–39. MR 0261114
  • [4] H. S. Tsien, The Poincaré-Lighthill-Kuo method, Advances in applied mechanics, vol. IV, Academic Press Inc., New York, N.Y., 1956, pp. 281–349. MR 0079929
  • [5] Wolfgang Wasow, On the convergence of an approximation method of M. J. Lighthill, J. Rational Mech. Anal. 4 (1955), 751–767. MR 0073785
  • [6] Craig Comstock, On Lighthill’s method of strained coordinates, SIAM J. Appl. Math. 16 (1968), 596–602. MR 0227559, https://doi.org/10.1137/0116048
  • [7] Nicolas Minorsky, Nonlinear oscillations, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962. MR 0137891

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 34.53

Retrieve articles in all journals with MSC: 34.53

Additional Information

DOI: https://doi.org/10.1090/qam/273146
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society