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A geometric approach to two-timing

Published online by Cambridge University Press:  24 October 2008

A. D. Gilbert
Affiliation:
University of Edinburgh

Extract

This paper is concerned with a particular method used in the calculation of asymptotic expansions of solutions to problems which contain a small parameter ∈. The physical idea underlying the method is that the parameter introduces new scales (of length and/or time) into the problem, and that the solution should incorporate these (see, for example, Cole (1)). This may be done in various ways: two-timing (two-scaling, multiple-scaling) does it by demanding that the solution is a function of all relevant scales. Consider, for example, a linear oscillator whose frequency is slowly adjusted, governed by

and subject to some initial condition. Account of the slowly varying frequency is taken by the dependence of Ω on ∈t, where 0 < ∈ ≪ 1, and we assume that Ω > 1 for all arguments. The rapid oscillatory behaviour anticipated in the solution is customarily described by a phase function Θ(isin;t)/∈, a generalization of Ωt the case that Ω is constant, while slow changes due to frequency variation occur on a scale ∈t.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Cole, J. D.Perturbation methods in applied mathematics. (New York, Blaisdell, 1968)Google Scholar
(2)Courant, R. and Hilbert, D.Methods of mathematical physics, vol. I (New York, Interscience, 1965).Google Scholar
(3)Courant, R. and Hilbert, D.Methods of mathematical physics, vol. II (New York, Interscience, 1962).Google Scholar
(4)Erdélyi, A.J. Inst. Math. Appl. 4 (1968), 113119.CrossRefGoogle Scholar
(5)Fink, J. P., Hall, W. S. and Hausrath, A. R.J. Differential Equations 15 (1974), 459498.CrossRefGoogle Scholar
(6)Levine, L. E. and Obi, W. C.Bull. Amer. Math. Soc. 82 (1976), 771774.CrossRefGoogle Scholar
(7)Luke, J. C.Proc. Roy. Soc. London, Ser. A 292 (1966), 403412.Google Scholar
(8)Reiss, E. L.SIAM Rev. 13 (1971), 189196.CrossRefGoogle Scholar
(9)Snow, R. E.J. Math. Anal. Appl. 54 (1976), 525.CrossRefGoogle Scholar
(10)Whitham, G. B.J. Fluid Mech. 44 (1970), 373395.CrossRefGoogle Scholar