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A repeated transformation in the asymptotic solution of linear differential systems

Published online by Cambridge University Press:  14 November 2011

M. S. P. Eastham
M. S. P. Eastham
Affiliation:
King's College (University of London), Strand, London WC2 2LS
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Synopsis

A sequence of transformations of the type Y = (I + o(1))Z is developed for the system Y′(x) = {Λ(x) + R(x)}Y(x), where Λ is diagonal. The transformations bring in the derivatives of R in succession until the Levinson form is obtained when a given derivative is reached. This theory covers rapidly varying coefficients and it extends results which are known for constant Λ.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 1-2 , 1986 , pp. 173 - 188
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Cassell, J. S.. An extension of the Liouville-Green asymptotic formula for oscillatory secondorder differential equations. Proc. Roy. Soc. Edinburgh Sect. A 100 (1985). 181–190.CrossRefGoogle Scholar
2Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
3Devinatz, A.. An asymptotic theorem for systems of linear differential equations. Trans. Amer. Math. Soc. 160 (1971), 353–363.Google Scholar
4Eastham, M. S. P.. Asymptotic theory for a critical class of fourth-order differential equations. Proc. Roy. Soc. London Ser. A 383 (1982), 465–476.Google Scholar
5Eastham, M. S. P.. Asymptotic formulae of Liouville–Green type for higher-order differential equations. J. London Math. Soc. (2) 28 (1983), 507–518.Google Scholar
6Eastham, M. S. P.. The asymptotic solution of linear differential systems. Mathematika 32 (1985), 131–138.Google Scholar
7Harris, W. A. and Lutz, D. A.. On the asymptotic integration of linear differential systems. J. Math. Anal. Appl. 48 (1974), 1–16.Google Scholar
8Harris, W. A. and Lutz, D. A.. Asymptotic integration of adiabatic oscillators. J. Math. Anal. Appl. 51 (1975), 76–93.CrossRefGoogle Scholar
9Harris, W. A. and Lutz, D. A.. A unified theory of asymptotic integration. J. Math. Anal. Appl. 57 (1977), 571–586.CrossRefGoogle Scholar
10Hartman, P.. Ordinary Differential Equations, 2nd ed. (Boston: Birkhäuser, 1982).Google Scholar
11Hartman, P. and Wintner, A.. Asymptotic integrations of linear differential equations. Amer. J. Math. 77 (1955), 48–86, 932.Google Scholar
12Hinton, D. B.. Asymptotic behaviour of solutions of (ry (m)) (k) ± qy = 0. J. Differential Equations 4 (1968), 590–596.CrossRefGoogle Scholar
13Levinson, N.. The asymptotic nature of solutions of linear differential equations. Duke Math. J. 15 (1948), 111–116.Google Scholar