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Uniform asymptotic solution of second order linear differential equations without turning varieties

Author: Gilbert Stengle
Journal: Math. Comp. 23 (1969), 1-22
MSC: Primary 34.50
MathSciNet review: 0247197
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Abstract: The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form

$\displaystyle { \in ^{2n}}y'' = a(t, \in )y,{\text{ }}({\text{ }})' = d/dt,$

where $ n$ is a positive integer, $ t$ and $ \in $ are real variables ranging over $ \left\vert t \right\vert \leqq {t_0},0 < \in \leqq { \in _0}$, and $ a$ is a function infinitely differentiable on the closure of this domain. We require that $ a(t, \in )$ satisfy conditions which can be regarded as generalized nonturning-point conditions. These conditions imply the absence of secondary turning points, and reduce in the simplest case to the condition $ a(t,0) \ne 0$, but also include cases (the interesting ones) in which $ a(0,0) = 0$

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  • [1] Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
  • [2] H. L. Turrittin, Asymptotic expansions of solutions of systems of ordinary linear differential equations containing a parameter, Contributions to the Theory of Nonlinear Osciallations, vol. II, Princeton University Press, Princeton, 1952, pp. 81–116. MR 0050754
  • [3] Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0203188
  • [4] Masahiro Iwano and Yasutaka Sibuya, Reduction of the order of a linear ordinary differential equation containing a small parameter, Kōdai Math. Sem. Rep. 15 (1963), 1–28. MR 0149034
  • [5] G. Stengle, Asymptotic Solution of a Class of Second Order Differential Equations Containing a Parameter, Courant Institute of Mathematical Sciences, IMM-NYU 319, 1964.
  • [6] M. A. Evgrafov and M. V. Fedorjuk, Asymptotic behavior of solutions of the equation 𝑤′′(𝑧)-𝑝(𝑧,𝜆)𝑤(𝑧)=0 as 𝜆→∞ in the complex 𝑧-plane, Uspehi Mat. Nauk 21 (1966), no. 1 (127), 3–50 (Russian). MR 0209562
  • [7] Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
  • [8] Bernard Malgrange, Le théorème de préparation en géométrie différentiable. IV. Fin de la démonstration, Séminaire Henri Cartan, 1962/63, Exp. 22, Secrétariat mathématique, Paris, 1962/1963, pp. 8 (French). MR 0160237
  • [9] J. G. Semple and G. T. Kneebone, Algebraic curves, Oxford University Press, London-New York, 1959. MR 0124801
  • [10] G. Bliss, Algebraic Functions, Colloq. Publ., Vol. 16, Amer. Math. Soc., Providence, R. I., 1933.
  • [11] L. V. Kantorovich and G. P. Akilov, Functional analysis in normed spaces, Translated from the Russian by D. E. Brown. Edited by A. P. Robertson. International Series of Monographs in Pure and Applied Mathematics, Vol. 46, The Macmillan Co., New York, 1964. MR 0213845
  • [12] J. G. van der Corput, Introduction to the neutrix calculus, J. Analyse Math. 7 (1959/1960), 281–399. MR 0124678,

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Article copyright: © Copyright 1969 American Mathematical Society

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