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A rigorous proof of an exponentially small estimate for a boundary value arising from an ordinary differential equation

Published online by Cambridge University Press:  14 November 2011

J. G. B. Byatt-Smith  and
A. M. Davie
J. G. B. Byatt-Smith
Affiliation:
Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, U.K.
A. M. Davie
Affiliation:
Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, U.K.
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Synopsis

The equation

has a solution y(t) which is non-oscillating on the interval (0, ∞) and has the asymptotic expansion

Each term of this expansion is even in t so that formally is zero to all orders of ɛ. The estimate of has been obtained by Byatt-Smith [3] who corrects (2) in the complex plane near t = i where the series ceases to be valid. This requires asolution of the equation

the equation for the first Painlevé transcedent. Here we prove rigorously that this method gives the correct asymptotic estimate

where

The proof involves converting (1) and (3) to integral equations. The existence and uniqueness of these integral equations are established by use of the contraction mapping theorem. We also prove that the appropriate solution to (3) provides a uniformly valid approximation to (2) over a suitably defined region of the complex plane.

We also consider the connection problem for the oscillatory solutions of (1) which have asymptotic expansions

where Ã+, Ã− φ+, and φ− are constants. The connection problem is to determine the asymptotic expansion at +∞ of a solution which has a given asymptotic expansion at −∞. In other words, we wish to find (Ã+,φ+) as a function of Ã−and φ−. We prove that there is a unique solution to the connection problem, provided Ã− is small enough, and obtain bounds on the estimate of Ã+

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 243 - 258
Copyright
Copyright © Royal Society of Edinburgh 1990

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