Published online by Cambridge University Press: 14 November 2011
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+
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