Approximate diagonalization in differential systems and an effective algorithm for the computation of the spectral matrix

Abstract

1. IntroductionIn a recent paper [3], an extended Liouville–Green formulaformula herewas developed for solutions of the second-order differential equationformula hereHere γM(x) ∼Q−¼(x), QM(x) ∼Q−½(x) and εM(x)→0 as x→∞, while M([ges ]2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M−1. The general form of (1·1) had been obtained previously by Cassell [5], [6], [7] and Eastham [10], [11, section 2·4]. In particular, the proof of (1·1) in [10] and [11] depended on the formulation of (1·2) as a first-order system and then on a process of M repeated diagonalization of the coefficient matrices in a sequence of related differential systems.

Asymptotic Solution of a Differential System Related to the Generalized Hypergeometric Equation

“…However, we can use a technique which has been exploited recently [2], §2 (see also [3], §1.7) for improving perturbation terms -that is, replacing them by terms of a smaller order of magnitude. The idea here is to make the approximate identity transformation…”

“…A system of this type, but with different Λ 1 and R 1 , was considered recently in [2] where is was shown that a sequence of transformations of (9.4) can be defined, by means of which the perturbation R 1 is successively replaced by perturbations of smaller magnitude as x → ∞. Thus in [2] we have (1) term in the asymptotic solution of (9.6) corresponding to (8.11).…”

“…Thus in [2] we have (1) term in the asymptotic solution of (9.6) corresponding to (8.11). Thus the product in (9.5) provides sub -dominant terms in the asymptotic solution of (9.4).…”

“…Thus the product in (9.5) provides sub -dominant terms in the asymptotic solution of (9.4). The definition of each P s in [2] requires the existence of a further derivative of ρ, and therefore the determination of sub -dominant terms is related to the differentiability of ρ. If ρ is infinitely differentiable, as of course is ρ(x) = x in the G -function situation, (9.5) will yield a full asymptotic expansion.…”

“…In the final section of the paper we discuss a number of extensions of our methods, which include the possibility of replacing the powers of x in (1.3) by more general functions of x. This takes us away from situations where Meijer G -functions can be used and allows us to discuss the relationship between the two approaches to the asymptotic solution of (1.1) in terms of the recent work of Brown et al [2] for improving the accuracy of asymptotic results derived from (1.9).…”