Liouville–Green asymptotics for almost-diagonal second-order matrix differential equations

Article type: Research Article

Authors: Spigler, Renato | Vianello, Marco

Affiliations: Dipartimento di Matematica, Università “Roma Tre”, 1, Largo S. Leonardo Murialdo, 00146 Roma, Italy E-mail: spigler@mat.uniroma3.it | Dipartimento di Matematica Pura e Applicata, Università di Padova, 7, Via Belzoni, 35135 Padova, Italy E-mail: marcov@math.unipd.it

Abstract: A WKB (or Liouville–Green) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y″=(D(t)+G(t))Y, on [a,+∞), where D(t) is a nonsingular diagonal matrix. A basis for the right-module of its solutions can be represented explicitly, and precise computable bounds for the error terms involved are given. The double asymptotic nature with respect to both, t and some parameters that might affect the matrix coefficient, is shown. Examples and applications are given.

Keywords: Liouville–Green asymptotics, WKB approximation, almost-diagonal systems of ordinary differential equations, generalized Airy functions

Journal: Asymptotic Analysis, vol. 48, no. 4, pp. 267-294, 2006