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

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.