Liouville–Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville–Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection–diffusion equations, are given.