On Multivariable Asymptotic Expansions

In this paper we consider the damped linear oscillator with small damping $\varepsilon $. We obtain uniform asymptotic expansions of the solution as $\varepsilon \to 0$ that are uniformly valid for all time $t \geqq 0$, by the multitime method. We show how to determine the expansion coefficients without resorting to intuitive arguments. This is done by considering the remainder in the expansion of the solution and by requiring that it be made small in a way that is precisely defined in the paper. This analysis also yields proofs of the uniform asymptotic convergence of the expansions. We find that there are a minimum number of time scales, namely two, that are required to obtain a uniform asymptotic expansion. For a fixed number of terms in the expansion there are a maximum number of time scales, namely three, that give uniform expansions with the smallest estimate of the remainder. Finally we show how to apply the analysis to obtain uniformasymptotic expansions of a mixed, initial boundary value problem for the damped wave equation.