# Equation level matching: An extension of the method of matched asymptotic expansion for problems of wave propagation

## TL;DR

This paper proposes an equation-level matching method as an alternative to traditional matched asymptotic expansions, simplifying the process and extending applicability to complex wave propagation problems.

## Contribution

It introduces a novel equation-level matching approach that avoids solving intermediate asymptotic equations, enabling analysis of wave problems where traditional methods fail.

## Key findings

- Allows matching without explicit solutions of asymptotic equations
- Effective in wave problems with differing time behaviors
- Simplifies the process of deriving approximate solutions

## Abstract

We introduce an alternative to the method of matched asymptotic expansions. In the "traditional" implementation, approximate solutions, valid in different (but overlapping) regions are matched by using "intermediate" variables. Here we propose to match at the level of the equations involved, via a "uniform expansion" whose equations enfold those of the approximations to be matched. This has the advantage that one does not need to explicitly solve the asymptotic equations to do the matching, which can be quite impossible for some problems. In addition, it allows matching to proceed in certain wave situations where the traditional approach fails because the time behaviors differ (e.g., one of the expansions does not include dissipation). On the other hand, this approach does not provide the fairly explicit approximations resulting from standard matching. In fact, this is not even its aim, which to produce the "simplest" set of equations that capture the behavior.