Matched Asymptotic Expansions for Coupled Equations

We have shown the basic steps for the method of matched asymptotic expansion. The matching principle is based on the use of the overlapping region and an intermediate variable in that region. In what follows is another simple method of matching for a class of problems, which takes the following form 

These types of coupled first order equations arise frequently in chemical engineering, where there are two dependent variables and their dynamics are coupled through the nonlinear functions / and g, but the dynamic evolution of one variable is faster than the other (in this case, x is faster). 

The outer variable for this problem is i, which is sometimes called slow time in the literature. The appropriate measure of the change of the dependent variable in the initial short period is an inner variable (sometimes called fast time), which is defined as 

In effect, the time variable is broken into two regions, the outer region and the inner region. In the inner region, the evolution of the x-variable is observed, whereas the evolution of the variable y is only seen in the outer region. The expansions associated with the inner and outer regions are called the inner and outer expansions, respectively. 

let us construct the outer expansion with the independent variable t. The outer asymptotic expansions are assumed to take the following form 

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