Boundary layers and inner–outer expansions

Study $\varepsilon u''+u'=0$ on the unit interval with values zero at the left and one at the right, for positive parameter. Setting the parameter to zero leaves $u'=0$, a first-order equation whose constant solution cannot meet both different endpoint values. This order reduction warns that a regular expansion across the whole closed interval is impossible. It does not tell us yet which endpoint supplies the layer or what width to use. Those follow by rescaling the complete equation.

Near a candidate left layer of width $\delta$, write $x=\delta X$ and assume an order-one change in the function across order-one inner coordinate. The second derivative term scales as $\varepsilon/\delta^2$ and the first derivative as $1/\delta$. Their balance requires $\delta=\varepsilon$. On this scale the small coefficient is no longer negligible. This conclusion uses the actual first-order derivative coefficient and the assumed order-one amplitude; different equations, turning coefficients, or smaller boundary mismatches can require different scalings.

The inner equation has solutions equal to a constant plus a decaying exponential in $X$. It can therefore approach a bounded outer constant as $X$ grows away from the left boundary. A similar coordinate measured inward from the right would give an exponentially growing mode for this sign of transport, making an order-one right layer incompatible with a bounded matching value. The right boundary consequently determines the outer constant one, and the left inner solution restores the zero value there.

The leading inner function is $1-e^{-X}$. It tends to one as the inner coordinate grows, consistent with the outer constant. This is a matching statement involving two different limiting descriptions: fixed positive position for the outer region and fixed stretched position for the inner region. The full function may approach one at every fixed interior point while retaining zero exactly at the boundary. Uniform convergence to that constant on the closed interval is therefore impossible.

The outer constant is accurate away from the left layer but has error one at the left boundary. The inner exponential resolves distances comparable with the parameter and in this simple case also provides a uniformly accurate composite expression by itself. An overlap region satisfies both $x\to0$ and $x/\varepsilon\to\infty$, for example $x=\sqrt\varepsilon$. There the inner function approaches its matching constant and the outer description remains meaningful. Choosing a coordinate without an overlap can produce incompatible local approximations.

Substitution into the equation and boundary data checks different aspects of an approximation. Here the proposed expression satisfies the differential equation exactly and one endpoint exactly, while the other endpoint mismatch is exponentially small. The exact comparison controls how that mismatch affects the whole solution. In a nonlinear or variable-coefficient problem, small boundary and equation residuals can be amplified; a uniform stability estimate is still needed to turn a formally matched construction into a quantitative theorem.