Matched asymptotic expansions

Consider $\varepsilon u''+u'=1$ with both endpoint values zero. The outer equation is $u'=1$, and the right endpoint selects $u_{\mathrm{out}}=x-1$. Near the left boundary use $X=x/\varepsilon$. The transformed equation is $U_{XX}+U_X=\varepsilon$, so its leading homogeneous solution with the left value zero can be matched to a constant. The large-inner-coordinate limit must equal the small-position limit of the outer solution.

The common leading value is minus one, so the inner solution is $U_{\mathrm{in}}=-1+e^{-X}$. An overlap has small original position but large inner coordinate. For $x=\varepsilon^\alpha$ with any fixed exponent strictly between zero and one, both requirements hold. The outer correction to minus one is $\varepsilon^\alpha$, while the inner exponential is $\exp[-\varepsilon^{\alpha-1}]$, tending to zero faster than any algebraic power. This explicit scaling makes the matching domain visible instead of comparing two limits at incompatible fixed coordinates.

The leading composite is the sum of outer and inner functions, expressed in the same original coordinate, minus their common constant. It therefore becomes $x-1+e^{-x/\varepsilon}$. At the left endpoint the terms cancel exactly, while far from the layer the exponential disappears and the outer behavior remains. If the common part were not subtracted, the approximation would be displaced by an order-one constant throughout the overlap. Higher-order matching can involve polynomials, logarithms, or several shared terms rather than only a constant.

One can derive an exact solution here by first solving the first-order equation for the derivative and integrating. This supplies a check independent of the formal matching procedure. The exact solution is $x-(1-e^{-x/\varepsilon})/(1-e^{-1/\varepsilon})$. Comparing it with the composite shows an exponentially small uniform error. The striking accuracy is a feature of this equation and retained terms, not a theorem that every leading composite has exponentially small error.

At higher order, re-expanding an inner expression in outer variables and an outer expression in inner variables helps identify which shared terms must be removed. The limits should be taken in an overlap where both retained hierarchies are ordered; terms small in one fixed-coordinate limit can become large after rescaling. A formal matching condition can determine constants while leaving a genuine proof of a uniform remainder open. Record the order retained in each region rather than claiming that two expressions match merely because their leading constants agree.

A residual audit checks the differential equation, each boundary condition, and any junction regularity. A stability estimate then converts those defects into an error bound in the desired norm. Several layers, turning points, or nonlinear instabilities may require more than one inner region or may leave no overlap for the initially chosen scales. Such failures are diagnostic: revise the scaling or the asymptotic form instead of forcing a composite from incompatible pieces. The usefulness of matching comes from preserving the relationship between regions, not from hiding a boundary mismatch in a plot.