= Additive composite expansion
= Composite asymptotic expansion
{synonym}
An <additive composite expansion> combines an <inner expansion> and an <outer expansion> by subtracting their common behavior in the <overlap region>:
$$
y_{\mathrm{comp}}=y_{\mathrm{out}}+y_{\mathrm{in}}-y_{\mathrm{overlap}}.
$$
The subtraction removes the part counted twice. Near the <boundary layer>, the <outer expansion> cancels the overlap contribution; away from the <boundary layer>, the <inner expansion> cancels it. An error estimate is needed to establish accuracy across the whole domain.<Asymptotic Analysis and Singular Perturbation Theory by John K. Hunter>[§4.3.5, p. 58: uniform solution and overlap subtraction]{ref}
For the <singular perturbation> problem $\epsilon y''+y'=0$, $y(0)=0$, $y(1)=1$, the <outer solution> is one, the leading <inner expansion> is $1-e^{-x/\epsilon}$, and their overlap is one. Thus $y_{\mathrm{comp}}=1-e^{-x/\epsilon}$. With $q=e^{-1/\epsilon}$, comparison with the exact normalized solution gives
$$
y_\epsilon(x)-y_{\mathrm{comp}}(x)
=\frac{q(1-e^{-x/\epsilon})}{1-q},\qquad
\sup_{0\leq x\leq1}|y_\epsilon-y_{\mathrm{comp}}|=q.
$$
The <supremum norm> error is therefore exponentially small on the entire interval, even though the <outer solution> alone has error one. This is an error bound for function values; errors in <derivatives> have additional powers of $\epsilon^{-1}$.
The composite satisfies the <differential equation> exactly but has right <boundary condition> $y_{\mathrm{comp}}(1)=1-q$. Dividing it by $1-q$ restores that <boundary condition> and recovers the exact solution. This example makes both the strength and the limit of matching explicit: it gives a global approximation, while finite-$\epsilon$ normalization can still carry information beyond every algebraic order.
Back to article page
