= Outer expansion
{wiki}
= Outer solution
{synonym}
An <outer expansion> describes the solution away from a thin or distant region resolved by an <inner expansion>. It is matched to that <inner expansion> in their common <overlap region>. Its range of validity must be specified: an approximation at each fixed positive $x$ need not have <uniform convergence> down to an endpoint.
For the <singular perturbation> problem $\epsilon y''+y'=0$, $y(0)=0$, $y(1)=1$, try the <asymptotic expansion> $y\sim y_0+\epsilon y_1+\epsilon^2y_2+\cdots$ at fixed $x>0$. Substitution gives $y_0'=0$ and $y_n'=-y_{n-1}''$ for $n\geq1$. Expanding the right <boundary condition> gives $y_0(1)=1$ and $y_n(1)=0$ for $n\geq1$. Therefore $y_0=1$ and every higher coefficient vanishes. The missing <boundary layer> is exponentially small at each fixed positive $x$, invisible to all these powers of $\epsilon$.
The exact solution supplies a quantitative limit. For every fixed $a\in(0,1)$,
$$
\sup_{a\leq x\leq1}|y_\epsilon(x)-1|
=\frac{e^{-a/\epsilon}-e^{-1/\epsilon}}{1-e^{-1/\epsilon}}
\leq\frac{e^{-a/\epsilon}}{1-e^{-1/\epsilon}}\longrightarrow0.
$$
This is <uniform convergence> on $[a,1]$. On $[0,1]$ the same <supremum norm> of the error is exactly one. The <inner expansion> supplies the endpoint information that this <outer expansion> cannot encode.
Back to article page
