Ordinary differential equations Definition

Method of matched asymptotic expansions

Builds on Singular perturbation Boundary layer Asymptotic analysis

Definition

Let $y_\varepsilon$ solve a singularly perturbed problem on an interval $[a, b]$ that has a boundary layer at $a$ of thickness $\delta(\varepsilon) \to 0$ . The method of matched asymptotic expansions builds an approximation in four steps.

Outer expansion: solve the reduced equations order by order in $\varepsilon$ , with the boundary conditions away from $a$ , to get $y^{\mathrm{out}}(x)$ .
Inner expansion: set $X = (x - a) / \delta(\varepsilon)$ and rewrite the problem in $X$ . Solve it order by order with the boundary conditions at $a$ , to get $Y^{\mathrm{in}}(X)$ , which may contain free constants.
Matching: choose the free constants so that the two expansions agree in the overlap. At leading order this is Prandtl's rule $\lim_{X \to \infty} Y^{\mathrm{in}}(X) = \lim_{x \to a^+} y^{\mathrm{out}}(x)$ . This common value is the common part.
Composite expansion: $y^{\mathrm{comp}}(x) = y^{\mathrm{out}}(x) + Y^{\mathrm{in}}((x - a)/\delta(\varepsilon)) - (\text{common part})$ .

Each expansion is an asymptotic expansion valid in its own region, and the composite one is meant to be valid uniformly on $[a, b]$ .

Remarks

Example: $\varepsilon y'' + y' + y = 0$ on $[0, 1]$ , $y(0) = 0$ , $y(1) = 1$ . Outer: $y' + y = 0$ with $y(1) = 1$ gives $y^{\mathrm{out}} = e^{1 - x}$ . Inner: with $X = x / \varepsilon$ , the leading equation is $Y'' + Y' = 0$ with $Y(0) = 0$ , so $Y^{\mathrm{in}} = A (1 - e^{-X})$ . Matching gives $A = y^{\mathrm{out}}(0) = e$ , so the common part is $e$ . Composite: $y^{\mathrm{comp}} = e^{1 - x} - e^{1 - x/\varepsilon}$ ; it satisfies $y(0) = 0$ exactly and $y(1) = 1 - e^{1 - 1/\varepsilon}$ , which is exponentially close to $1$ .

Non-example: putting the outer solution and the inner solution side by side without step 3 fails, since the constant $A$ stays undetermined. The layer must also be placed correctly: a trial layer at $x = 1$ leads to inner solutions that grow like $e^{X}$ away from the boundary and cannot match.

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Asymptotic analysis concept
Boundary layer concept
Singular perturbation concept
Method of matched asymptotic expansions definition

Tag	`CAYQ1S3RM8`
URI	`m2://CAYQ1S3RM8`
Kind	definition
Writing	W0 stub
MSC	34E05, 34E15
Defines	Method of matched asymptotic expansions
Uses	Singular perturbation, Boundary layer, Asymptotic analysis
Changed	2 November 2026

Method of matched asymptotic expansions

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