Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 3 / b

Solution

Away from the thin layer, set

ϵ = 0

. The outer equation is

(1 + x)^{3} y_{out}^{'} + y_{out} = 0,

and the boundary condition selected by this first-order problem is the right-end condition. Thus

y_{out} (x) = exp [\frac{1}{2 ( 1 + x ) ^{2}} - \frac{1}{8}],

which satisfies

y_{out} (1) = 1

but has

y_{out} (0) = e^{3/8}

The missing left condition is supplied by an outflow boundary layer. Put

X = x / ϵ

and

y = Y (X)

ϵ

, the equation is

Y_{XX} + (1 + ϵ X)^{3} Y_{X} + ϵ Y = 0.

At leading order,

Y_{0, XX} + Y_{0, X} = 0.

The conditions

Y_{0} (0) = 1

and

Y_{0} \to y_{out} (0) = e^{3/8}

X \to \infty

give

Y_{0} (X) = e^{3/8} + (1 - e^{3/8}) e^{- X} .

The common overlap is

e^{3/8}

y_{comp} (x) = exp [\frac{1}{2 ( 1 + x ) ^{2}} - \frac{1}{8}] + (1 - e^{3/8}) e^{- x / ϵ} .

It satisfies both endpoint values up to exponentially small or higher-order errors and is uniformly valid to

O (1)

0 \leq x \leq 1

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