Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 336 / 3 / a

Solution

The reduced outer equation is

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

. Imposing the right boundary condition gives

y_{out} = \frac{2}{1 + x} + ϵ [\frac{2}{( 1 + x ) ^{3}} - \frac{1}{2 ( 1 + x )}] + O (ϵ^{2}) .

The condition at

x = 0

requires

X = x / ϵ

. The first two inner terms are

Y_{0} = 2 - e^{- X}, Y_{1} = - 2 X + \frac{3}{2} + (\frac{X ^{2}}{2} - \frac{3}{2}) e^{- X} .

They match

2 + ϵ (- 2 X + 3/2)

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

[- 1, 1]

1 + x

vanishes at the left endpoint. The ordinary

O (ϵ)

exponential layer is replaced by a turning-point endpoint region of width

O (ϵ)

, where all three terms in the equation enter the leading balance.

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