Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 336 / 2 / b

Solution

Let

X = x / ϵ

a (X) = 1/ d (X)

, and seek

u = u_{0} (x, t) + ϵ u_{1} (x, X, t) + \dots

with periodic correctors. The leading cell equation makes

u_{0}

independent of

X

. At the next order, continuity of microscopic heat flux gives

a (X) (u_{0 x} + u_{1 X}) = J (x, t) .

Averaging over one period and using

⟨ u_{1 X} ⟩ = 0

yields

u_{0 x} = J \int_{0}^{1} d (X) d X = J \overline{d} .

1/ \overline{d}

u_{0 t} = \frac{1}{d} u_{0 xx} .

For the sawtooth profile, the two triangular areas give

\overline{d} = \int_{0}^{p} \frac{2 X}{p} d X + \int_{p}^{1} \frac{2 ( 1 - X )}{1 - p} d X = p + (1 - p) = 1.

The homogenized problem is therefore the unit-diffusivity heat equation. Its half-line step solution is

u (x, t) = T_{0} erfc (\frac{x}{2 t}) = T_{0} [1 - erf (\frac{x}{2 t})] .

It has the required initial and boundary limits, and direct differentiation verifies

u_{t} = u_{xx}

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