Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 2 / 5A / ii

Solution

Integrate the first equation over

(1 - ε, 1 + ε)

. The integral of the ordinary cooling term tends to zero as

ε ↓ 0

T_{1} (1^{+}) - T_{1} (1^{-}) = - 1.

After the jump there is no further forcing, so Newton's law of cooling gives

T_{1} (t) - T_{\infty} = [(T_{0} - T_{\infty}) e^{- a} - 1] e^{- a (t - 1)} .

Equivalently,

T_{1} (t) = T_{\infty} + (T_{0} - T_{\infty}) e^{- a t} - e^{- a (t - 1)}, t > 1.

The last term has jump minus one at time one and then decays at the same exponential rate as the unforced temperature.

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