Paper 2, Section II, B

The temperature $T$ in an oven is controlled by a heater which provides heat at rate $Q(t)$. The temperature of a pizza in the oven is $U$. Room temperature is the constant value $T_{r}$.

$T$ and $U$ satisfy the coupled differential equations

$\begin{aligned} \frac{d T}{d t} &=-a\left(T-T_{r}\right)+Q(t) \\ \frac{d U}{d t} &=-b(U-T) \end{aligned}$

where $a$ and $b$ are positive constants. Briefly explain the various terms appearing in the above equations.

Heating may be provided by a short-lived pulse at $t=0$, with $Q(t)=Q_{1}(t)=\delta(t)$ or by constant heating over a finite period $0<t<\tau$, with $Q(t)=Q_{2}(t)=\tau^{-1}(H(t)-H(t-$ $\tau)$, where $\delta(t)$ and $H(t)$ are respectively the Dirac delta function and the Heaviside step function. Again briefly, explain how the given formulae for $Q_{1}(t)$ and $Q_{2}(t)$ are consistent with their description and why the total heat supplied by the two heating protocols is the same.

For $t<0, T=U=T_{r}$. Find the solutions for $T(t)$ and $U(t)$ for $t>0$, for each of $Q(t)=Q_{1}(t)$ and $Q(t)=Q_{2}(t)$, denoted respectively by $T_{1}(t)$ and $U_{1}(t)$, and $T_{2}(t)$ and $U_{2}(t)$. Explain clearly any assumptions that you make about continuity of the solutions in time.

Show that the solutions $T_{2}(t)$ and $U_{2}(t)$ tend respectively to $T_{1}(t)$ and $U_{1}(t)$ in the limit as $\tau \rightarrow 0$ and explain why.

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