Paper 2, Section II, B

Differential Equations | Part IA, 2018

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

TT and UU satisfy the coupled differential equations

dTdt=a(TTr)+Q(t)dUdt=b(UT)\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 aa and bb are positive constants. Briefly explain the various terms appearing in the above equations.

Heating may be provided by a short-lived pulse at t=0t=0, with Q(t)=Q1(t)=δ(t)Q(t)=Q_{1}(t)=\delta(t) or by constant heating over a finite period 0<t<τ0<t<\tau, with Q(t)=Q2(t)=τ1(H(t)H(tQ(t)=Q_{2}(t)=\tau^{-1}(H(t)-H(t- τ)\tau), where δ(t)\delta(t) and H(t)H(t) are respectively the Dirac delta function and the Heaviside step function. Again briefly, explain how the given formulae for Q1(t)Q_{1}(t) and Q2(t)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=Trt<0, T=U=T_{r}. Find the solutions for T(t)T(t) and U(t)U(t) for t>0t>0, for each of Q(t)=Q1(t)Q(t)=Q_{1}(t) and Q(t)=Q2(t)Q(t)=Q_{2}(t), denoted respectively by T1(t)T_{1}(t) and U1(t)U_{1}(t), and T2(t)T_{2}(t) and U2(t)U_{2}(t). Explain clearly any assumptions that you make about continuity of the solutions in time.

Show that the solutions T2(t)T_{2}(t) and U2(t)U_{2}(t) tend respectively to T1(t)T_{1}(t) and U1(t)U_{1}(t) in the limit as τ0\tau \rightarrow 0 and explain why.

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