The Heat Equation — Question 5

Question 5 – Solution

Strategy. The temperature equation determines a ratio of material parameters; a flux measurement provides an independent scale.

Step 1: Verify the field and recover diffusivity. Differentiation gives $$u_t=-\lambda B e^{-\lambda t}\cos(\pi x/L),\qquad u_{xx}=-\frac{\pi^2}{L^2}B e^{-\lambda t}\cos(\pi x/L).$$ Also $u_x=-(\pi B/L)e^{-\lambda t}\sin(\pi x/L)$, so $u_x=0$ at both ends. Matching the equation at points where the cosine is nonzero gives $$\boxed{\alpha=\frac{k}{C}=\frac{\lambda L^2}{\pi^2}.}$$ The same equality then verifies the equation everywhere, including the node.

Step 2: Identify the nonuniqueness. For every choice $C_*>0$, the pair $$\boxed{C=C_*,\qquad k=\frac{\lambda L^2}{\pi^2}C_*}$$ produces exactly the observed field. Multiplying both parameters by the same positive factor leaves the equation and insulated boundary conditions unchanged. More exact observations of this same temperature field cannot resolve that scaling ambiguity.

Step 3: Use the independent flux data. At the midpoint initially, $q=-ku_x=kB\pi/L$. Therefore $$\boxed{k=\frac{q_mL}{B\pi},\qquad C=\frac{k}{\alpha}=\frac{q_m\pi}{B\lambda L}.}$$ With $q_m$ in $\mathrm{W/m^2}$, $B$ in kelvins and $\lambda$ in $\mathrm{s^{-1}}$, these have units $\mathrm{W/(mK)}$ and $\mathrm{J/(m^3K)}$, respectively. Positivity of the measured quantities ensures physically positive parameters.

Step 4: Choose informative sensor locations. At $x=L/2$, $u=T_a$ for all time: both $u_t$ and $u_{xx}$ vanish, so a ratio of these derivatives would be $0/0$. But $|u_x|$ is largest there, giving a strong flux signal. At any fixed $x\ne L/2$ in $[0,L]$, the excess temperature is nonzero and $$\boxed{\lambda=\frac 1{t_2-t_1} \log\left|\frac{u(x,t_1)-T_a}{u(x,t_2)-T_a}\right|,\quad t_2>t_1.}$$ The absolute values allow sensors on either side of the node; the ratio eliminates the unknown local amplitude.