Heat Equation with Non-Zero Temperature Boundaries — Question 2

Question 2 – Solution

Strategy. The sudden boundary change creates an initial corner mismatch; subtracting the stationary profile isolates its decaying sine expansion.

Step 1: Construct the boundary step response. The stationary field is $S=1-x$. For $v=u-S$, the endpoints are zero and $v(x,0)=x-1$. Integration gives $2\int_0^1(x-1)\sin(n\pi x)\,dx=-2/(n\pi)$. Hence $$\boxed{u(x,t)=1-x-\frac 2\pi\sum_{n\ge 1} \frac{e^{-n^2\pi^2t}\sin(n\pi x)}{n}.}$$

Step 2: Verify the appropriate initial trace. For $t\ge\tau>0$, Gaussian decay makes every finitely differentiated transient series uniformly convergent. It satisfies the heat equation and zero endpoint values; adding $S$ gives the required boundary data. The square-summable coefficients and Parseval prove $v\to x-1$ in $L^2$, hence $u\to 0$ in $L^2$. At each interior point, the Fourier limit is zero after reconstruction; Gaussian damping preserves that limit by summation by parts. At $x=0$, the positive-time value is always one, so its limit is one. Uniform convergence to zero on the closed interval is therefore impossible. No corner-continuous solution is claimed.

Step 3: Integrate the boundary-driven heating. For positive time, $$H(t)=\frac 12-\frac 4{\pi^2}\sum_{n\text{ odd}} \frac{e^{-n^2\pi^2t}}{n^2},\qquad H'(t)=4\sum_{n\text{ odd}}e^{-n^2\pi^2t}>0.$$ The initial norm trace gives $H(0+)=0$ and the final heat is $1/2$. Differentiating the spatial series gives $H'=[u_x]_0^1=j(0,t)-j(1,t)$, where $j=-u_x$. The endpoint supplies heat; no interior source is needed.

Step 4: Quantify the singular initial input. At the left endpoint, $$j(0,t)=1+2\sum_{n\ge 1}e^{-n^2\pi^2t}.$$ With $h=\sqrt t$, the decreasing, integrable Gaussian gives $h\sum_{n\ge 1}e^{-\pi^2(nh)^2}\to\int_0^\infty e^{-\pi^2s^2}\,ds$ by upper/lower integral bounds. Thus $\sqrt t\,j(0,t)\to 1/\sqrt\pi$, or $\boxed{j(0,t)\sim 1/\sqrt{\pi t}}$. The singular rate is integrable in time and is consistent with vanishing initial heat. Substituting $t=0$ into the differentiated series would be invalid.