= Solution
The cooled endpoints impose <Dirichlet boundary conditions> $\theta(0,t)=\theta(L,t)=0$. The spatial modes are $\sin(n\pi x/L)$. Integrating the initial profile on its two straight segments gives
$$
b_n=\frac2L\left[\int_0^{L/2}x\sin\frac{n\pi x}{L}dx+\int_{L/2}^L(L-x)\sin\frac{n\pi x}{L}dx\right]=\frac{4L}{n^2\pi^2}\sin\frac{n\pi}{2}.
$$
The <heat flow from a triangular initial temperature> with height $L/2$ is therefore
$$
\boxed{\theta_D(x,t)=\frac{4L}{\pi^2}\sum_{n=1}^\infty\frac{\sin(n\pi/2)}{n^2}\sin\frac{n\pi x}{L}\,e^{-\nu(n\pi/L)^2t}.}
$$
Every nonzero mode decays, so $\theta_D\to0$ as $t\to\infty$. There is no nonzero constant mode compatible with these endpoint temperatures.
Back to article page
