= Solution
Substitute the proposed <Fourier sine series> into the <heat equation>:
$$
\sum_{n\geq1}b_nT_n'(t)\sin(k_nx)
=-D\sum_{n\geq1}b_nk_n^2T_n(t)\sin(k_nx).
$$
Projecting onto each sine using <orthogonality> gives $T_n'=-Dk_n^2T_n$ for every mode with $b_n\ne0$. The <initial condition> fixes $T_n(0)=1$, since $b_n$ already describes the initial amplitude. Hence
$$
\boxed{T_n(t)=e^{-Dk_n^2t},\qquad
\Theta(x,t)=\sum_{n=1}^{\infty}b_n e^{-D(n\pi/L)^2t}\sin\!\left(\frac{n\pi x}{L}\right)}.
$$
For a mode with $b_n=0$, the value of $T_n$ is immaterial and the same convention is convenient. The positive <thermal diffusivity> $D$ damps every mode, with higher frequencies decaying faster. For square-integrable initial data the damping justifies termwise differentiation for $t>0$ and recovery of the initial data in <L2 norm convergence> as $t\downarrow0$.
\Image[/past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-resit-fourier-heat.png]
{title=Odd Fourier modes reconstruct a square wave, while heat diffusion damps short spatial scales first}
The left panel illustrates the <Fourier series> in part (b). The right panel is a self-generated example with $D=L=1$ and $\Theta_0(x)=\sin(\pi x)+\tfrac12\sin(3\pi x)$, showing the two <normal modes> decaying at rates $\pi^2$ and $9\pi^2$.
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