= Solution
Take positive <heat> flux upwards, along increasing $z$, and denote <thermal conductivity> by $k$. <Fourier's law> gives $q_s=-kT_z|_{z=a}$. Since $\cos(n\pi)=(-1)^n$, <differentiation> of the literal printed expression gives
$$
\boxed{q_{s,\rm printed}(t)=\frac{k\Delta T}{a}
\left[1-2\sum_{n=1}^\infty e^{-n^2t/\tau_0}\right].}
$$
Its negative young-age flux is another sign of the unphysical initial profile. For the explicitly corrected hot initial plate, the <surface heat flux of a cooling plate> is instead
$$
\boxed{q_s(t)=\frac{k\Delta T}{a}
\left[1+2\sum_{n=1}^\infty e^{-n^2t/\tau_0}\right].}
$$
The old-age limit is $k\Delta T/a$. For $t\ll\tau_0$, replacing the Gaussian sum by its leading <integral> gives $q_s\sim k\Delta T/\sqrt{\pi\kappa t}$, recovering the <half-space cooling model>. Neither $k$ nor $\kappa$ is numerically specified, so the age-dependent flux remains symbolic as requested.
Back to article page
