 Solution

After bath cooling followed by insulated equilibration, the remaining mean temperature is conserved: integrating the insulated heat equation givesFor constant density, specific heat capacity and cross-section, this is conservation of energy. Its Fourier cosine series has a constant mode equal to the initial integral average; every nonconstant mode decays. Therefore the eventual uniform temperature isFor each mode with an odd integer index in part (c), $L^{-1}{\int }_0^L\sin ((2j+1)\pi x/L)dx=2/[(2j+1)\pi ]$. Integrating the bath-cooled series yieldsFor $t_f>0$, uniform convergence justifies termwise integration; the $t_f=0$ result also follows by dominated convergence of the summable positive coefficients. As a check, ${\sum }_{j\ge 0}(2j+1{)}^{-2}={\pi }^2/8$ gives $T_f=T_0$ for zero bath time, and $T_f\to 0$ as $t_f\to \infty$. More bath time strictly lowers the retained average temperature when $T_0>0$ and $\kappa >0$.