Bath cooling followed by insulated equilibration ($T_{\infty }=\frac{1}{L}{\int }_0^{L}T(x,t_b)dx$)

Let a homogeneous plate $0<x<L$ begin at uniform temperature $T_0$, cool with zero-temperature Dirichlet boundary conditions until time $t_b$, then switch to insulated Neumann boundary conditions. For thermal diffusivity $\kappa >0$, the Fourier sine series during cooling is $T=(4T_0/\pi ){\sum }_{n\ge 1,n\text{ odd}}\sin (n\pi x/L)e^{-\kappa (n\pi /L{)}^{2}t}/n$. Integrating the insulated heat equation shows that its integral average is conserved. All nonconstant Fourier cosine series modes then decay, proving that the final uniform temperature is $T_{\infty }=(8T_0/{\pi }^2){\sum }_{n\ge 1,n\text{ odd}}{e}^{-\kappa (n\pi /L{)}^2{t}_b}/n^2$. Its limits are $T_0$ at zero bath time and zero at infinite bath time. Instantaneous changes of boundary condition can create initial-boundary corner incompatibilities without invalidating the positive-time solution.