 Solution

Each spatial eigenfunction acquires the temporal factor $e^{-\kappa {\lambda }_{mn}t}$ under the heat equation. Expanding the initial constant using the preceding coefficients gives the cooling of an initially uniform square:For $t>0$ the exponential factors justify differentiating the series and enforce the zero boundary temperature. As $t\downarrow 0$, it tends to $T_0$ at every interior point and in L2 space; the instantaneous boundary cooling is incompatible with continuity at the initial boundary corners, which is expected for this idealization.

The slowest decaying nonzero mode is $(1,1)$, givingat each fixed interior point when $T_0\ne 0$. The next contributing modes have $(m,n)=(1,3),(3,1)$ and decay as $e^{-10\kappa {\pi }^{2}t/{\ell }^2}$, making the leading-mode approximation exponentially accurate.