= Solution
Write a nonzero separated solution as $T=X(x)Y(y)S(t)$. Division of the <heat equation> by this product gives
$$
\frac{S'}{\lambda S}=\frac{X''}{X}+\frac{Y''}{Y}.
$$
The two spatial factors solve independent constant-coefficient <eigenvalue> problems. Zero endpoint values in $x$ require
$$
X_n(x)=\sin(n\pi x/a),\qquad X_n''=-(n\pi/a)^2X_n,\quad n=1,2,\ldots.
$$
The insulating conditions require
$$
Y_m(y)=\cos(m\pi y/b),\qquad Y_m''=-(m\pi/b)^2Y_m,\quad m=0,1,2,\ldots.
$$
The zero cosine mode is the nonzero constant function and must be retained; there is no nonzero zero-index sine mode. The temporal equation is $S'=-\lambda\pi^2(n^2/a^2+m^2/b^2)S$. Superposing the complete sine-cosine basis yields
$$
\boxed{T(x,y,t)=\sum_{n=1}^\infty\sum_{m=0}^\infty A_{nm}\sin\frac{n\pi x}{a}\cos\frac{m\pi y}{b}\exp\left[-\lambda\pi^2\left(\frac{n^2}{a^2}+\frac{m^2}{b^2}\right)t\right].}
$$
The constants are fixed by expansion of the initial profile. <Orthogonality> gives sine norm $a/2$; the cosine norm is $b$ at $m=0$ and $b/2$ otherwise. This is the <heat equation on a rectangle with two insulated sides>.
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