= Solution
For a separated nonzero solution $T=f(x)g(y)$, substitution and division by $fg$ gives
$$
a\frac{f'}f=\frac{g''}g=-\lambda^2.
$$
The homogeneous <Dirichlet boundary conditions> impose $g(0)=g(1)=0$. For a nonzero spatial factor, <integration by parts> gives $\lambda^2\int_0^1g^2dy=\int_0^1(g')^2dy$, so the separation value is nonnegative. The zero value gives only a trivial linear $g$ under these boundary conditions. The nontrivial separated factors, for $a\ne0$, are therefore
$$
\boxed{g_n(y)=B_n\sin(n\pi y),\qquad
f_n(x)=A_n e^{-n^2\pi^2x/a},\qquad n=1,2,\ldots}.
$$
For the intended forward <diffusion equation>, assume $a>0$. The <superposition principle> gives a <Fourier sine series>, whose coefficients must expand the initial ramp in the interior:
$$
b_n=2\int_0^1y\sin(n\pi y)dy=\frac{2(-1)^{n+1}}{n\pi}.
$$
Consequently the standard heat-flow solution is
$$
\boxed{T(x,y)=\frac2\pi\sum_{n=1}^\infty
\frac{(-1)^{n+1}}n\,e^{-n^2\pi^2x/a}\sin(n\pi y),\qquad a>0,\ x>0}.
$$
It satisfies both lateral <boundary conditions> for $x>0$, and tends to $y$ as $x\downarrow0$ for $0<y<1$, also converging to the ramp in the <L2 norm>. The exponential coefficient decay justifies differentiating the series for positive $x$.
There is a genuine incompatibility in the printed data: at $(0,1)$ the lateral condition demands zero while the initial condition demands one. Thus \b[no function satisfies every stated value at that corner]. The displayed series solves the usual consistent interpretation with initial data on $0<y<1$ and lateral data for $x>0$; it cannot be continuous at the upper initial corner. This discrepancy is present in the original PDF.
The sign of $a$ is also unstated. If $a=0$, the spatial equation with two zero endpoints forces $T=0$, incompatible with the ramp. If $a<0$, the separated Fourier coefficients grow like $e^{n^2\pi^2x/|a|}/n$ and fail to be square summable for every $x>0$; these data therefore have no forward solution in the standard <L2 space> class. The positive-$a$ assumption is essential to the stated heat-flow interpretation.
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