I think this could be done like this
heateq = D[fT[x, t], t] - a D[fT[x, t], x, x]
Do a Laplace-Transformation, giving an ODE for the PDE
fLT = LaplaceTransform[heateq, t, s]
Solve the equivalent ODE
sol = DSolve[-T0 + s fTL[x] - a D[fTL[x], x, x] == 0, fTL[x], x] // Flatten
fT1 = fTL[x] /. sol
The solution must be finite
fT2 = fT1 /. C[1] -> 0
Constant flow of heat at x = 0 gives
c2 = Solve[-a D[fT2, x] == a/s /. x -> 0, C[2]] // Flatten
and
fT4 = fT2 /. c2
Transform back in the x-t-domain
f1xt = FullSimplify[InverseLaplaceTransform[fT4, s, t], x > 0 && t > 0]
and show the result for different times
t1 =.
werte = {a -> .1, k -> .1, T0 -> 2};
f1xtn = f1xt /. werte /. t -> t1
t1 = 100;
Plot[f1xtn, {x, 0, 5}, PlotRange -> {0, 10}]
Info:
https://www.amazon.de/Conduction-Science-Publications-Carslaw-1986-04-10/dp/B00M0R1W0I/ref=sr_1_1?qid=1662652984&refinements=p_27%3AH.S.+And+J.C.+Jaeger+Carslaw&s=books&sr=1-1&text=H.S.+And+J.C.+Jaeger+Carslaw
