Solving heat equation for heat-pulse in a point on the surface

Chestermiller

Two reasons. First of all, in the OP's relationship, κ (kappa) is the thermal diffusivity, not, as he stated, the thermal conductivity k. Secondly, the term involving Q is not quite right. The units don't properly give temperature. I'm too lazy to look up what the correct expression for what this term should be, but, at the very least, there should be a k (thermal conductivity) in the denominator (if U has units of temperature and Q has units of energy). I'm guessing that the denominator should be a constant times [itex]kt\sqrt{\kappa t})[/itex]. This would give the correct units for temperature.

Chestermiller

Try "Conduction of Heat in Solids" [H. S. Carslaw, J. C. Jaeger] . Another possibility is to solve it yourself. Of course, there won't be a nice similarity solution like the one you have given.

rollingstein

+1 for that book. If it is documented anywhere it is in there.

I'll add a possibility of trying to take help from one of the Symbolic solvers like Maxima / Mathematica etc. after you frame the problem.