3D Heat equation with elementary Dirichlet BC

[Flo31]

TL;DR

3D Heat equation with dirichlet condition in semi infinite domain

Hi,

I am looking for the solution of the following heat conduction problem (see figure below):

• the geometry is the semi-infinite domain such that (x,y)∈R² and z∈[0,∞[ ;
• the thermal diffusivity is constant;
• the domain is initially at a temperature of 0;
• At t>0, a small square of the surface ((x,y)∈R²) is instantaneously brought at a temperature of u₀. The rest of the boundary is maintained at a temperature of 0.

The solution to such problem is fairly easy to get with Green's functions when Neumann boundary conditions are imposed(i.e. imposed heat flux on the surface z=0). I would be surprised that no analytical solution to the same problem with Dirichlet conditions does not exist, although I don't manage to find it. How such boundary can be treated?

I looked into the solutions given by A.V. Liukov Analytical Heat Diffusion Theory (1968), but nothing looks similar to this. The issue, here is the combination of having Dirichlet BC and that the value of temperature on z=0 depends on x and y. Using the properties of door function may help (similarly to the dirac distribution for describing a point source in Green's problem), but I am still not sure how to tackle the problem.

Please let me know your opinion on this problem.
Thanks,
Florian

 

[hunt_mat]

Why isn't taking Laplace transforms viable here?