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- 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):

I looked into the solutions given by A.V. Liukov

Please let me know your opinion on this problem.

Thanks,

Florian

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)∈R
^{2}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)∈R
^{2}) is instantaneously brought at a temperature of u_{0}. The rest of the boundary is maintained at a temperature of 0.

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