Temperature distribution due to a periodic heat flux

[ehsan666]

I have a half-space where at z=0, I have a periodic heat flux q=A*sin(2w*t) on circular region with radius a, where the rest of surfaces can be assumed adiabatic. I would like to find the temperature distribution due to this boundary condition. I also would like to find the frequency-dependent response of the system.

I would appreciate your suggestions how to tackle the problem. I already started working on it, by considering an axisymmetric model and Laplace transformation for T.

Thanks,
Ehsan

 

[IRstuff]

A drawing or picture would be beneficial

TTFN
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[ehsan666]

Here is top and side views of the problem.
In the problem, we have a homogeneous solid with known thermal conductivity and specific heat, Q(t) (periodic heat flux) is a known function as well. We wanted to find the steady-state temperature distribution.