boundary condition for Heat Transfer

sylvain

I am trying to write a program (based on finite difference method) able to predict the Temperature within the skin of a human being when a laser is pointing toward the skin. The energy associated with the laser decays exponentially with the depth.

I am having troubles to find and/or solve the PDE at the boundary. The surface of the skin is submitted to convection, radiation and a heat flux from the laser energy.

My problem is when I write my energy balance on the first control volume (to solve for the temperature at the surface, the boundary) I end up with a hyperbolic equation (1st order derivative in time, and first order derivative in space). The PDE for the rest of the domain (within the skin) are parabolic as expected.

Should my PDE at the boundary be hyperbolic? If yes, do I use a first order upwind scheme to solve it? If yes, to express the first order difference in space what temperature do I use (T at the surface and the first node within the domain temperature, or the temperature of a node OUT of the domain???).

Thanks