Suppose I am am investigating heat flow using the equation:
$$\partial_{t}T=\partial_{x}(k(x)\partial_{x}T)+Q$$
where $Q$ is a constant. I am investigating on a finite interval $[0,L]$ and suppose I can make several measurements of the temperature at the edges $T(t,0)=T_{0}(t)$ and $T(t,L)=T_{L}(t)$ say. I understand that I can use this to find the function $k(x)$.
The way to do this seems to be to take the Laplace transform of the heat equation and solve the corresponding Sturm-Louville equation in the $x$ variable and to differentiate the solution multiple times to find the function $k(x)$.
Is this the basic idea for doing this? I've not really studied this before and am a bit confused with how to proceed as all the stuff on inverse problems seems to be drowning in finding a-priori bounds of things.
If anyone has any references to this I would be most grateful.