I have the following problem: given a rod of length $L$, the heat equation describing the behaviour of the temperature $u(x,t)$ on this rod, is: $\partial_t u(x,t)=k(x)\partial_{xx}u(x,t)$ in which the function $k(x)$ is unknown. We have also the following boundary and initial conditions: $u(x,0)=T_0$ $u(0,t)=u_0(t)$ $u(L,t)=u_L(t)$ and we have also the following functions in two points $x_0$ and $x_1$ on the rod: $u(x_0,t)=u_{x_0}(t)$ and: $u(x_1,t)=u_{x_1}(t)$
The question now is: is it possible to find the function $k(x)$ with only this informations available?