Heat equation with an unknown diffusion coefficient

Question

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?

Answer 1

Not always. Suppose the boundary conditions are

$$ u(0,t) = u(L,t) = T_0 $$

Then $u \equiv T_0$ is a solution to the heat equation for any diffusion coefficient $k(x)$.

Answer 2

Note that the boundary conditions are inhomogeneous. So we should first apply the method in http://maths.swan.ac.uk/staff/vl/pdes-course.pdf#page=38 , i.e. let $u(x,t)=v(x,t)+u_0(t)+\dfrac{x}{L}(u_L(t)−u_0(t))$ ,

Then $\partial_tu(x,t)=\partial_tv(x,t)+\partial_tu_0(t)+\dfrac{x}{L}(\partial_tu_L(t)−\partial_tu_0(t))$

$\partial_xu(x,t)=\partial_xv(x,t)+u_L(t)−u_0(t)$

$\partial_{xx}u(x,t)=\partial_{xx}v(x,t)$

$\therefore\partial_tv(x,t)+\partial_tu_0(t)+\dfrac{x}{L}(\partial_tu_L(t)−\partial_tu_0(t))=k(x)\partial_{xx}v(x,t)$

$\partial_tv(x,t)-k(x)\partial_{xx}v(x,t)=-\partial_tu_0(t)-\dfrac{x}{L}(\partial_tu_L(t)−\partial_tu_0(t))$

with $v(0,t)=0$ , $v(L,t)=0$ and $v(x,0)=T_0-u_0(0)-\dfrac{x}{L}(u_L(0)−u_0(0))$