Temperature distribution two semi-infintie bodies

[Drokz]

Hi there, I have to solve the following problem:

A semi-infinite body x<0 has thermal conductivity $$K_1$$, density $$\rho_1$$, and specific heat $$C_1$$. It is initially at temperature $$T_0$$. At time t=0, it is placed in thermal contact with the semi-infinite body x>0, which has parameters $$K_2$$, $$\rho_2$$, $$C_2$$, and is initially at temperature T=0. Find $$T_2(x,t)$$, the temperature in the second body.


$$\frac{\delta^2 T}{\delta x^2} = \frac{C \rho}{K}\frac{\delta T}{\delta t}$$

Does someone know what boundary conditions I have to apply? Tried using the Heaviside step function, but that got difficult when Fourier transforming it.

Thought about considering it as two different problems, but then I have only one boundary condition $$T(\infty, t) = 0$$?

Thanks in advance for your help.

 

[turin]

Thought about considering it as two different problems, but then I have only one boundary condition $$T(\infty, t) = 0$$?

Is there something wrong with this? This is what I would assume.