The question goes as follows;
Two homogeneous rods have the same cross section, specific heat c, and density $\rho$ but different heat conductivities $k_1$ and $k_2$. The rods are welded together so that the temperature $u$ and heat flux $ku_x$ are continuous. The left-hand rod has its left end maintained at temperature $T1$. The right-hand rod has its right end maintained at temperature $T2$ degrees.
At t = 0 the temperature $u_1(x,0)=0$ for $0<x<1$ and $u_2(x,0)=0$ for $1<x<2$. Determine $u_1(x,t)$ and $u_2(x,t)$.
The boundary conditions at $x=1$ follow from continuity, as stated in the question. However, I do need boundary conditions at $x=0$ and $x=2$. This is where I get confused. The question states that the ends are maintained at $T_1$ and $T_2$ respectively, but the initial conditions state that the temperature at $t=0$ equals $0$.
What are the correct boundary conditions? And how to solve the problem (I was thinking about Neumann or Dirichlet methods)?