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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)?

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  • $\begingroup$ As stated, the boundary and initial conditions are incompatible. There are techniques to work around this, generally either based on finding a weak solution, or based on finding a strong solution to a "close" problem (one where the boundary condition is initially compatible with the initial condition and then rapidly changed to the desired boundary condition). Such weak solution techniques can actually be done using separation of variables...if the boundary conditions are nice enough. (For example, I have previously solved the problem $u_t=u_{xx},u(x,0)=0,u(0,t)=0,u(1,t)=1$.) $\endgroup$ – Ian Sep 25 '17 at 14:47
  • $\begingroup$ (Cont.) I'm not so sure that this "weld" boundary condition is nice enough for that to work. Because if you think about it, if you had already solved for $u_2$, then you would have a time-dependent Robin condition in the problem for $u_1$. Time-dependent BCs tend to be fairly difficult to handle analytically. $\endgroup$ – Ian Sep 25 '17 at 14:50

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