Two homogeneous rods have the same cross-section, specific heat $c$, and density $ρ$ but different heat conductivities $κ_1$ and $κ_2$ and lengths $L_1$ and $L_2$. Let $k_j$$=$$κ_j$/$cρ$ be their diffusion constants. They are welded together so that the temperature u and the heat flux $κu_x$ at the weld are continuous. The left–hand rod has its left end maintained at temperature $T_1$. The right–hand rod has its right end maintained at temperature $T_2$ degrees.
a) Find the equilibrium temperature distribution in the composite rod.
b) 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 $u_2(x,t)$.
a) The heat equation is $u_{xx} = 0$. Solving on the interval $0≤x≤L_1$ we have $u_1(x,t)=ax+b$, and on the interval $L_1≤x≤L_1+L_2$ we have $u_2(x,t)=cx+d$. There are four boundary conditions, what happens at the two endpoints, and the condition that both $u$ and the flux are continuous.
As the temperature is $T_1$ at $0$ we have $b=T_1$.
As the temperature is $T_2$ at $x=L_1+L_2$ have $c(L_1+L_2)+d=T$.
As $u$ is continuous at $x=L_1$ we have $aL_1+b=cL_1+d$.
As the flux is continuous at $x=L_1$ we must have $κ_1a=κ_2c$.
Solving a system with these 4 boundary conditions I can find the coefficients for $u_1$ and $u_2$, but then how can I solve part $b$? How can I find $u_1$ and $u_2$ knowing the initial conditions? Can someone help me?