Two homogeneous rods have the same cross section, c and $\rho$ are the same. lengths $L_1$ and $L_2$, $\kappa_1$, $\kappa_2$ are different. $k_j = \frac{\kappa_j}{c\rho}$. Left side of the left rod has temperature $T_1$ and right side of the right rod $T_2$.
I needed to find the equilibrium temperature distribution in the composite rod.
I divided them in two parts one for $0\leq x\leq L_1$ and for $L_1\leq x\leq L_2$ repectively $u_1(x)$ and $u_2(x)$.
$u_1(x)=c1x+c2$ and $u_2(x)=d1x+d2$.
After calculations I come to the conclusion:
$u_1(x)=\frac{k_2(T_1-T_2)}{L_2k_1+k_2}x+T_1$
$u_2(x)=\frac{k_1(T_1-T_2)}{L_2k_1+k_2}x+\frac{T_1(k_1L_2+k_1L_1)+T_2(k_1L_2+k_1L_1)}{L_2k_1+k_2}+T_2$
Now the question is: at $t=0$, $u_1(x,0)=0$ and $u_2(x,0)=0$. Determine $u_1(x,t)$ and $u_2(x,t)$
Any help on how to start this? Forgot how to begin with this.
Do I need seperation of variables? I only know how to do it with homogeneous boundary conditions, but these arent. The initial condition is $0$ though for both rods.
Do I need to do the same process, but then use $\phi_1(0)=T_1$ and $\phi_1(L_1)=\phi_2(L_1)$ for rod 1?