Heat Equation without Fourier Series

The problem:

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Two rods of length $\displaystyle L_1$ and $\displaystyle L_2$, which have heat diffusion constants $\displaystyle k_1$ and $\displaystyle k_2$, respectively, are welded together.

The left end of the left rod (length $\displaystyle L_1$) is maintained at temperature $\displaystyle 0$, the right end of the right rod is kept at temperature $\displaystyle T$. The temperature $\displaystyle u(x,t)$ and heat flux $\displaystyle ku_x$ are continuous across the weld.

I need to find the equilibrium temperature distribution across the welded rod of length $\displaystyle L_1+L_2$.

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I started setting up the data:

$\displaystyle u_t = k_1u_{xx}$

$\displaystyle v_t = k_2v_{xx}$

$\displaystyle u(0,t)=0$

$\displaystyle v(L_2,t)=T$

$\displaystyle u(L_1,t)=v(0,t)$

$\displaystyle k_1u_x(L_1,t)=k_2v_x(0,t)$

Since nothing was given for $\displaystyle u(x,0)$ and $\displaystyle v(x,0)$, I assume they're

$\displaystyle u(x,0)=f(x)$

$\displaystyle v(x,0)=g(x)$

If we call the solution $\displaystyle w(x,t)$, it seems that

$\displaystyle w(x,t)=\bigg\{\begin{array}{lr}u(x,t)&:x\leq L_1\\v(x-L_1,t)&:x>L_1\end{array}$

$\displaystyle w(0,t)=0$

$\displaystyle w(L_1+L_2,t)=T$

$\displaystyle w(x,0)=\bigg\{\begin{array}{lr}f(x)&:x\leq L_1\\g(x-L_1)&:x>L_1\end{array}$

with $\displaystyle f(L_1)=g(0)$ and $\displaystyle k_1f'(L_1)=k_2g'(0)$

We found in class that the general solution for the heat equation on a rod of infinite length with $\displaystyle u(x,0)=f(x)$ is

$\displaystyle \frac{1}{\sqrt{4\pi kt}}\int_{-\infty}^{\infty}\exp\left(-\frac{(x-y)^2}{4kt}\right)f(y)\,dy$

However, this doesn't seem to help because we have finite length and boundary conditions. And here I get lost (before I really even begin!) because there is just a ridiculous amount of information and I'm not really sure what to do with it.

Help is appreciated!