The problem:

------------

Two rods of length

and

, which have heat diffusion constants

and

, respectively, are welded together.

The left end of the left rod (length

) is maintained at temperature

, the right end of the right rod is kept at temperature

. The temperature

and heat flux

are continuous across the weld.

I need to find the equilibrium temperature distribution across the welded rod of length

.

------------

I started setting up the data:

Since nothing was given for

and

, I assume they're

If we call the solution

, it seems that

with

and

We found in class that the general solution for the heat equation on a rod of infinite length with

is

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!