Thread: 3D Boundary Value Heat Transfer with constant flux and nonzero boundary

1. 3D Boundary Value Heat Transfer with constant flux and nonzero boundary

I'm having trouble getting started on this problem. Here's the question:

My issue is in setting up the governing partial differential equation in 3 dimensions. What I've tried so far is setting du/dt equal to the Laplacian of u(x,y,z,t) + g(x,y). I'm not entirely sure if I can do this, though. It leads me to assume u(x,y,z,t) = X(x)Y(y)Z(z)T(t), but I'm not sure how to cancel out the inhomogeneous boundary conditions.

I know that if g were a constant I could let v = u - g, but since it isn't its derivative with respect to x and y won't go away and v will stay inhomogeneous.

Sorry if that sounds like word vomit - its late and I've been up reading textbooks that only seem to have one dimensional examples. If anybody could shed some light on how to tackle this problem I would be very thankful - I'm mostly looking for a starting point and hopefully I can take it from there.

2. Nothing in the problem says that heat is being generated inside the block. Your differential equtaions should be

$u_t = u_{xx} + u_{yy} + u_{zz}$

Then set up the apporxiate BC's.