Boundary Conditions for 1D heat flow in Wire with source

[phil ess]

I'm trying to understand how to set up the problem of a 1D wire that is insulated at one end and has a heat source at the other. I know the heat law, from my textbook:

du/dt = B d²u/dx² + q(x,t) 0 < x < L, t > 0

Where q(x,t) is the source of heat.

The problem is, I want the heat source to be only at one end of the bar, but every example I find of this kind of problem has a heat source that is a function of x, which I don't understand, since the heat generated should be 0 for all x except x = 0, and I don't know how to deal with that.

Also, what do I do for the insulated boundary on the other side? I want the heat to continue to build without escaping.

This is so frustrating! Any help is greatly appreciated!

ps. The actual problem I am trying to solve is to find the pressure P(x,t) in a 1D finite reservoir, with a constant source of water at one end, but as far as I can tell, this problem is analogous, and I figured more people would be familiar with the heat equation example.

 

[gtoguy97]

Thanks!The solution to this problem is to use a Dirichlet boundary condition at x = 0, which states that u(0,t) = q(0,t), where q(x,t) is the heat source. This will ensure that the heat source is only present at the end of the bar. At x = L, you can use a Neumann boundary condition which states that du/dx = 0. This will ensure that there is no heat transfer across the boundary, thus preventing any heat from escaping.