Working with initial and boundary conditions in PDEs

**albertlee** · September 9th 2010, 07:54 PM

The problem is stated here:
A homogeneous body occupying the solid region D is completely insulated. Its initial temperature is f(x). Find the steady-state temperature that it reaches after a long time. (Hint: No heat is gained or lost)

Now, it was asked many months ago:
Working with initial and boundary conditions in PDEs | MathFax.com

However, it's no longer on mathhelpforum and the solution there doesn't have the math expressions any more.

Can any one tell me how to solve it?

All I have now is:

du/dn = 0
u(x, 0) = f(x)
u_t = k*u_xx

Intuitively though, I think the solution is a constant.

please help

thanks

**Danny** · September 10th 2010, 06:54 AM

If steady state then $u_t = 0$ so $u_{xx} = 0$ so $u = c_1 x + c_2.$
Now you say that at the boundary there's no heat flow so $u_x = 0$ at the boundaries. Thus

$u_x = c_1 = 0$ so $u = c_2$ is your solution (as you said).