Thread: Working with initial and boundary conditions in PDEs

1. Working with initial and boundary conditions in PDEs

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.

thanks

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

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

3. err... Danny

are you just assuming that the body is 1Dimensional?

4. I am considering that you gave me

$\displaystyle u_t = k u_{xx}$

and

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

implying a ($\displaystyle 1+1$) dimensional problem.