Working with initial and boundary conditions in PDEs

• Apr 16th 2009, 09:55 AM
dlbsd
Working with initial and boundary conditions in PDEs
Hello,

I have this problem:

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)

From what I understand the initial condition is written as $u(x, t_o) = f(x)$
and since the region is insulated the neumann condition is interpreted to be $\partial{u}/\partial{n} = 0$

At this point, how would i a derive a means to define a steady-state temperature?
• Apr 17th 2009, 11:39 PM
Coomast
Quote:

Originally Posted by dlbsd
Hello,

I have this problem:

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)

From what I understand the initial condition is written as $u(x, t_o) = f(x)$
and since the region is insulated the neumann condition is interpreted to be $\partial{u}/\partial{n} = 0$

At this point, how would i a derive a means to define a steady-state temperature?

Start from the basic conductivity equation (in 1D). This is:
$\frac{\partial u}{\partial t}=\frac{\lambda}{\rho c_p}\left(\frac{\partial^2 u}{\partial x^2} \right)$
$\frac{\partial u}{\partial t}=0$
$\frac{\partial^2 u}{\partial x^2}$
$u=A\cdot x+B$