1
$\begingroup$

This question comes from the text book Partial Differential Equations An Introduction, Walter A. Strauss. I was trying to do some of text book questions as I am doing a course in PDE's and had a bit of trouble with one of the questions.

Question

A homogeneous body occupying a 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)

Solution

So u know that the heat equation is given by $$ u_t = ku_{xx}$$ where $k\neq0$. Since we know the the temperature is in steady state (i.e. there is no heat lost or gained) then
$$ u_t= 0 $$ substituting this back into our PDE we find $$ 0 = ku_{xx} \implies 0 = u_{xx} .$$ Since the body is completely insulated we know $$ 0 = u_{x} $$ Finally we can can say $$ u = constant $$but i am a bit stuck on how the find the steady state temperature now that I have the general solution to u ? Any help would be appreciated!

$\endgroup$
3
  • 1
    $\begingroup$ You should also have the initial condition that gives you the total amount of the heat energy in your body. You can use it to determine the constant since, due to the fact that the boundaries are insulated, there is no loss of the energy with time. $\endgroup$
    – Artem
    Sep 23, 2016 at 23:42
  • $\begingroup$ No sorry that is the whole question $\endgroup$
    – Sam
    Sep 24, 2016 at 1:01
  • 1
    $\begingroup$ You state the initial temperature is given as $f(x)$. Then $\int_D f(x) \, dx$ is the total initial heat. $\endgroup$
    – Jeff
    Sep 24, 2016 at 3:08

0

Reset to default

You must log in to answer this question.