1 dimensional heat flow boundary conditions

hfitzgerald

1. The problem statement, all variables and given/known data
n is given by:
∂²Θ/∂x²=1/α² ∂Θ/∂t
, where Θ(x, t) is the
temperature as a function of time and position, and α²
is a constant characteristic for the
material through which the heat is flowing.
We have a plate of infinite area and thickness d that has a uniform temperature of 100◦C.
Suddenly from t = 0 onwards we put both sides at 0◦C (perhaps by putting the plate between
two slabs of ice).
Write down the four boundary conditions for this plate.


2. Relevant equations

I can't think of any relevant equations to this

3. The attempt at a solution
so far I have got
Θ(0, t)=0
Θ(d, t)=0 where d is the thickness of the bar.
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

HallsofIvy

Well, so far, all you have done is write down the problem!

theta_xx= (1/a^2)theta_t
theta(0, t)= theta(d, t)= 0, theta(x, 0)= 100.

Now, attempt a solution. What methods have you learned for solving such problems? Most common are "separation of variables" and "Fourier series", both of which will work here, but no one can make any suggestions until we know which methods you know and where you are stuck with this problem.

hunt_mat

I would try as separation of variables method, so write:
[tex]
\theta (t,x)=T(t)X(x)
[/tex]