Hi... I've been set this question as homework, and I'm really struggling with it - I'd really appreciate any pointers that anyone could give me - I'm NOT asking for the answer, but instead would appreciate any methodological hints.

It's about a bar adjusting to ambient temperature.

The equation is:

$\displaystyle \frac{\partial y}{\partial t} =\kappa \frac{\partial^2 y}{\partial x^2}, t > 0, x \in[0, \pi] $ with boundary conditions:

$\displaystyle y(0,t) = T_0$

$\displaystyle \frac{\partial y}{\partial x}(\pi,t) - \kappa y(\pi,t)=-\kappa T_0$

$\displaystyle T_0$ is just a constant, but using the substitution $\displaystyle v(x,t) = y(x,t) - T_0$, I've gotten:

$\displaystyle \frac{\partial v}{\partial t} =\kappa \frac{\partial^2 v}{\partial x^2}, t > 0, x \in[0, \pi] $

with boundary conditions:

$\displaystyle v(0,t) = 0$ (first part of the Dirichlet conditions)

$\displaystyle \frac{\partial v}{\partial x}(\pi,t) = \kappa v(\pi,t)$

I'm also going to make the ansatz that $\displaystyle v(x,t) = X(x)T(t)$, implying:

$\displaystyle \lambda = \frac{\partial^2 X}{\partial x^2} = \frac{1}{\kappa} \frac{\partial T}{\partial t}$

I'm aware that there are obviously differing cases of $\displaystyle \kappa$ to deal with (positive, negative, and zero), but I'm stuck when dealing with the boundary conditions - I'm assuming I have to somehow show that the Dirichlet conditions hold for each case (or that it is trivial), but I can't see how to use $\displaystyle \frac{\partial v}{\partial x}(\pi,t) - \kappa v(\pi,t)=0$ to show that $\displaystyle v(\pi,t) = 0$ for each case of $\displaystyle \kappa$.

Does anybody have any ideas? I just can't see how to use this information to solve the equations - without proper boundary conditions, I can't solve anything at all, and I've been at this for a few hours now.