Consider the heat flow in a one-dimensional rod of length 4 with cross-section of 1 and constant thermal conductivity of 1. Assume that the rate at which the heat energy flows out of the rod at "that" end is twice the temperature difference between the temperature at the right end and the air temperature of 23 degrees celcius. Derive the boundary conditionsat the right end.

I am not sure what to make of " that " end for now, leaning toward the right end, but i'll do both ends. I just want to check if i am getting the right boundary conditions here.

Right end : Energy leaving right end is given by $\displaystyle -\lambda A \frac{\partial u}{\partial x}(4,t) = 2(u(4,t)-23)$ which becomes

$\displaystyle -\frac{\partial u}{\partial x}(4,t) = 2(u(4,t)-23)$ and then... not sure. I assume they want 2 boundary conditions at the right end because they ask for boundary conditions.

I would say the other boundary condition could be $\displaystyle u(4,t) = u(4,0)-t\frac{\partial u}{\partial x}(4,t)$

Am i on the right track here ? I don't have much experience with boundary conditions so i don't have much of a feel for what " fits ". Which boundary conditions would you put here ? Any help is much appreciated !