I have a question about a Robin Boundary Condition on a BVP. The BVP is given by:
$$\begin{cases} u_t=u_{xx} & x\in(0,1), t>0\\ u(0,t)-u_x(0,t) = 20 & u(1,t)=10\\ u(x,0)=e^x & x\in(0,1) \end{cases}$$
Now I tried to find the Steady State Solution, which is $u(x,t)=v(x)$, $v''(x)=0$, with the conditions $v(0)-v'(0) = 20$ and $v(1)=10$. Then, I got that this gives: $$v=-5x+15$$ Now, I am asked if $v(0)=20$, and if not, why does it make physical sense that it is not $20$?
I answered that $v(0)=15$, and this makes sense as the steady state does not depend on time, so if we were to consider a bar with temperature $u$, then if the bar on the left hand side is exposed to a temperature of $20$ and the right hand side is exposed to temperature of $10$, then at any point by the Diffusion Equation the temperature of the bar should be the average of the two.
Am I thinking about this correctly? Or is there another way to interpret this?