just a question regarding neumann conditions, I seem to have forgotten these things already. I think this question is answerable by a yes or a no.
So given the 2D heat equation,
$$ \frac{\partial T}{\partial t} = \alpha \left [\frac{\partial T^{2}}{\partial x^{2}} + \frac{\partial T^{2}}{\partial y^{2}} \right ] $$
If I assign a neumann condition at say, x = 0;
$$ \left.\begin{matrix} \frac{\partial T}{\partial x} \end{matrix}\right|_{x=0}=k $$
Does it still follow that at both $T(x,y,t)$ and $\frac{\partial T}{\partial t}$, the condition holds?
Does this follow that everytime, the correct value of $\frac{\partial T^{2}}{\partial x^{2}} + \frac{\partial T^{2}}{\partial y^{2}}$ at $\frac{\partial T}{\partial t}$ follows the condition $\left.\begin{matrix} \frac{\partial T}{\partial x} \end{matrix}\right|_{x=0}=k$ ?
Thank you!