I am given a physics problem and I am struggling with interpreting it as a boundary value problem for Laplace equation. I do know how to solve BVPs, but I don't know how to build BVP from a physical problem.
A two-dimensional rectangular plastine of height $a$ and width $b$ gives off heat of density $q$. Its right and top sides are heat-isolated, and its bottom and left sides are forced to zero temperature. Find the stationary temperature distribution.
I can only intuitively suggest that the corresponding BVP is $$\Delta u = -q \frac{\partial u}{\partial t}, \;\; u|_{y=a} = u|_{x=b} = u|_{x=0} = u|_{y=0} = 0$$
but that's for sure wrong as then the answer is $u=0$ trivially.
The correct boundary condition at $y=a$ and $x=b$ are: $$ \frac{\partial u}{\partial x}\bigg |_{y=a}=0\\ \frac{\partial u}{\partial y}\bigg |_{x=b}=0\\ $$ which imply that there is no conduction of heat across these sides (heat-insulation). The other 2 boundary conditions are as stated.