I am struggling with the following problem:
One edge of the square plate with insulated faces is kept at uniform temperature $u_{0}$ and the other three edges are kept at temperature zero. Without solving a boundary value problem, but by superposition of solutions of like problems to obtain the trivial case in which all four edges are at temperature $U_{0}$, show why the steady temperature at the center of the given plate must be $U_{0}/4$.
What I tried:
Laplace equation of a PDE with four edges have the boundary conditions of the form
$$u(0,y)=g_{1}(y)$$ $$u(L,y)=g_{2}(y)$$ $$u(x,0)=f_{1}(x)$$ $$u(x,H)=f_{2}(x)$$
But all the boundary conditions here are non-homogenous, to get a homogeneous boundary conditions in order to solve the PDE we must split the solution into four parts and the add up the solutions of the four parts after solving each individually with the boundary conditions of each part being homogeneous (This can be done due to the linearity property). An example of the boundary conditions for one part is given below.
$$u(0,y)=0$$ $$u(L,y)=g_{2}(y)$$ $$u(x,0)=f_{1}(x)$$ $$u(x,H)=f_{2}(x)$$
Is my explnation correct and could it be improved upon?