How can one solve a Parabolic PDE (like the wave or diffusion equations) if the boundary conditions were given **over ranges**?
Here is an example: How to solve the equation $u_{xx}+u_{yy}-\alpha^{2}u_{t}=0$ over the following boundary conditions, assuming zero initial condition, variables $x\in [0,a], y\in [0,a]$, and $b<a$:

$u=0$ for $x=0$;

$u=0$ for $y=0$;

$u=0$ for $x=a\ \ \text{and}\ y\in[0,b]$;

$u=0$ for $y=a\ \ \text{and}\ x\in[0,b]$;

$u=0$ for $x=[(a+b)-y]\ \ \text{for}\ x\in[b,a]$ and $y\in[b,a]$

Note that these boundary ranges look geometrically like a square with a cut at its far corner. Also note that $b$ should **not** be assumed to be very small (as in $b<<a$) as to blindly grant the usage of perturbation techniques (though this would also be of interest, if generalisations can be shown).

I tried asking this question first on stackexchange, but no answer was given (since March 2014): https://math.stackexchange.com/questions/713298/solving-a-parabolic-pde-with-boundary-conditions-given-over-ranges