Firstly, my main question boils down to speaking about the initial conditions and boundary conditions. I was given: $$ u(0,y,t) = u(\pi,y,t) = u(x,0,t) = u(x,\pi,t) = 0 $$ but then the initial condition was: $$ u(x,y,0) = 1 $$ Aren't the initial and boundary conditions inconsistent in such a case? For example, what is the value of ##u(0,0,0)##? Based on the boundary conditions, it should be 0, but based on the initial conditions, it should be 1, no? Why exactly is this okay, if it is okay?