I was presented this problem in PDE class involving heat equation on unit circle in polar coordinates using separation of variables, giving the following heat equation problem:
$ u_t = 9\Delta u ; \space \space \space x^2+y^2 < 1 ; \space \space \space t>0 $
$ u(x,y,t) = t ; \space \space \space x^2+y^2=1 ; \space \space \space t \geq 0 $
$ u(x,y,0) = 0 ; \space \space \space x^2+y^2 \leq 1 $
The first thing I noticed was there is a certain symmetry in $ \theta $ because the boundary and initial conditions are symmetric in $ \theta $ am I right on this? What really prevents me doing this is the following: If I wish to separate variables then without symmetry it is $ u(r,\theta,t) = R(r)\Theta(\theta)T(t) $ and with symmetry I need to omit the $ \theta $ dependence now separating the variables gives obviously ODE's with initial values so the conditions translate to $ R(1)\Theta(\theta)T(t) = t $ and $ R(r)\Theta(\theta)T(0) = 0 $ the second one seems to imply (because it holds for all r and $ \theta $ and R and $ \Theta $ are not identically zero) that $ T(0)=0 $ but the ODE for T (the max order derivative WRT t is one) is :
$ T'+\lambda T = 0 $
and the general solution to this is well known $ T(t) = Ce^{-\lambda t} $ and the condition I thought $ T(0)=0 $ implies the only solution is $ T=0 $ so the solutions are all identically zero because they are multiplied by $ T(t) = 0 $ but obviously this is wrong? Am I missing something here? Do you think there is a problem with my attempts? Thanks