I am given the heat equation with a source term in polar coordinates.
$$ u_{t} = k ( \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u}{\partial r}) + \frac{1}{r^{2}}\frac{\partial^2 u}{\partial \theta^2}) + g(r,\theta,t) $$
The boundary conditions are such that: $$ u(1,\theta, t)= 0, u(r,0, t)= u(r,\frac{\pi}{2}, t)=0 $$
The initial condition is some function such that $ u(r,\theta, 0) = f(r,\theta,t)$.
I've attempted solving for the solution through eigenfunction expansion, but I may need a different method to solve the above PDE.
Is there a better way to go about this than through eigenfunction expansion?