Is there any software that can solve 2nd order separable linear PDEs? (heat equation, wave equation, etc.)
For example, I'd like to get the analytic solution to the equation
$\partial_t B + G = R (\partial_{xx} B + \partial_{yy} B), \qquad B(x,y,t),G(t) = \text{known function of $t$},R=\text{constant}$
With Dirichlet BCs and zero initial condition: $B = 0 \in \partial \Omega, B(x,y,t=0) = 0$ on a domain $0 \le x \le L_x,0 \le x \le L_y, 0 \le t$.
I tried using mathematica, but it does not seem to provide any useful information:
My solution strategy without software is using Fourier series, I expect to see a convolution for the non-homogeneous part. Or using a Laplace transform. This is not a homework problem, I'm a researcher and often run into these equations and end up spending a lot of time deriving solutions every time I run into a new one.
I appreciate any help or suggestions.