I'm trying to solve numerically the heat equation on a rectangle, with homogeneous Dirichlet boundary conditions and a source term.
$u_t = u_{xx} + u_{yy} + Q(x,y,t,u) \qquad \text{in} \quad 0<x<1, \quad 0<y<2, \quad t>0$
with
$u=0 \quad \text{on} \quad x=0,1, \quad y=0,2.$
while
$u (t=0) = u_0(x,y) $
I want to test my code with an exact solution but I can´t find any for this particular case. Do you know any book, website, resource in which example solutions to this equation can be found?
Thank you