I have the following PDE

PDE: u_t=A^2(u_xx+u_yy)+f(x,y,t) such that A is a constant
BC: u_x(1,y,t)=0
u_x(0,y,t)=0
 u_y(x,1,t)=0
u(x,0,t)=0
IC: u(x,y,0)=0

I've set up my ODEs using seperation of variables to get

X''/X=k_1<br />
Y''/Y=k_2<br />
T'/(A^2)T=k_1+k_2

where k_1 and k_2 are constants.



How do I account for my source term (f(x,y,t))? I'm reading up on eigenfunction expansion but so far it's only for the dimensional case.