# Need help with two dimensional heat equation with source

• March 31st 2010, 11:04 AM
Lionheart814
Need help with two dimensional heat equation with source
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

$X''/X=k_1
Y''/Y=k_2
T'/(A^2)T=k_1+k_2$

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.