Two dimensional heat equation with source

[Lionheart814]

I have the following PDE

PDE: u_t=A²(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 separation of variables to get

X''/X=k₁ Y''/Y=k₂ T'/(A²)T=k₁+k₂

where k₁ and k₂ 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.

 

[lippyka]

The source term (f(x,y,t)) can be accounted for by expanding it in a series of eigenfunctions. The technique is known as the Eigenfunction Expansion Method. The idea is to expand the source term as a linear combination of the eigenfunctions of the homogeneous part of the equation. For example, if we have the following PDE:PDE: ut=A2(uxx+uyy)+f(x,y,t) BC: ux(1,y,t)=0ux(0,y,t)=0uy(x,1,t)=0u(x,0,t)=0IC: u(x,y,0)=0We can represent the source term (f(x,y,t)) as an expansion of the eigenfunctions of the corresponding homogeneous equation:f(x,y,t)=∑mn αmnφmn(x,y)θmn(t)where φmn(x,y) are the eigenfunctions of the homogeneous equation and θmn(t) are the corresponding time-dependent coefficients.