2-D Poisson Equation Boundary Value Prob

[Chingon]

Homework Statement

Solve the equation:
∂²F/∂x² + ∂²F/∂y² = f(x,y)

Boundary Conditions:
F=F_o for x=0
F=0 for x=a
∂F/∂y=0 for y=0 and y=b

Homework Equations

How can I find Eigengunctions of F(x,y) for expansion along Y in terms of X?

The Attempt at a Solution

I can't imagine what the Fourier transform of the generic f(x,y) looks like. Once this is done I'm supposed to be left with an ODE which is solvable.

 

[Chingon]

Ok, so based on my boundary conditions, I think I have to expand f(x,y) as a double Fourier cosine expansion, so:

f(x,y)= 1/2Ʃ(C₂cos(n∏x/a) + C₃cos(n∏y/b)) + ƩƩC₄cos(m∏x/a)cos(n∏y/b)

Also, I can let F(x,y)=u(x,y)=X(x)Y(y)

Plugging that into the original Poisson equation I end up with:
1/X(x)∂²X/∂x² + 1/Y(y)∂²Y/∂y² = f(x,y)/X(x)Y(y)

I think I'm doing something wrong by having the right-hand side of the Poisson equation equal to f(x,y)/F(x,y). It's also not clear to me if I can solve for any of the Fourier expansion coefficients or reduce the equation...

 

[Chingon]

Ok, so since ∂F/∂y = 0, that means it has a homogeneous solution of the form
Y=Ʃcos(n∏y/(b/2)) correct?

So that would cancel out the Y terms in the Fourier Expansion of f(x,y)?

 

[pasmith]

Homework Statement

Solve the equation:
∂²F/∂x² + ∂²F/∂y² = f(x,y)

Boundary Conditions:
F=F_o for x=0
F=0 for x=a
∂F/∂y=0 for y=0 and y=b

As with inhomogeneous linear ODEs, Poisson's equation can be solved by taking a particular solution $F_p$ with
$$\nabla^2 F_p = f(x,y)$$
subject to $F_p = 0$ on the boundary, and adding a complimentary function $F_c$ where
$$\nabla^2 F_c = 0$$
subject to the given boundary conditions for $F$.

To find $F_p$, you will want to find a family of eigenfunctions $\phi_{nm}(x,y)$ such that
$$\nabla^2 \phi_{nm} = k_{nm}\phi_{nm}$$
and such that $\phi_{nm}$ vanishes on the boundary. You can then take a linear combination of these so that
$$F_p(x,y) = \sum_{n}\sum_m A_{nm} \phi_{nm}(x,y)$$
with the $A_{nm}$ chosen so that $\nabla^2 F_p = f(x,y)$.