Poisson's Eq. with separable variables?

[badkitty]

Greetings-
In trying to solve a thermal stress problem, I have encountered an inhomogeneous differential equation of the following general form:
$$\nabla^2 \Phi(r,z) = F_r(r)F_z(z)$$
Solving the homogeneous case is no problem, as it is kind of a classic. Is there a route to finding a particular solution for the inhomogeneous case? Since my "charge density" (it's really temperature) is separable, I expected this to be straightforward.

It may, in fact, be straightforward, but it is still beyond my ken.

Thanks,
-BK

 

[the_wolfman]

You can solve the equation using Green's Functions.

The idea is to solve the equation
$\nabla^2 G(r,r',z,z') = \delta(r - r',z-z')$

Once you know G you can integrate to find $\Phi$

$\Phi(r,z)= \int d^3r' F_r(r')F_z(z') \delta(r - r',z-z')$

 

[jasonRF]

Greetings-
In trying to solve a thermal stress problem, I have encountered an inhomogeneous differential equation of the following general form:
$$\nabla^2 \Phi(r,z) = F_r(r)F_z(z)$$
Solving the homogeneous case is no problem, as it is kind of a classic. Is there a route to finding a particular solution for the inhomogeneous case? Since my "charge density" (it's really temperature) is separable, I expected this to be straightforward.

It may, in fact, be straightforward, but it is still beyond my ken.

Thanks,
-BK

You didn't really give us much info. What is you domain? What are the boundary conditions? Often a particular solution can be found by techniques like a) expanding the source term in eigenfunctions of the homogeneous problem, or b) integral transform techniques. Sometimes you can use the method of images to find the Green's function mentioned by the_wolfman. Again, without more info I'm not sure what more I can say to help you. Also, are F_r and F_z arbitrary or specific given functions?

jason