Is a 3D or 2D Poisson's equation separable or non sperable?

Question

Can someone please explain to me if a 2/3D Poisson's equation is separable or non separable? Thank you

Answer 1

May be what you are looking for is, solving poission's equation using separation of variable method. Here is a link,http://farside.ph.utexas.edu/teaching/em/lectures/node66.html, to explore more about it.

Answer 2

For a homogenous linear PDE to be separable, you must be able to write the differential operator as the sum of two or more differential operators involving non-empty, non-overlapping subsets of the variables in the original PDE. Then, if you solve the eigenvalue problem for these differential operators (which should be easier since the number of variables in each has reduced - in many cases, we simply get ODEs), and assuming appropriate conditions (e.g. the differential operators are Hermitian, the eigenfunctions satisfy appropriate completeness relations, etc.), you can write the solution to the original PDE as a linear combination of products of the separated eigenfunctions with appropriate coefficients. Then, the original PDE, together with boundary conditions, constrain the allowed eigenvalues, any constants of integration in the eigenfunctions and the aforementioned coefficients of the linear combination, hopefully giving enough information to specify a unique solution.

For instance, the Laplace equation is separable since we can decompose the Laplacian into the sum of the second derivatives with respect to $x,y,z$ in Cartesian coordinates, or a radial and an angular part in spherical coordinates. In the first case the eigenfunctions are complex exponentials (or, equivalently sines and cosines) in $x,y,z$ separately, whereas in the second case we get eigenfunctions are complex exponentials (or, equivalently sines and cosines) in $r$ and the famous spherical harmonics $Y_{l,m}$ in $\theta,\phi$. The choice of decomposition here is suggested by the boundary conditions - if we know our solution on the walls of a rectangular box, we should probably choose Cartesian coordinates; if we know them on the surface of a sphere, we should probably go with spherical coordinates.

I'm a little unsure what is meant by separable in the case of an inhomogenous linear PDE such as the Poisson equation, but I suppose it just means that the underlying homogenous linear PDE, in this case the Laplace equation is separable: since, in this case, we could expand the inhomogenous term in terms of the eigenfunctions from the homogenous case (in the case of the Laplace equation and Cartesian coordinates, this is just a 3D Fourier series expansion), and then we still get the desired simplification of the problem as we did in the homogenous case.

Thus, I would probably categorise the 3D Poisson equation as separable. I would go ahead and put it into the software like this. You can always go back and check that the differential equation and boundary conditions are indeed satisfied (approximately satisfied, if this is a numerical solver).