How to Derive the Finite Difference Laplacian in Various Coordinates?

AI Thread Summary
The discussion focuses on deriving a finite difference expression for the Laplacian in various coordinate systems, specifically Cartesian, cylindrical, and spherical. The original poster initially sought help with integration but realized a simpler approach might exist through expansion techniques. They successfully derived the finite difference Laplacian in Cartesian coordinates but struggled with incorporating the radial factor in the expression. The problem was ultimately resolved, and there is a suggestion to share the solution for future reference. Sharing solutions can greatly assist others facing similar challenges.
mojomike
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Updated: Finite difference of Laplacian in spherical

Homework Statement



I understand the problem a little better now and am revising my original plea for help. I don't actually need to integrate the expression. Integration was just one technique of arriving at a finite difference expression for the Cartesian Laplacian. I'm now convinced there is a simpler way to do it, possibly with some kind of expansion.

Here's what I'm seeking: A finite difference, 1-dimensional expression of the Laplacian, expressed so that it can be made to work in cartesian(n=0), cylindrical(n=1), or shperical(n=2) coordinates.

2 φ(r) dr =

\frac{1}{r^{n}}\frac{\partial}{\partial r}r^{n}\frac{\partial}{\partial r}\Phi(r) dr<br />

Homework Equations

see above

The Attempt at a Solution



I have solved the finite difference laplacian in Cartesian coordinates, which is pretty easy. I just don't know how to express that double derivative with the rn factor in the middle in finite difference form. I'll continue to update my request as I make progress, so nobody thinks I'm loafing here. Thanks again.

,Mike S
 
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Problem resolved. Thanks anyway!
 
Dear Mike,

Since you've resolved the problem, perhaps you could also post the solution? This way, when someone with a similar problem searches for it, they will find an answer instead of just the question.

Cheers,
 
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