Solve Laplace Equation in Oblate/Prolate Spheroidal Coordinates

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SUMMARY

This discussion focuses on solving the Laplace equation in oblate and prolate spheroidal coordinates using the separation of variables method. Participants emphasize that the scalar Laplace equation can be approached by expressing the solution as a sum of products of functions of one variable. The conversation highlights the necessity of utilizing Legendre polynomials and circular functions, as indicated by Mathworld, which confirms that these coordinate systems allow for solutions via separation of variables. The discussion also points out the complexity of the algebra and calculus involved in this process.

PREREQUISITES
  • Understanding of Laplace's equation and its applications
  • Familiarity with oblate and prolate spheroidal coordinates
  • Knowledge of separation of variables method in partial differential equations
  • Proficiency in Legendre polynomials and circular functions
NEXT STEPS
  • Study the separation of variables technique in depth
  • Explore the properties and applications of Legendre polynomials
  • Review the derivation of solutions for Laplace's equation in various coordinate systems
  • Investigate numerical methods for solving partial differential equations
USEFUL FOR

Mathematicians, physicists, and engineers who are working on problems involving Laplace's equation, particularly in specialized coordinate systems, will benefit from this discussion.

Aamodt
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Hi, I'm trying to solve the Laplace equatio in oblate and prolate spheroidal coordinates, but it's proving to be too much for me to handle, can anyone help me out?
You can see the equations I'm using in:
http://mathematica.no.sapo.pt/index.html
 
Last edited:
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I "Cannot find server". Can you attach a written document with your work...?

Daniel.
 
I have corrected the problem, you can now access the web page with the equations, thanks for the warning.
 
Aamodt said:
Hi, I'm trying to solve the Laplace equatio in oblate and prolate spheroidal coordinates, but it's proving to be too much for me to handle, can anyone help me out?
You can see the equations I'm using in:
http://mathematica.no.sapo.pt/index.html
Laplaces equation for what (scalar, vector, tensor rank-2?). Using what method (numerical solution, separation of variable, integral transforms?).
I would guess that you intend to solve the scalar laplace equation using separation of variables. So you presume the solution can be written in the form of a sum of terms that are products of functions of one variable. Then the partial differential equation implies that the functions of one variable satisfy some strum louiville problem.
Mathworld says your two systems are among the 13 where laplaces equation can be solved by separation of variables and that solutions involve Legendre polynomials and circular functions. In any case you are looking at some messy algebra and calculus.

http://mathworld.wolfram.com/LaplacesEquation.html
 

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