Tips & Recommendations for Solving Laplace Equation Potentials.

Click For Summary
SUMMARY

This discussion focuses on solving Laplace's equation to find potentials in Cartesian, Cylindrical, and Spherical coordinates, with a particular emphasis on the Spherical case involving Legendre Polynomials. The user seeks guidance on determining the parameter l in the Legendre Polynomials, specifically how it relates to the order of the equation and the physical context of the problem. Key techniques mentioned include the Method of Frobenius and the concept of Orthogonality. The discussion highlights the need for clarity in questions posed to receive effective assistance.

PREREQUISITES
  • Understanding of Laplace's equation and its applications in physics.
  • Familiarity with Legendre Polynomials and their properties.
  • Knowledge of the Method of Frobenius for solving differential equations.
  • Basic concepts of Orthogonality in mathematical functions.
NEXT STEPS
  • Study the derivation and properties of Legendre Polynomials in detail.
  • Explore the Method of Frobenius and its application in solving differential equations.
  • Research the physical significance of the parameter l in the context of spherical harmonics.
  • Examine examples of Laplace's equation solutions in Spherical coordinates to understand practical applications.
USEFUL FOR

Students and professionals in physics and engineering, particularly those focusing on potential theory and mathematical methods for solving differential equations.

Fjolvar
Messages
156
Reaction score
0
Question on Solving Laplace Equation Potentials.

Hello, I'm learning how to solve Laplace's equation to find Potentials in Cartesian, Cylindrical, and Spherical Coordinates and let's just say it's not going as smoothly as I'd like. In particular, I'm having difficulty with the Spherical case which involves Legendre Polynomials, Method of Frobenius, Orthogonality, etc. Can anyone recommend any sources or even simply give me a hint/tips on how to approach these types of problems? Any advice would be greatly appreciated. Thank you.
 
Last edited:
Physics news on Phys.org
Perhaps my post is too vague, so let's try this: In the Spherical case, how do you determine Pl(X) in the Angular Equation of V(r,\vartheta) where \Theta(\vartheta) = Pl(cos(\vartheta))..

What determines l (lower case L) in the Legendre Polynomials when solving for Pl(X)..

I know that when l=0, Pl(X) = 1. When l=1, Pl(X) = X, etc. So what does l depend on and how does it relate to the order of the equation and the physics of a problem?
 
Last edited:
Your textbook or teacher shold be able to answer that, but your question here is still too vague.
 

Similar threads

Undergrad Boundary conditions for a stream function in a hydrodynamics problem
  • · Replies 5 ·
Replies
5
Views
1K
How to Solve the Laplace Equation for Potential Flow Around a Sphere?
  • · Replies 1 ·
Replies
1
Views
2K
Confused about the nature of Laplace vs Poisson equation in BVP
  • · Replies 11 ·
Replies
11
Views
3K
Graduate Solving Laplace's equation in polar coordinates for specific boundary conditions
  • · Replies 3 ·
Replies
3
Views
3K
Potential function of a flow around a stagnation point
  • · Replies 19 ·
Replies
19
Views
3K
Graduate Regular vs stable orbits in spherically symmetric potentials
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad What Are the Methods for Solving Laplace's Equation in 3D Coordinates?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad PDEs: Laplace's Equation over a Parallelogram
  • · Replies 10 ·
Replies
10
Views
3K
MHB Solve Laplace equation on unit disk
  • · Replies 33 ·
2
Replies
33
Views
5K
Graduate Laplace transform in spherical coordinates
  • · Replies 3 ·
Replies
3
Views
3K