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On Laplace's Equation
Posted:
Sep 14, 2011 4:04 PM
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Resent-From: <bergv@illinois.edu>
From: Anamitra Palit <palit.anamitra@gmail.com>
Date: September 14, 2011 8:57:51 AM MDT
To: "sci-math-research@moderators.isc.org"
<sci-math-research@moderators.isc.org>
Subject: On Laplace's Equation
Let us consider Laplace's equation with azimuthal symmetry
The solution is given by:
psi[r,theta]=summation over n [from zero to infinity] [An r^n+ Bn r^{-
(n+1)}]Pn[Cos(theta)] ----------- (1)
An and Bn are constants.Pn stands for Legendre-polynomials. "n" is a
suffix.
Now we look for a an elementary solution of the same equation in the
form
psi=An f(psi) r^n +Bn r^{-(n+1}}
Substituting the above "psi" into the original equation[Laplace's
Equation with azimuthal symmetry] we may obtain expressions for f.
So we have a different solution now.The general solution obtained by
summing up these new elementary solutions may cover a broader range of
boundary conditions, uncovering new types of solutions.
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