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On Laplace's Equation
Posted:
Sep 14, 2011 4:04 PM


ResentFrom: <bergv@illinois.edu> From: Anamitra Palit <palit.anamitra@gmail.com> Date: September 14, 2011 8:57:51 AM MDT To: "scimathresearch@moderators.isc.org" <scimathresearch@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 Legendrepolynomials. "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.



