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Elliptic Boundary Value Problem: Lapalce and Polar Coordinates

(lap) means the Laplacian
Vrr means the second derivative of V with respect to r
V(theta theta) means the second derivative of V with respect to theta

Solve:
(lap)V(r,theta)= Vrr+(1/r)Vr+(1/r^2)V(theta theta)=0
0 < r < 1, -(pi) < theta < pi

V(1,theta) = {1, -(pi/2) < theta < (pi/2)
{0, elsewhere

Please show all work including the derivation of any eigenvalues or eigenvectors.
Thank you

Solution Summary

This shows how to solve an elliptic boundary value problem with Laplace and polar coordinates

$2.19