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Elliptic Boundary Value Problem: Lapalce and Polar Coordinates

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(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

(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

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Solution Preview

Please see the attached doc file.

We have:

We consider:

Therefore by substitution we have:

and we can simplify as:

This must be a constant. The constant can be positive, negative or zero. In these kinds of problem, the positive choice works. We call that k^2. Then:

k must be a whole number according ...

Solution Summary

This shows how to solve an elliptic boundary value problem with Laplace and polar coordinates. The means for the second derivative with respect to theta are given.