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Consider, in the xy-plane, the upper half disk of radius R, with temperature u governed by...

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Advance Math

Consider, in the xy-plane, the upper half disk of radius R, withtemperature u governed by Laplace's equation, and withzero-Dirichlet B.C. on the (bottom) flat part of the disk, andNeumann B.C. ur=f(theta), on its (top) curved boundary.

(a)Use the Method of Separation of Variables to completelyderive the solution of the BVP. Show all details of the procedure,and of your work. Do not start with with the eigenfunctions, theare to be derived. The final solution must be summarized andcircled at the end of your work, showing the final completeexpression for u(r,theta) and any coefficents and necessaryconditions for f(theta).

(b) Obtain the complete simplified solution of the BVP above,with R=2 and f(theta) = 100.

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