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

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

(a)Use the Method of Separation of Variables to completely derive the solution of the BVP. Show all details of the procedure, and of your work. Do not start with with the eigenfunctions, the are to be derived. The final solution must be summarized and circled at the end of your work, showing the final complete expression for u(r,theta) and any coefficents and necessary conditions for f(theta).

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

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Expert Solution

Colby Messinger answered 1 year ago

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