TEAM PROJECT. Series for Dirichlet and Nemnann (a) Show

Chapter 12, Problem 12.9

(choose chapter or problem)

TEAM PROJECT. Series for Dirichlet and Nemnann (a) Show that lin = 1'71 cos lie. "n = rn sin ne, II = 0, I, ... , are solutions of Laplace's equation -V2 u = 0 with ,211 given by (5). (What would Un be in Cartesian coordinates'? Experiment with small II.) (b) Dirichlet problem (See Sec. 12.5) Assuming that term wise differentiation is permissible. show that a solution of the Laplace equation in the disk r < R satisfying the boundary condition u(R, e) = I(e) (f given) is x [ (r)n u(r, B> = 00 + ~l an Ii cos lie (20) ( r)n ] + bn R sin nO where (In' bn are the Fourier coefficients of f (see Sec. 11.I). (c) Dirichlet problem Solve the Dirichlet problem using (20) if R = I and the boundary values are u(O) = -100 volts if -7r < 0 < O. u(O) = 100 volts if 0 < e < 7r. (Sketch this disk, indicate the boundary values.) (d) Neumann problem Show that the solution of the Neumann problem y211 = 0 if r < R, llN(R, e) = f(B) (where LIN = iJ"/iJN is the directional de11vative in the direction of the outer normal) is u(r, 0) = Ao + L rn(An cos IlO + Bn sin lie) n~1 with arbitrary Ao and I TI" --n-_--cl f f(A) cos nA de, 7rIlR -TI" I " Bn = n-l f fee) sin lie de. 7rIlR _ .. (e) Compatibility condition Show that (9), Sec. 10.4, impo~es on f(O) in (d) the "compatibility condition" (f) Neumann problem Solve y 2u = 0 in the annulus I < r < 3 if liTO, 0) = sin 0, U,(3, e) = o.