TEAM PROJECT. Series for Dirichlet and Neumann Problems A) Show that un rncos ne, un = rn sin ne, n = 0, 1, are solutions of Laplace's equation V2u = 0 with V2u given by (5). (What would un be in Cartesian coordinates? Experiment with small n)
B) Dirichlet problem:
Assuming that termwise differentiation is permissible, show that a solution of the Laplace equation in the disk r < R satisfying the boundary condition u(R, 0) = f(e) (R and fgiven) is
u(r) = ao + Σ (an(/rR)n cos ne
where an, b are the Fourier coefficients of f.
C) Dirichlet problem. Solve the Dirichlet problem using (20) if R = 1 and the boundary values are u(e) = -100 volts if -T<< 0, u (8) = 100 volts if 0<