Laplace equations in circular regions: Dirichlet problems on annular regions
show that the steady state solution in the annular region is
u(r,theta)= a_0*((ln(r)-ln(R_1)/(ln R_1-ln(R_2))+ sigma(n=0..infinity)*[a_n*cos(n*theta)+b_n*sin(n*theta)]*(R_1/r)^n*((R_2^2n-r^2n)/(R_2^2-R_1^2n) )
, where u=0 at R1 and u=f_1(theta) at R2. and (R1 < r < R2) where a_0 , a_n, and b_n are the Fourier coefficients of f1(theta). [ Hint: Proceed as in the solution derived and used the condition R(R2)=0]
I think I need to the determined the coefficients , a_0, a_n and b_n. a_0 is theta indepedent. a_n and b_n are theta dependent.
Given: R(r)= c1+c2*ln(r/a), where c1 and c2 are constants
not sure how to proceed.