# Laplace equations in circular regions: Dirichlet problems on annular regions

• Apr 21st 2009, 09:23 AM
pentazoid
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

u_0(r,0)=a_0

u_n(r,theta)=(r/a)^n*(a_n*cos(n*theta)+b_n*sin(n*theta))

0=c1+c2*ln(R_2/a)

f1(theta)=c1+c2ln(R_2/a)

not sure how to proceed.
• Apr 23rd 2009, 06:27 AM
pentazoid
anyone not understand the problem?