Laplace equation solution in polar coordinates

Dear all,

In my research I need to solve a polar form of Laplace equation. No big deal, I have solved but I am not able to satisfy the boundary conditions.

For the past three months I have tried every thing but no success. I am so desperate please help me:

t:=theta

a: is a constant

U(r,t)

Utt+(r^2)*Urr+(r)*Ur=-a*(r^2)

B.C.1: U(R,t)=0 for -t1<t<t1 and R is a constant

B.C.2: (sin(t)/r)*Ut-(cos(t))*Ur=0 when r=(R-h)/cos(t) for -t1<t<t1 and h is a constant

My solution:

u(r,t)=(A*r^k+B*r^(-k))*(C*sin(kt)+D*cos(kt))-(a/4)*r^2

The first part is a general solution which since the equation is linear can be extended with a sigma and the second part having a is a particular solution.

A,B,C,D, and k should be satisfied.

Any feedback will be highly appreciated:

Thanks

Amin Mohebbi

Ph.D. candidate, UNL

Nebraska- Lincoln