Laplace solution in polar coordinates

[alphadyn]

Hello, its been a pleasure finding you

I have an asignment due to the end of this week and due to some problems, i hadn't found time to get to it so far.

I have to calculate the exact solution of the Laplace equation in polar coordinates, in a hollow disk in the domain Ω

where R1<=r<=R2 and 0<=θ<=2π

and with the following boundary conditions

φ(R1,θ)=0 and φ(R2,θ)=sin2θ

I would be in debt to anyone who can assist me or refer me to a link that can be helpful[edit] unfortunately i didn't see soon enough the 2nd sticky :( please move my post to the right thread..I'm sorry for the inconvinience..

 

[HallsofIvy]

What is Laplace's equation in polar coordinates? You should be able to separate it into equation for the $\theta$ and r dependencies separately.

 

[alphadyn]

Its form would be the following:

$$\frac{\partial^2f}{\partial r^2}+\frac{1}{r^2} \frac{\partial ^2f}{\partial \theta^2} + \frac{1}{ r} \frac{\partial f}{\partial r}= 0$$

From this stage, can you tell me how do I proceed?[edit] Suppose the solution I'm seeking is in the following form

$$f(r,\theta) =C_0lnr + D_0 + \sum_{0}^\infty(C_nr+\frac{D_n}{r^n}).(A_ncosn\theta + B_nsinn\theta)$$

how do i find the constants that occur by my B.C? I know my questions are a bit amateur style but I'm quite confused