Numerical PDE's II - Circular Domain

[Nusc]

Homework Statement

Approximate the solution of:
$$<br /> \frac{\partial^{2}u }{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}} = 0, 1<r<3, 0<\theta<\pi,<br /> <br /> u(1,\theta) = 1-\cos(\theta),u(2,\theta)=u(r,0)=0 u(r,\pi)=4-2r<br />$$

Homework Equations

$$<br /> \delta r = \frac{r_{outer} - r_{inner}}{N}<br /> \delta \theta = \frac{\theta_{end} - \theta_{start}}{M}<br />$$
n,m are positve integers

$$\delta r_i = r_{inner} + i\delta r$$
$$\delta theta_j = \theta_{start} + j\delta \theta$$

The Attempt at a Solution

$$\frac{u_{i+1,j} - 2U_{i,j} + U_{i-1,j}}{(\delta r)^2} + \frac{1}{r_i}\frac{u_{i+1,j}-u_{i-1,j}}{2\delta r} +\frac{u_{i,j+1} - 2U_{i,j} + U_{i,j-1}}{(\delta \theta)^2} = f_{i,j}$$

Multiply this equation by $$(\delta r)^2$$
$$u_{i+1,j} - 2u_{i,j} + u_{i-1,j} + \frac{1}{r_i}\frac{u_{i+1,j}-u_{i-1,j}}{2}\delta r + \frac{1}{r_i}\frac{\delta r}{\delta \theta}(u_{i+1,j}-2u{i,j}+u{i,j-1} = f_{i,j}(\delta r)^2<br />$$

What do I do next>

 

[Integral]

And the question is?

 

[Nusc]

Nevermind, I got it.

But does anyone know if such a problem with those given BC's can be applied elsewhere?
When would you model something with a circular annulus?