# Numerical PDE's II - Circular Domain

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

2. Homework Equations
$$\delta r = \frac{r_{outer} - r_{inner}}{N} \delta \theta = \frac{\theta_{end} - \theta_{start}}{M}$$
n,m are positve integers

$$\delta r_i = r_{inner} + i\delta r$$
$$\delta theta_j = \theta_{start} + j\delta \theta$$
3. 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 $$(\delta r)^2$$
$$u_{i+1,j} - 2u_{i,j} + u_{i-1,j} + \frac{1}{r_i}\frac{u_{i+1,j}-u_{i-1,j}}{2}\delta r + \frac{1}{r_i}\frac{\delta r}{\delta \theta}(u_{i+1,j}-2u{i,j}+u{i,j-1} = f_{i,j}(\delta r)^2$$

What do I do next>

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