Laplace's equation problem in Polar Coordinates (Edit)

Question

Is there public code in Matlab for solving the Laplace equation in polar coordinates in a circular domain?

I tried a lot but my level of Matlab and Mathematica is not good enough, but still not quite understand the scheme. Excuse my English but I'm from Peru.

Consider the equation of Laplace

$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2u}{\partial\theta^2}=0$

with $0\leq r\leq R$ y $0\leq \theta \leq 2\pi$. We will use a mesh $r_i=i\Delta R$ and $\theta_j=j\Delta\theta$. We approximate the equation with the squeme:

$\frac{1}{r_i}\left(r_{i+1/2}\frac{u_{i+1j}-u_{ij}}{\Delta r}-r_{j-1/2}\frac{u_{ij}-u_{i-1j}}{\Delta r}\right)\frac{1}{\Delta r}+\frac{1}{r_i^2}\frac{u_{ij+1}-2u_{ij}+u_{ij-1}}{\Delta\theta^2}=0$

where $u_{ij}M$ and $f_{ij}$ are mesh functions $(r_i,\theta_j)=(i\Delta r,j\Delta\theta)$. The functions are periodic in the angular index $j$ with period $J=2\pi/\Delta\theta$ and $u_{0j}$ is independent of the value of $j$.

To derive the condition of the source Laplace equation integrate on the disk $D$ with radius $\varepsilon$, obtaining

$\int\int_D frdrd\theta=\int\int_D\frac{1}{r}\left[\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)\right] + \frac{1}{r^2}\frac{\partial^2u}{\partial\theta^2}rdrd\theta$

$\int_0^2\pi \frac{\partial u}{\partial r}\varepsilon d\theta$

Now, choosing $\varepsilon$ = $\Delta j/2$ and approximating this relation by

$f(0)\left(\frac{\Delta r}{2}\right)^2\pi=\sum\frac{u_{1j}-u_0}{\Delta r}\frac{\Delta r}{2}\Delta\theta$

$u_{0j}$ is independent $j$, let's call $u_0$ we have

$u_0=\frac{1}{J}\sum u_{ij}-f(0)\left(\frac{\Delta r}{2}\right)^2$ as $f(0)=0$ then $u_0=\frac{1}{J}\sum u_{ij}$

Answer 1

You could use pdepe and turn Laplace's equation into a BVP, then solve it with a multiple shooting method. This solution is not a turnkey solution, but it does use MATLAB built-in functions.

Alternately, you could set up the equations you've written in matrix form; these are algebraic and you should be able to solve them using an LU decomposition (if they're linear; use the matrix backslash operator) or a Newton-Raphson-like approach (if they're nonlinear; use fsolve).