# Numerical solution of the Laplace equation on circular domain

I was solving Laplace equation in MATLAB numerically. However I have problems when the domain is not rectangular.

The equation is as follows:

$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$

domain is circular

$$x^2 + y^2 < 16$$

and boundary condition $$u(x,y)= x^2y^2$$

How should I start with solving this equation numerically ?

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I have the same problem as you had. Could you help?. It's urgent. I need this program to solve some problems, but I'm bad using matlab. –  user40196 Sep 13 '12 at 2:54
@BlancaHernándezGalván Please don't post comments as answers. –  Bill Dubuque Sep 13 '12 at 3:16

## 1 Answer

Perhaps begin by rewriting the problem in polar coordintates: $$\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}}=0$$ $$r^2<16$$ $$\left. u(r,\theta)\right|_D=r^4\cos^2\theta\sin^2\theta$$

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But be careful around $r=0$ and that the function is periodic in $\theta$, $u(r,\theta)=u(r,\theta+2\pi)$. –  Daryl Sep 1 '12 at 8:45