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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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    @BlancaHernándezGalván Please don't post comments as answers.2012-09-13

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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 $\theta$, $u(r,\theta)=u(r,\theta+2\pi)$.2012-09-01