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