Brief: I am struggling with applying boundary condition to Poisson equation and being inspired by this work:
https://dept.math.lsa.umich.edu/~sijue/greensfunction.pdf
For eq. $\text{$\Delta $u}[\textbf x]=f[\textbf x], \textbf x \epsilon D$ where $\text{$\Delta $u}$ - denotes laplacian in Cartesian coordinates. Following with notation it is derived in polar coordinates to be:
$r=|\textbf x|=\sqrt{x^2+y^2}$
So in the paper the solution with Green's function with applied method of images in polar coordinates becomes like this:
$$u[r,\theta ]=\int _0^{2\pi }\frac{1}{2\pi }\frac{1-r^2}{1+r^2-2 r\:\text{Cos}[\theta -\phi ]}h[\phi ]d\phi $$ ...and that was the final result. But now I want to test that formula then my assumption since Green's function contains scalar module of vector r so radius r is constant during integration over ϕ. (For now I ignore 2nd body integral assuming forcing term=0) Then following with that I define example function for boundary of unit disc (star like shape):
$$h[\phi ]=1+\frac{1}{10} \text{Sin}[6\,\phi ]$$
After integration over ϕ and substituting range [0, 2π] to antiderivative of integral (and simplification) I get this:
$$u[r,\theta ]=\frac{\left(-1+r^{12}\right) \text{Sin}[6\, \theta ]}{20 r^6}$$
Then I observed my solution is indeterminate at the origin of coordinates. So question is:
- What am I missing when applying result from mentioned work since I expect a solution should be definite in the interior of domain of the disc. How to resolve this issue. Is that possible to get proper solution for interior of the domain ?
- Or my guess... it has something to do with delta function under the hood of Green's function and is not possible to get around of such problem ?