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Laplace’s equation with source
Suppose I solve Laplace’s equation in a region $R$ without sources and I get the values of the solution $u(r,\theta)$.
Suppose I decide to add a point source $P$ in $R$, and solve Laplace’s equation again, I suppose the solution would be a mixture of those values of $u$ and the value of the solution in $P$ should be added.
My question is, how do you know which values are the ones of the mixture? I mean, there is a function $f$ that doesn’t have any informations about $u$, then, when you add a point source, how do you know if $f$ has an argument in $P$ or just a constant in the point source?
Thank you in advance
A:
To address the question in the title: if the point source is the same as the point the source in the solution is being added to the solution, then the source constant is the same as the value of the solution in the point source. If the point source is a different shape or size, there is no way to know what the source constant should be.
Consider a solution $u(\textbf{x})$ to a given Laplace equation, and a point source $\textbf{x}_0$ that generates a solution $\hat{u}(\textbf{x})$.
You’d like to know how to find the coefficients $c_i$ that go to make up the solution when you add the point source to the solution.
The coefficient $c_i$ for the point source is simply the solution of the equation
$$
\left(-\Delta – \hat{\lambda_0
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