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I was wondering what are the techniques ideas to solve the following problem, consider Poisson equation for a delta source $$\nabla^2 f(x,y,z)=\delta(x-x_0)\delta(y)\delta(z)$$ where $x,y,z$ are Cartesian coordinates, $0<x_0<1$, $\nabla^2$ is the Laplacian and $\delta$ is Dirac distribution.

How do I solve this equation with boundary conditions, specifically, $f(0,y,z)=f(1,y,z)=0$?

I have seen Green function methods where there is one boundary $x=0$, the solution is obtained by making an ansatz of the form of the Green function. Here I also have a boundary at $x=1$. The solution in this case is expressed in terms of an infinite sum but I do not know where to start.

Mauricio
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  • My take on this question is to multiply the PDE by $f(x,y,z)$ and then integrate on $[0,1]\times\mathbb{R}\times\mathbb{R}$ and then use integration by parts on the LHS. (or for $3-d$ it's called Green integration formula. https://en.wikipedia.org/wiki/Green%27s_identities). – MathematicalPhysicist Mar 08 '22 at 19:45
  • @MathematicalPhysicist which equation in the wiki should I be looking at? – Mauricio Mar 09 '22 at 19:45

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