Poisson's equation

Poisson's equation is a non-homogeneous generalization of Laplace's equation

\nabla 2 𝑓 = 𝜑

where $\nabla 2$ is the Laplacian.

Solution

Assume $𝜑$ has compact support. Let $\nabla 2 𝑓 𝐻 = 0$ solve Laplace's equation. Invoking Green's function for the Laplacian $𝐺$ , solutions take the form

𝑓 (⃗ 𝐫) = (𝐺 * 𝜑) (⃗ 𝐫) = - 1 4 𝜋 ∭ ℝ 3 𝜑 (⃗ 𝐫') 𝔯 𝑑 𝜏'

where $𝔯 = ‖ ⃗ 𝐫 - ⃗ 𝐫' ‖$ .

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