Green's function

Let $𝐿$ be some translation-invariant linear differential operator. A Green's function $𝐺$ is a solution to the differential equation¹ #m/def/anal/fun

𝐿 𝐺 = 𝛿

where $𝛿$ is the Dirac delta — hence it may be thought of as the convolution kernel of $𝐿$ . Green's functions can be used to solve inhomogenous differential equations by Convolution of the source function. Specifically, the differential equation $𝐿 𝑓 = 𝜌$ is solved by $𝑓 = 𝐺 * 𝜌 + 𝑓 𝐻$ (plus a homogenous solution), which can be made unique after applying boundary conditions.

Proof

Since

𝐿 [𝐺 * 𝜌] (𝑡) = 𝐿 \int ℝ 𝑛 𝐺 (𝑡 - 𝜏) 𝜌 (𝜏) 𝑑 𝜏 = \int ℝ 𝑛 𝐿 [𝐺 (𝑡 - 𝜏)] 𝜌 (𝜏) 𝑑 𝜏 = \int ℝ 𝑛 𝛿 (𝑡 - 𝜏) 𝜌 (𝜏) 𝑑 𝜏 = 𝜌 (𝑡)

as claimed.

If $𝐿$ is not translation-invariant, then one replaces $𝐺 (𝑥 - 𝑥')$ with $𝐺 (𝑥, 𝑥')$ .

Properties

If $𝐺$ is a Green's function and $𝐿 𝑔 = 0$ then $𝐺 + 𝑔$ is a Green's function.

Examples

Heaviside function (Green's function for $𝐷$ )
Green's function for the Laplacian

#state/tidy | #lang/en | #SemBr

In physics there is a convention to write $𝐿 𝐺 = - 𝛿$ . ↩

Green's function

Properties

Examples

Footnotes