Differential equations MOC

Green’s function

Let be some translation-invariant linear differential operator. A Green’s function is a solution to the differential equation¹ fun

where is the Dirac delta — hence it may be thought of as the convolution kernel of . Green’s functions can be used to solve inhomogenous differential equations by Convolution of the source function. Specifically, the differential equation is solved by (plus a homogenous solution), which can be made unique after applying boundary conditions.

Proof

Since

as claimed.

If is not translation-invariant, then one replaces with .

Properties

1. If is a Green’s function and then is a Green’s function.

Examples

• Heaviside function (Green’s function for )
• Green’s function for the Laplacian

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Footnotes

1. In physics there is a convention to write . ↩