Wave equation
Duhamel's principle

The main use of Green's functions is to solve initial value problems by Duhamel's principle, both for the homogeneous and the inhomogeneous case.

Given the Green function G {\displaystyle G}, and initial conditions u (0, x), ∂ t u (0, x) {\displaystyle u(0,x),\partial _{t}u(0,x)}, the solution to the homogeneous wave equation is u = (∂ t G) ∗ u + G ∗ ∂ t u {\displaystyle u=(\partial _{t}G)\ast u+G\ast \partial _{t}u} where the asterisk is convolution in space. More explicitly, u (t, x) = ∫ (∂ t G) (t, x − x ′) u (0, x ′) d x ′ + ∫ G (t, x − x ′) (∂ t u) (0, x ′) d x ′. {\displaystyle u(t,x)=\int (\partial _{t}G)(t,x-x')u(0,x')dx'+\int G(t,x-x')(\partial _{t}u)(0,x')dx'.} For the inhomogeneous case, the solution has one additional term by convolution over spacetime: ∬ t ′ < t G (t − t ′, x − x ′) s (t ′, x ′) d t ′ d x ′. {\displaystyle \iint _{t'<t}G(t-t',x-x')s(t',x')dt'dx'.}
change_case:capital_word_frequency[3]