= Jump conditions for a third-order Green function
For $\mathcal L=D^3+a_2D^2+a_1D+a_0$ with smooth coefficients and leading coefficient one, a <Green function> solving $\mathcal L G=\delta_\xi$ is continuous together with its first derivative at an interior source, and its second derivative jumps by one. Distributional differentiation shows why: a jump in $G$ would create a delta second derivative, a jump in $G'$ a delta first derivative, and a jump in $G''$ the required delta. For a causal initial-value Green function, take the left branch zero and choose the right branch with initial values $0,0,1$ at the source.
Back to article page
