Adjoint Green function

An adjoint Green's function solves the corresponding adjoint point-source problem, with boundary or terminal conditions chosen so that integration by parts gives a representation for a solution of the original operator. For $L={\partial }_t-{\partial }_x^2+c$ on the real line, the formal adjoint is $L^{\ast }=-{\partial }_t-{\partial }_x^2+c$. The backward kernel with terminal Dirac delta at $(\xi ,\tau )$ is the translated heat kernel with constant absorption, defined for $t<\tau$. Its pairing with a homogeneous forward solution is constant in $t$; taking the terminal limit evaluates that solution at $(\xi ,\tau )$.