(HEAT EQUATION).

The following properties are common to Green's functions for the heat equation on domain .

- Auxiliary problem. Every GF satisfies an auxiliary problem, which
includes a Dirac delta generation term in the differential equation, homogeneous boundary conditions of the same type as the original boundary
value problem, and a homogeneous initial condition.
- Causality. In domain , for , and
for . This is called the causality relation, because the GF
exhibits zero response until after the heat impulse appears.
- Reciprocity.
. This follows from the heat
equation which is second order in space and first order in time.
- Time dependence. The time dependence of is always ,
so the functional form of a one-dimensional GF could be written
.
- Units. The transient GF takes its units from the (spatial) Dirac delta function, which depends on the dimensionality of the problem. For the heat equation in rectangular coordinates, for one-dimensional problems, for two-dimensional problems, and for three-dimensional problems.