PROPERTIES OF THE GF
(HEAT EQUATION).

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

1. 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.

2. Causality. $\$In domain $R$, $G\geq 0$ for $t-\tau >0$, and $G=0$ for $t-\tau <0$. This is called the causality relation, because the GF exhibits zero response until after the heat impulse appears.

3. Reciprocity. $\ G(x,t\,\left\vert \,x^{\prime },\tau \right. )=G(x^{\prime },-\tau \,\left\vert \,x,-t\right. )$. This follows from the heat equation which is second order in space and first order in time.

4. Time dependence. The time dependence of $G$ is always $(t-\tau )$, so the functional form of a one-dimensional GF could be written $% G(x,x^{\prime },t-\tau )$.

5. 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, $\left[ G\right] =m^{-1}$ for one-dimensional problems, $\left[ G\right] =m^{-2}$ for two-dimensional problems, and $\left[ G\right] =m^{-3}$ for three-dimensional problems.