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# PROPERTIES OF THE GF (HEAT EQUATION).

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

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 , for , and for . This is called the causality relation, because the GF exhibits zero response until after the heat impulse appears.

3. Reciprocity. . This follows from the heat equation which is second order in space and first order in time.

4. Time dependence. The time dependence of is always , so the functional form of a one-dimensional GF could be written .

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, for one-dimensional problems, for two-dimensional problems, and for three-dimensional problems.   Next: Dirac delta function Up: What is Green's Function Previous: Another Interpretation of G
2004-01-21