Another Interpretation of G

There is a second physical interpretation of the GF for transient problems, as follows: the GF may also be considered as the temperature response to a concentrated initial condition. Consider the following boundary value problem:

$$\begin{eqnarray*} \alpha \frac{\partial ^{2}P}{\partial x^{2}} &=&\frac{\partial... ...ight. ) &=&0 \\ P(x=0,t\,\left\vert \,x^{\prime }\right. ) &=&0 \end{eqnarray*}$$

The above equations are nearly identical to the equations defining G, if the time at which the heat source is released has been set to $\tau =0$. Barton (1989) calls function $P(x,t\,\left\vert \,x^{\prime }\right. )$ the propagator; it's relation to the GF is given explicitly by

$$G(x,t\,\left\vert \,x^{\prime },\tau \right. )=H(t-\tau )P(x,t\,\left\vert \,x^{\prime }\right. )$$

where $H(t)$ is the Heaviside unit-step function. Several GF in this Library have been found with this view point.