22.7 Finding the Green's functions

We can, if we want, write the solution for $v$ in other ways that may be more efficient numerically. The solution was, rewritten more concisely in terms of the eigenvalues and eigenfunctions:

$$v(x,t) = \sum_n \left[ \int_{\tau=0}^t \bar q_n(\tau) ... ...\tau + \bar f_n e^{-\kappa \lambda_n t} \right] X_n(x).$$

The first part is due to the inhomogeneous term $\bar q$ in the partial differential equation, the second due to the initial condition $v(x,0)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\bar f(x)$

Look at the second term first, let's call it $v_f$,

$$v_f \equiv \sum_n \bar f_n e^{-\kappa \lambda_n t} X_n(x).$$

We can substitute in the orthogonality relationship for $\bar f(x)$:

$$v_f = \sum_n \frac{\int_0^\ell \bar f(\xi)X_n(\xi){ \rm... ..._n^2(\zeta){ \rm d}\zeta} e^{-\kappa \lambda_n t} X_n(x)$$

and change the order of the terms to get:

$$v_f = \int_0^\ell \left[ \sum_n \frac{X_n(\xi) X_n(x... ...ta} e^{-\kappa \lambda_n t} \right] \bar f(\xi) d \xi$$

We define a shorthand symbol for the term within the square brackets:

$$G(x,t,\xi) \equiv \sum_n \frac{X_n(\xi)X_n(x)}{\int_0^\ell X_n^2(\zeta){ \rm d}\zeta} e^{-\kappa \lambda_n t}$$

Since this does not depend on what function $\bar f(x)$ is, we can evaluate $G$ once and for all. For any $\bar f(x)$, the corresponding temperature is then simply found by integration:

$$v_f(x,t) = \int_0^\ell G(x,t,\xi) \bar f(\xi) d \xi$$

Function $G(x,t,\xi)$ by itself is the temperature $v(x,t)$ if $\bar f$ is a single spike of heat initially located at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\xi$. Mathematically, $G$ is the solution for $v$ if $\bar f(x)$ is the ``delta function'' $\delta(x-\xi)$.

Now look at the first term in $v$, due to $\bar q$, let's call it $v_q$:

$$v_q \equiv \sum_n \int_{\tau=0}^t \bar q_n(\tau) e^{-\kappa \lambda_n(t-\tau)} { \rm d}\tau X_n(x)$$

We plug in the orthogonality expression for $\bar q_n(\tau)$:

$$v_q = \sum_{n=0}^\infty \int_{\tau=0}^t \frac{\int_0^\... ...\zeta} e^{-\kappa \lambda_n(t-\tau)} { \rm d}\tau X_n(x)$$

and rewrite

$$v_q = \int_{\tau=0}^t \int_0^\ell \left[ \sum_n \fra... ...-\tau)} \right] \bar q(\xi,\tau) { \rm d}\xi {\rm d}\tau$$

We see that

$$v_q(x,t) = \int_{\tau=0}^t \int_0^\ell G(x,t-\tau,\xi) \bar q(\xi,\tau) { \rm d}\xi {\rm d}\tau$$

where the function $G$ is exactly the same as it was before. However, $G(x,t-\tau,\xi)$ describes the temperature due to a spike of heat added to the bar at a time $\tau$ and position $\xi$; it is called the Green's function.

The fact that solving the initial value problem ($\bar f$), also solves the inhomogeneous partial differential equation problem ($\bar q$) is known as the Duhamel principle. The idea behind this principle is that function $\bar q(x,t)$ can be ``sliced up'' as a cake. The contribution of each slice $\tau$ $\raisebox{-.3pt}{$\leqslant$}$ $t$ $\raisebox{-.3pt}{$\leqslant$}$ $\tau+{\rm d}\tau$ of the cake to the solution $v$ can be found as an initial value problem with $\bar q(x,\tau){ \rm d}\tau$ as the initial condition at time $\tau$.