1D Green's function: from interval to infinite line Let's consider two problems for diffusion equation.
The first one:
$$
u_t = a^2u_{xx},\qquad 0<x<l,\quad 0<t\leq T
$$
$$
u(x,0) = \phi(x), \qquad 0 \leq x \leq l 
$$
\begin{equation}
u(0,t)=0,\quad u(l,t)=0, \quad 0 \leq t \leq T
\end{equation}
and the second one:
$$
u_t = a^2u_{xx},\qquad -\infty <x<+\infty,\quad t>0
$$
$$
u(x,0) = \phi(x), \qquad -\infty < x < +\infty
$$
For both these cases there are well known expressions for Green's function:
$$
G_1(x,\xi,t) = \frac{2}{l}\sum_{n=1}^{\infty} \exp\left({-\left(\frac{\pi n}{l}\right)^2}a^2t\right)\sin\frac{\pi nx}{l}\sin\frac{\pi n\xi}{l}
$$
$$
G_2(x,\xi,t) = \frac{1}{\sqrt{4\pi a^2 t}}\exp{\left(-\cfrac{(x-\xi)^2}{4a^2t}\right)}
$$
Thus we obtain solutions for first (finite) problem:
\begin{equation}
u(x,t)=\int\limits_0^lG_1(x,\xi,t)\phi(\xi)\,d\xi
\end{equation}
and for the second (infinite):
\begin{equation}
u(x,t)=\int\limits_{-\infty}^{+\infty}G_2(x,\xi,t)\phi(\xi)\,d\xi
\end{equation}
Is there any way to obtain Green's function for infinite case from Green's function for range $[0,l]$ (for example by stating  $\quad l\rightarrow+\infty$, but I have not achieved any success with this idea)?
Or may be second solution from first?
 A: Yes, $G_2$ can be obtained as a limit of $G_1$ but I will do a slightly easier limit first.
Write
$${G}_1(x,\xi,t)=\frac{2}{l}\sum_{q} \exp\left(-q^2a^2t\right)\sin q x\sin q\xi,$$
where $q=\frac{\pi n}{l}$, $n=1,2,\ldots$ Now since $q_{j+1}-q_j=\frac{\pi}{l}$, for $l\rightarrow\infty$ we have
$$\frac{\pi}{l}\sum_{q}\rightarrow \int_0^{\infty}dq,$$
and therefore
\begin{align}\lim_{l\rightarrow\infty}{G}_1(x,\xi,t)&=\frac{2}{\pi}\int_0^{\infty}\exp\left(-q^2a^2t\right)\sin q x\sin q\xi\;dq=\\
&=\frac{1}{\sqrt{4\pi a^2t}}\left[\exp{\left(-\cfrac{(x-\xi)^2}{4a^2t}\right)}-
\exp{\left(-\cfrac{(x+\xi)^2}{4a^2t}\right)}\right].
\end{align}
As you may guess, this is the Green function for the heat equation on the half-line. Now the modifications for the whole line will be as follows:


*

*Instead of $G_1(x,\xi,t)$ one should start with translated Green function $$\tilde{G}_1(x,\xi,t)=G_1(x-l/2,\xi-l/2,t).$$ It corresponds to the solution of the heat equation on $[-l/2,l/2]$.

*Similarly transform the sum over $n$ into an integral over $q$ (you will need to consider odd and even values of $n$ separately). Evaluating two resulting gaussian integrals, you will recover $G_2(x,\xi,t)$.
