Verifying properties of Green's function

[psie]

TL;DR Summary

I want to verify properties of the Green's function and its derivatives, such as continuity, discontinuity and being a solution to a linear homogeneous ODE.

I'm reading about fundamental solutions to differential operators in Ordinary Differential Equations by Andersson and Böiers. There is a remark that succeeds a theorem that I struggle with verifying. First, the theorem:

Theorem 6. Let $$L(t,\lambda)=\lambda^n+a_{n-1}(t)\lambda^{n-1}+\ldots+a_1(t)\lambda+a_0(t)\quad\text{and }D=\frac{d}{dt}.\tag1$$
Denote by ##E(t,\tau)## the uniquely determined solution ##u(t)## of the initial value problem
\begin{align}
&L(t,D)u=0 \tag2\\
&u(\tau)=u'(\tau)=\ldots=u^{(n-2)}(\tau)=0,\quad u^{(n-1)}(\tau)=1. \tag3
\end{align}
Then,
$$y(t)=\int_{t_0}^t E(t,\tau)g(\tau)d\tau\tag4$$
is the solution of the problem
\begin{align}
&L(t,D)y=g(t) \tag5\\
&y(t_0)=y'(t_0)=\ldots=y^{(n-1)}(t_0)=0. \tag6
\end{align}

If the leading coefficient in ##(1)## is not ##1## but ##a_n(t)##, then the last condition in ##(3)## reads ##u^{(n-1)}(\tau)=1/a_n(\tau)## and ##(4)## changes to $$y(t)=\int_{t_0}^t E(t,\tau)\frac{g(\tau)}{a_n(\tau)}d\tau.\tag7$$ Put ##\overline{E}(t,\tau)=\frac{E(t,\tau)}{a_n(\tau)}## and define $$F(t,\tau)=\begin{cases} \overline{E}(t,\tau) &\text{when } t\geq\tau \\ 0 &\text{when } t<\tau.\end{cases}\tag8$$ then ##F(t,\tau)## satisfies the following properties:

1. ##\frac{d^kF}{dt^k}(t,\tau)## is a continuous function of ##(t,\tau)## when ##k=0,1,\ldots,n-2.##
2. ##\frac{d^{n-1}F}{dt^{n-1}}(t,\tau)## is continuous when ##t\neq\tau##, and has a step discontinuity of height ##1/a_n(\tau)## across the line ##t=\tau##.
3. ##L(t,D)F(t,\tau)=0,\quad t\neq \tau##.

The authors note that this is easily verified by noting that ##E(t,\tau)## solves the IVP ##(2)## and ##(3)##, yet I have hard time verifying this to myself.

First of all, I'm confused about them writing ##\frac{d^k}{dt^k}## instead of ##\frac{\partial^k}{\partial t^k}##. Is this because we view the function as a function of ##t## only? If so, then 1. makes very little sense to me. How can the ##k##th derivative (##0\le k\le n-2##) of ##F## with respect to ##t## be continuous?

Second, I do not see how either 2. or 3. follows from the fact ##E(t,\tau)## solves ##(1)## and ##(2)##. I'd be very grateful if someone could share their understanding on the matter.

 

[pasmith]

[itex]\tau[/itex] is regarded as a parameter of the IVP (2,3), so [itex]d/dt[/itex] here means [itex]\partial/\partial t[/itex].

We know that both [itex]\bar{E}[/itex] and 0 are [itex]n - 1[/itex] times differentiable with respect to [itex]t[/itex]: 0 trivially, and [itex]\bar E[/itex] because it is the solution of the IVP (2,3), so its first [itex]n - 1[/itex] derivatives with respect to [itex]t[/itex] exist and are continuous in [itex]t[/itex]. Continuity of [itex]\bar E[/itex] and its [itex]t[/itex]-derivatives in [itex](t,\tau)[/itex] jointly follows from the fact that if you write [tex]\bar E(t,\tau) = A_1(\tau)u_1(t) + \dots + A_n(\tau)u_n(t)[/tex] for [itex]n[/itex] linearly independent solutions [itex]u_k[/itex] of (2), then the [itex]A_k[/itex] can be shown to be continuous in [itex]\tau[/itex].

[itex]F[/itex] is defined as either [itex]\bar E[/itex] or zero so [itex]F[/itex] and its first [itex]n - 1[/itex] derivatives with respect to [itex]t[/itex] can fail to be continuous in [itex](t,\tau)[/itex] only at the boundary of the regions where those definitions are applied, ie. when [itex]t = \tau[/itex]. By construction [tex]
\frac{\partial^k F}{\partial t^k}(\tau,\tau) = \begin{cases} 0 & k = 0, \dots, n-2 \\
1/a_n(\tau) & k = n - 1. \end{cases}[/tex] but [tex]
\lim_{t \to \tau^{-}} \frac{\partial^k F}{\partial t^k}(t,\tau) = 0.[/tex]