Verifying properties of Green's function

In summary, "Verifying properties of Green's function" discusses the mathematical framework and techniques used to confirm the essential characteristics of Green's functions in solving differential equations. The paper emphasizes the significance of boundary conditions, symmetry, and the behavior at singular points. It also highlights methods for testing the correctness of Green's functions through integral representations and their relationship to physical phenomena in various fields such as physics and engineering.
  • #1
psie
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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.
 
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  • #2
[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]
 
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