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chipotleaway
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Questions about Green's functions for ODEs, jump conditions
I'm having a hard time understanding Green's functions which have been introduced quite early on in the course, and which I think hasn't been well motivated. I can't find any other resource which explains this at this level (have only covered most of 1st order linear DEs before this) so the only thing I have to go by are the lecture slides.
The (one-sided) Green's function is been introduced in the notes as the solution to the DE:
[itex]y'+ay=\delta(t-\tau)[/itex] with [itex]y(t_0)=0[/itex] and [itex]\tau > t_0[/itex]
the solution being
[itex]y(t)=e^{-a(t-\tau)}[/itex] for [itex] t_0 < \tau < t[/itex]
[itex]y(t)=0[/itex] for [itex] t_0 \leq t < \tau[/itex]
and this is referred to as the one-sided Green's function.
Two methods for calculating the Green's function are described. First is to obtain the integral solution to the initial value problem
[itex]y'+ay=g(t), y(t_0)=0[/itex] and just 'read off' the Green's function...but this apparently is not instructive since it gives no insights into what it means.
Second method is to calculate it for
[itex]y'+ay=\delta(t-\tau)[/itex] with [itex]y(t_0)=0[/itex] and [itex] \tau>t_0[/itex] in the two domains [itex]t_0 \leq t < \tau[/itex] and [itex]t > \tau[/itex] with the jump condition [itex][y(t)]^{t=t^+}_{t=t^-}=1[/itex] which supposedly will give the Green's function.
---
Firstly, what is the point of the Green's function?
What exactly is a 'jump condition' and how do I work with it?
I'm having a hard time understanding Green's functions which have been introduced quite early on in the course, and which I think hasn't been well motivated. I can't find any other resource which explains this at this level (have only covered most of 1st order linear DEs before this) so the only thing I have to go by are the lecture slides.
The (one-sided) Green's function is been introduced in the notes as the solution to the DE:
[itex]y'+ay=\delta(t-\tau)[/itex] with [itex]y(t_0)=0[/itex] and [itex]\tau > t_0[/itex]
the solution being
[itex]y(t)=e^{-a(t-\tau)}[/itex] for [itex] t_0 < \tau < t[/itex]
[itex]y(t)=0[/itex] for [itex] t_0 \leq t < \tau[/itex]
and this is referred to as the one-sided Green's function.
Two methods for calculating the Green's function are described. First is to obtain the integral solution to the initial value problem
[itex]y'+ay=g(t), y(t_0)=0[/itex] and just 'read off' the Green's function...but this apparently is not instructive since it gives no insights into what it means.
Second method is to calculate it for
[itex]y'+ay=\delta(t-\tau)[/itex] with [itex]y(t_0)=0[/itex] and [itex] \tau>t_0[/itex] in the two domains [itex]t_0 \leq t < \tau[/itex] and [itex]t > \tau[/itex] with the jump condition [itex][y(t)]^{t=t^+}_{t=t^-}=1[/itex] which supposedly will give the Green's function.
---
Firstly, what is the point of the Green's function?
What exactly is a 'jump condition' and how do I work with it?
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