Green's functions for ODEs, jump conditions

[chipotleaway]

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:

y'+ay=\delta(t-\tau) with y(t_0)=0 and \tau > t_0

the solution being

y(t)=e^{-a(t-\tau)} for t_0 < \tau < t
y(t)=0 for t_0 \leq t < \tau

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

y'+ay=g(t), y(t_0)=0 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

y'+ay=\delta(t-\tau) with y(t_0)=0 and \tau>t_0 in the two domains t_0 \leq t < \tau and t > \tau with the jump condition [y(t)]^{t=t^+}_{t=t^-}=1 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?

 

[homeomorphic]

First off, good for you for realizing the rather silly and unnecessary lack of motivation in your class. My guess is that your professor's philosophy is that you'll learn the motivation later in due course, but are you really going to learn anything if it's all meaningless and unmotivated and therefore lacks any glue to make it stick? This is a good example of what I'm always complaining about on here.

I don't have time to go into great detail, so this will probably be a little vague, but if you have some more physical examples, you can see the point of a Green's function. I think maybe it's more natural in the context of partial differential equations, but don't let that scare you. I will describe it in purely physical terms.

My favorite example would be a point charge. That gives you a certain electric potential, which is just a function like k q/r, (k = constant, q = charge, r = radius). That's the Green's function here. Then, the idea is once you know what happens to a charge at a point, you can use that to figure out what happens with any distribution of charge. For examples, if you add two charges, you would just add the potentials from each one. And then if you had more and more point charges in between, you'd get a bigger and bigger sum. If you had a continuous charge distribution, this sum would become an integral. So, that's the idea. You "add" (integrate) all the contributions at different points to figure out the potential.

Another example (in the case of the heat equation) would be if you pumped in heat at exactly one point and found the temperature distribution. Then, if you wanted to see what happened if you added a continuously varying heat source, you could just do an integral (one for each point in space, but you can do them all at once) to get the answer.

These examples are based on a source at one point in space. You can also think of a source at one point in time. So, getting back to ODE's, you could consider a loaded spring. You can ask what would happen if you gave it a very sharp push at exactly one moment in time (the delta function). If you know how the system responded to that, you could figure how it responded to a series of sharp pushes by adding up the results, or to a more continuously varying push if you did an integral ("add" up all the contributions of the different pushes to find the response at each point in time).

 

[chipotleaway]

Thanks homeomorphic, last paragraph clarified the 'motivation' in my notes for me - I also really like the charge example as well.