need help with intuition behind greens formula (convolution integral)

Question

baru · Answer

lets take an example (swinging pendulum with air resistance)
mx''+cx'+ kx = f(t)

let w(t) be the impulse response to the above equation

then the convolution integral comes from considering f(t) to be a series of repeated impulses of magnitude f(t)dt (perhaps repeatedly hitting the pendulum with a small hammer?).

by superposition, the solution is the sum of the responses to each individual impulse(each blow of the hammer)

and since the interval is infinitely small, the sum is an integral

baru · Answer

is the interpretation correct?
anything to add/subtract to it?
is there any alternate explanation?

baru · Answer

@ganeshie8

kainui · Answer

I'm not super experienced with using them but my understanding of them is different, it's more like you're reconstructing your function out of eigenfunctions of your differential operator. This is both more and less scary than it actually sounds, I'll try to show what I mean with your example:

$$mx''+ cx'+kx = f$$
I factor out the differential operator on the left:
$$(mD^2 +cD+kI) x = f$$

So we can kinda like find all the eigenfunctions of this operator now:
$$(mD^2+cD+kI)g = \lambda g$$

and then you basically like somehow magically use the fact that the eigenvectors span the space and some spectral theory mumbo-jumbo and you basically reconstruct your f(t) function out of these eigenfunctions.

---

Like I said, I'm not really too familiar with this, but maybe this helps you get an intuition for it somehow. It might also be that this description coincides with the "little bumps over time" kind of picture in your mind so I'm not really sure.

baru · Answer

you lost me at "eigen function" :/
perhaps it will make more sense when i learn what that is...
thanks anyway :)

baru · Answer

"spectral theory mumbo-jumbo" XD lol

kainui · Answer

Well quick answer is an eigenfunction of an operator is a constant times that function, for instance:

$e^{3x}$ is an eigenfunction of $\frac{d}{dx}$ with eigenvalue 3. $\sin(3x)$ is not an eigenfunction of the $\frac{d}{dx}$ operator but it is an eigenfunction of the $\frac{d^2}{dx^2}$ operator with eigenvalue -9.

Yeah, so uhh that's what it _is_ but that's not nearly helpful enough to you now haha.

baru · Answer

nice! that makes some sense...
i'll keep this in mind

kainui · Answer

Basically the best step I can give you in the right direction for understanding this is if you've learned what eigenvectors are from linear algebra, a function is really a vector with uncountably infinite many dimensions.

So a regular vector you are used to in 3D is like v(1), v(2), v(3)

A function as a vector is now... well f(1), f(1.0003), f(8535), etc.... but you can do cool stuff so now the dot product between two vectors is an integral:

$$\int f(x) g(x) dx$$

which makes sense since the dot product before was like:

$$v(1)u(1)+v(2)u(2)+v(3)u(3) = \sum v(i)u(i)$$

hehehe ok I guess I'll stop here :P

baru · Answer

it sounds reasonable... 
but still to complicated for me right now
thanks :)