Green's functions, Delta functions and distribution theory

This lecture is part of a series on advanced differential equations: asymptotics & perturbations. This lecture introduces the Green's function, or fundamental solution, which can be used to solve Lu=f. The Dirac delta function and distribution theory is also introduced.

[muttleycrew]

Thanks for your public outreach to all nerds everywhere.

[TheRsmits]

Very nice, would be useful for students to see this before PDEs where Green's functions seem to come out of nowhere.

[VasilevArtem-gu]

the clearest explanation I've seen so far in the web. It even outstands cartoon videos (of which I'm a great fan) that explain the same topic

[robertzavala]

The graphic with 4 plots displaying the form forms of the Dirac delta function is very helpful. A brief explanation in the video also compares the Green's function to the Eigenfunction/value method. This comparison is a helpful presentation as the different differential equation solution methods often seem to yield such similar results that a student wonders why there are these different methods.

[americocunhajr]

Two observations in 11:35: (i) the lower limit of integration must be x0-ksi instead of x0+ksi; (ii) in the last limit, after apply mean value theorem, a 2ksi in the denominator is missing. But the class is excellent!

[jamesraymond]

Some day I gotta understand Green's function. This was too advanced for me, but I was very impressed by the graphics. First time I've seen this type of blackboard. which allows the teacher to point to different parts of the equations. Great job. Thank you.

[GhulamNabiDar]

excellent precise focussed lecture, covered much beyond in a single lecture.

[tianshugu]

Better than the lectures we've got in UBC math 401

[aboalgadah]

Great lecture. Thank you, Nathan. (I believe the factor of 2ksi should not be there after applying the mean value theorem, in the proof of sifting property)

[franciscojavierramirezaren]

Great content, thanx for sharing!! Happy new year!!🥳🥳👍

[sambroderick]

Blackboard is a U.W. thing, pioneered by Steve Brunton (e.g., data science) if I‘m not mistaken…

[rohinbardhan]

I have a confusion. I have seen Green's functions defined in other places as L(x)[G(x, y)] = delta (x-y). Could you tell me how your definition relates to that?
Thanks.

[ericsmith]

So, the impulse function reaches infinity as the horizontal width of it approaches zero. But the area of the function is finite, equals to 1.

[douglasstrother]

How did George Green and his contemporaries conceptualize the impulse function before the formal theory of distributions?

[adwaitrijal]

Why are we assuming the continuity criteria for G?
I've been searching the answer for this question for a while now, but haven't been able to find a satisfactory answer yet.

[duttaalt]

For people having exams and wanna learn about Green's function only, video starts at 12:00

[ericsmith]

In the Signal Processing class I was taking the impulse function had a y magnitude of 1, not infinity. I wonder if the mathematicians insisted on it being infinite after taking the limit as xsi was approaching 0.

[otterlyso]

When people pronounce xi as ksee, it sounds almost exactly the same as 'c' and is a constant distraction.

[jacobvandijk]

@ 3:28 IN PHYSICS, AN IMPULSE IS DEFINED AS THE PRODUCT OF A FORCE OVER A TIME-INTERVAL: Impulse = Force x Period = F(x) . dt. SO, HIS FUNCTION f(x) IS NOT A FORCE! YOU CAN TURN IT INTO A FORCE BY MULTIPLYING IT BY VELOCITY V ... AND DIVIDING dx BY V (= dt). NOW WE ARE DOING GOOD PHYSICS AGAIN ;-)

[samirelzein]

the idea of democratizing teaching online is to prove the basics before going lecturing. You need to not let the viewer imagine anything or go look for anything. Use google and all kinds of graphs to illustrate your ideas to get out of that rigid academic explanations.