Green's functions, Delta functions and distribution theory

WEB: faculty.washin...
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.

[@sdsa007]

Thank you! Your auto-white board and your emphasis on what is important in the equations helps me focus!

[@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.

[@douglasstrother6584]

That's exactly what it's like in J.D. Jackson's "Classical Electrodynamics".

[@VasilevArtem-g4u]

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

[@jamesraymond1158]

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.

[@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!

[@jean-philippesuter2550]

Also, in deriving that equation, he reverses the order of the integral and the limit without justifying the reversal.

[@robertzavala7064]

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.

[@GhulamNabiDar]

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

[@tianshugu9283]

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)

[@franciscojavierramirezaren4722]

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

[@demr04]

You and Steve are great teachers :)

[@dominicquick107]

best explanation out there

[@adwaitrijal4224]

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.

[@dinojericevic8623]

I am not sure,but i think that it comes from another integration of integration across jump.It than looses derivations(double integration removes second order derivative) and you get constant on the right side.It should imply that the ordinary G is continuous

[@the_informative_edge]

Continuity enables us to deal limit, i.e. Lim F(x)= F(Lim x).

[@pablocesarherreraortiz5239]

Because it's necessary to get the continuity of the solution |u(x) - u(y)| =||--> 0 in the domain. Here we expect a classical solution, that means u should be continous. Besides In 1D the distributional solutions agrees with the clasical solution the reason is the sobolev space H1 is embedded into continous functions. However in 2D,3D green functions are not necessary continuous and the classical solutions doesn't agree with the distributional solution. For example the green function of the 3D laplacian is G(x,y) = c/||x-y|| where x=(x1,x2,x3) and y=(y1,y2,y3) and c a constant and this green función is discontinous where x=y.

[@NoNTr1v1aL]

Absolutely amazing video!

[@kristiantorres1080]

beautifully explained! Thanks a lot!

[@asdfmy1234]

really really amazing video, thanks

[@miro.s]

Excellent and clear. Thanks!

[@sakthiganesh6155]

Amazing explanation

[@duttaalt]

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

[@juliogodel]

Thanks, very clear

[@Martin-nr2xf]

Amazing!

[@rohinbardhan222]

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.

[@sambroderick5156]

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

[@ЛюблюТебя-т1у]

So nice

[@douglasstrother6584]

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

[@Johnnius]

They didn't care about being rigorous.

[@douglasstrother6584]

"WHO INVENTED DIRAC'S DELTA FUNCTION?" by MIKHAIL G. KATZ & DAVID TALL Abstract: The Dirac delta function has solid roots in 19th century work in Fourier analysis by Cauchy and others, anticipating Dirac's discovery by over a century. You'll have to look this up since YeeTube won't let me include the link.

[@douglasstrother6584]

@@Johnnius Look up "WHO INVENTED DIRAC'S DELTA FUNCTION?" by MIKHAIL G. KATZ & DAVID TALL Abstract: The Dirac delta function has solid roots in 19th century work in Fourier analysis by Cauchy and others, anticipating Dirac's discovery by over a century. YeeTube won't let me include the link.

[@alsneed309]

they handwaved it away using the heaviside function and spamming integration by parts

[@him31anshu]

10:00 Really thank you.

[@ericsmith1801]

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.

[@jacobvandijk6525]

If dx = 2 and y = 1/2, then area = 1. If dx = 10 and y = 1/10, then area = 1. So in general, dx = a and y = 1/a gives area = 1. As a goes to 0 dx goes to 1/0 = infinity. And if y = 1, then dx = 1 (from from x = -1/2 to x = + 1/2, around x = 0).

[@ericsmith1801]

@@jacobvandijk6525 , True, but when dx = 1 then dy = 1 also, area is also 1.

[@jacobvandijk6525]

@@ericsmith1801 Right. As it should be. The area is always 1.

[@ericsmith1801]

@@jacobvandijk6525 Yeah, the 1700s when calculus was discovered it was replete with problems involving infinity and yet arriving at a finite solution.

[@jacobvandijk6525]

@@ericsmith1801 That's the point. It needs to be 1 all the time.

[@NeuralEngin33r]

fantastic, ty

[@ericsmith1801]

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.

[@DM-sl9hp]

If it was a discrete time class then the impulse is, as you say, unit amplitude. It is when you get into a continuous case that you get the height going to infinity.

[@ericsmith1801]

@@DM-sl9hp That explains it, thank you.

[@ericsmith1801]

@@DM-sl9hp Yes, this would be the case in DIGITAL signal processing, which is the discrete case.

[@jacobvandijk6525]

@ 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 ;-)

[@otterlyso]

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

[@tadeuszadach5732]

Oh no people are pronouncing greek words like the greeks

[@otterlyso]

@@tadeuszadach5732 Like modern Greeks. Let's say pee for π? Let's pronounce Latin letters like modern Italians? Let's do what is well established and avoids confusion?

[@tadeuszadach5732]

complaining to a university professor that his less bastardized pronunciation of foreign words makes you distracted is about the most american thing one can do in university. godspeed

[@otterlyso]

​@@tadeuszadach5732 Missing the point again. The OED and Webster agree on the pronunciation in English (ksai - as it has been for a long time) with no mention of ksee, but the point is that adopting pronunciations that make Greek letters sound confusingly like Latin letters (ksee - see) is a bad idea. Good luck with your physics career.

[@tadeuszadach5732]

@@otterlyso doesn't ex sound confusingly like es then?

[@김미라-x2t]

Wow

[@404errorpagenotfound.6]

Seems a hard way to solve a problem.

[@demr04]

I made a playlist of this lectures kzbin.info/aero/PLLpLlqKPoatVGkMLCx1wD0Vf8j_9_z-yy

[@samirelzein1978]

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.

[@marshacd]

The same occurred to me, for instance when the unit impulse function is defined. Why not show a graph of it, which is conceptually richer than the symbolic representation? B u t ... there is great benefit, I believe, in doing the exercise of visualizing the situation mentally. The professor might just suggest that the student sketch or imagine the graph. I have wondered if a lot of the wonderful computer graphics in use now don't to some extent short circuit conceptual development.

[@marshacd]

Well, the professor does sketch the pictures just a little later. Sorry for jumping the gun.

[@benheideveld4617]

Uselessly complicated displanation of Green’s function