Weak Solutions of a PDE and Why They Matter

What is the weak form of a PDE? Nonlinear partial differential equations can sometimes have no solution if we think in terms of functions, that's why we need to think in terms of distributions instead! Although they're not really physical, they are still hugely important in mathematics. The Navier-Stokes equations for example, which is the basis for all of fluid dynamics, only has weak solutions as far as we know. And weak solutions play a huge role in numerical solution of pdes, everything in computational fluid dynamics (CFD), to numerical astrophysics and relativity, structures engineering, all of these solve for weak solutions!

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Music:
It All Comes Together - Kyle Preston
Summer Again - Jay Denton

[traultdrum]

You’re shockingly good at explaining this, my dude.

[theultimatereductionist]

In differential algebra we study various transformations that allow us to express a solution of one nonlinear PDE or ODE in terms of simpler PDEs and ODEs. The discontinuity of functions that are solutions of nonlinear PDEs is essentially a generalization of an algebraic function u=u(x) where P(u, x)=0 where P is some non-differential expression, e.g. like a polynomial, but it can be transcendental as well, thus u(x) has discontinuities at branch points of P.

[shohamsen]

Okay, so the problem I have is that we have understood the strong form of most PDEs pretty well. Both from a point-wise definition everywhere to integral forms of conservation laws. These do not need the introduction of an arbitrary test function. Only the source term and the solution make an appearance. On the other hand, for the weak form, you have test functions that have nothing to do with the physics of the problem. So my question is, what do the test functions mean.

[fadiamuhieddine]

thank you i know you didn't get many views but the video is valuable and it was so useful :)

[MusicKnowte]

Wow, I am so elated watching this video. Everything makes since. I'm going to cite this video on my final report for my pde class.

[nirbhay.k]

Thoroughly impressed by the method of explanation! Great job.

[daniel-pineapple]

Thanks for the awesome video!
I have read about weak forms but I couldn't understand it at the time. After watching the video, I would like to check it out again!

[umedina]

WOW I loved the video!! Please upload more on the thermodynamic and more in depth mathematical part! just one question, the reason we are getting a decay in the weak equation solution is because this form introduces some type of diffusion into the equation?

[jbergamp]

Which formula did you use to represent the behaviour of the wave with the burger equation in 2D? f(x)=f(x+f(x)t) ?
Thanks

[marcomoldenhauer]

Hello, did you animate the graph with LaTex (see 00:40)? BR and THX for the Video

[香港地舖購物]

Good stuff ! Great video ! It will be interesting to see more similar topics being discussed intuitively (e.g. viscosity solution)

[wp]

Cool video!
But remember that fundamental laws of physics are integral laws, and only when things get sufficiently regular/smooth you can get differential equations as a local representation of physics laws

[She_]

Can you tell me how to generate the animation for showing the solution wave? I

[alexcastrotello]

Vertical function? Una función “vertical” no es una función, es una relación, ya que a un X se le asigna más de un valor de Y

[pawankhanal]

You are very very good. you have to make more videos like this man.

[MustafaKhan-wsuc]

This was awesome Alex!! Can't wait for the shaved eyebrows and pink dyed hair when you reach a 100k likes 👀. Btw, is there a computational speed up or gain when using the weak form of the PDE as opposed to the strong form when trying to find solutions that satisfy the two equations?

[prithvis.j.monteiro]

Ah, I just learnt something new in a really simple way! Love it BTBB <3

[ziedmasarweh]

Love this video! But it might be a bit too big brain for me right now lol

[adamradekmartinez]

This is all trivially obvious though, pretty straightforward.

[lukemcconnell]

99, 895 likes away from no eye brows and pick hair




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