Very well put together! The transitions, graphics, writing on the paper with different markers, walk and talk, etc. we’re all very well utilized, and the video definitely gained me some understanding of these ideas! Look forward to seeing more :)
I came in expecting just another mathematical explanation, but damn, this was pretty well put together! Especially the explanations of the burger's equation part.
From one small channel to another, I absolutely loved this video. Major respect man. I have been feeling really stressed about studies this week, but your enthusiasm and brilliance have completely turned my mood around. Very happy to get back to my books this evening. Thank you for your inspiring work. :)
Wow Alex, this was extremely well done! I really loved how you included the history behind the equation as well :) I haven't reviewed shocks in a long time, and this was a nice new way of thinking about them. Great job! Also, yay Shankar's QM textbook at 8:51!
You're a rockstar at explaining things, As a mathematician, I learn many new things about PDEs in a very interesting way. Please make more videos about physical applications of other PDEs like Sobolev and fractional Burgers equation.
Thank you! Glad you enjoyed the video :) And absolutely I already plan on making some videos like this, hopefully I'll get them out in the coming months!
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.
Hi! You should do more videos like this, longer, into details. It is amazing how you explained those concepts so easily and fun way. Love you, guys. Saludos desde Venezuela.
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.
This is a good question. I'll give you 3 different answers, you pick which you like the best. Perhaps this warrants making another video altogether. 1. If you think simply in terms of how to take a derivative when the function is not differentiable (ex. discontinuous), you naturally go towards distributions replacing functions. The generalization of a derivative is defined in terms of test functions. It recovers the same derivative for differentiable functions, but allows for more general answers. And when you replace all the derivatives in a PDE with "weak derivatives", you get the weak form. 2. You can think of the PDE as a residual, so the solution to the PDE is that which sets the residual equal to 0. Of course, if we really want a residual we need a scalar quantity, so you must take the inner product. The idea then is that the solution is the "function" such that no matter how you take an inner product with this PDE, the residual should be zero. The inner product of course is set by integrating and multiplying by a test function. The test function can even be seen like the weights of a weighted residual. No matter what the weights are, the solution should still give you a zero residual. But this formulation has the advantage of allowing you the freedom to set the space for inner products and test functions, not just L2. 3. This one is perhaps the most beautiful if you are familiar with Lagrangian mechanics. Set your lagrangian L to be equal to the PDE, i.e. L(u)=du/dt + d/dx(u^2/2). We want u such that L=0. By the principle of least action, we want to minimize the action S, i.e. the integral of the lagrangian. So we want the integral of L(u+ev) to be minimized for dS/de=0, where e is a small parameter defining a perturbation. If you go through some variational calculus, you will find that the condition which minimizes the action is identical to what we consider the weak form of the PDE! i.e. multiplying by a test function and integrating by parts is entirely equivalent to finding the solution which minimizes the total variation of the PDE, which in my opinion is quite beautiful.
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!
Wow I love this content man. Really digs deep into my desire to know early models of Fluid Mechanics. Nowadays at university they skip through all the uninspired bits to get straight into Navier - Stokes Equations only to directly solve it with CFD. 😅
Which I mean is the most relevant is not necessarily a complain. But understanding how models came to be can make wonders for understanding latter work
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?
Thank you :) Actually it’s quite interesting, although the integral of the solution is exactly conserved over the domain, there is a decay in energy (integral of solution squared). This comes from the shock, it’s easy to show that energy is conserved up until that point and then the energy will dissipate. The rankine hugoniot condition, which determines the speed of the shock wave, will introduce some dissipation. It’s similar to a shock wave in the Euler equations, entropy will always be conserved in an ideal fluid until a shock develops, and then you necessarily you have a drop in entropy across a shock.
Absolutely incredibly video, I couldn’t even wait to finish it before subscribing. It would also be neat if you could cover more numerical fluid flow and maybe even the navies stokes equations :)
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
It looks like while deriving the weak differential equation you used Fubini's Theorem before integration of parts for one of the components of the final integral-why can you change the order of integration? Due to our assumptions of phi do we know more information about the regularity of phi*u_t? Can you explain more about the function space that phi belongs to?
Yup correct, you do have to interchange the order of integrations, but this isn't really a problem as you are assuming that phi is integrable. Aside from that, phi can really be from any space you want really (though this is more important for numerical schemes). It is completely arbitrary, but the key is it must be smooth and have compact support, meaning Fubini's theorem is valid.
I don't know if you are still checking your RUclips comments. I am coming across your channel for the first time. Can you tell me how long did it take to make this video? I am particularly interested in how long it took for you to read the references and get tge knowledge organised before you start shooting or writing for the video.
I came here from r/mathematics. I am doing my MSc math and studying wave equations in pde. Not a huge fan of pde’s. But this was packed with lots of theory and history. (I was blown away when you read the pde). In our program structure, we are just studying the theory of pde’s solely and my faculty doesn’t really connect with physics, so most of it feels like ‘born out of thin air’ and missing story links. What do you think are the essential story links I should read before studying “solving wave equations”? Can you do a video on this?
Thank you! Difficult question to answer... So of course I have my own physics interpretation in that I always like to think of a PDE by applying a physical meaning on it. As you say, when you don't do that it can feel a bit made up. I don't necessarily know too much about a lot of wave equations, just the famous ones, so things like Burgers equation, shallow water, KdV, shrodinger, etc. They all have really interesting stories behind them but I'm not really too sure of any single reference that covers them all. The history for this video I got from the first chapter of Dafermos "Hyperbolic Conservation Laws in Continuum Physics", which has a few more details I didn't mention in this video you might like, but as you can guess only applies to hyperbolic PDEs. I wish I could give you a better reference sorry! For sure in the future though I'll be looking to cover different equations and talk about PDEs more, such as how to interpret PDEs by just looking at them, etc.
I have a question. I took a Introduction into CFD in my Mechanical Engineering bachelor's degree and we learnt about the Burger Equation. I think the weak form is more like Finite Element Method than Finite Volume or Finite Difference method, because we use explicit or implicit Upwind schemes to discretize the PDE. Am i correct or no?
Thank you Jose! You're absolutely correct, if you use the weak form you naturally derive the Finite element method, whereas if you start from the strong form you usually get finite difference, and finally if you use the integral form you get finite volume. Though of course once you get into the numerical side where you're discretizing things and working with matrices instead of continuous forms the lines become muddy and the distinctions aren't quite so clear anymore.
As long as you got the main idea - that weak solutions are more general because you pass the derivatives off to a test function, you're golden! That's already way more big brain than I was in first year
Great video! Would love to watch a video about the Klein-Gordon Equation, and Dirac Equation of the Electron (from Strauss' Intro to Partial Differential Equations - 13.5) If you could please, I'd love to see your take on that topic.
@@BeyondtheBigBang Gotcha. Lemme know if you're itching to tackle it - I studied up to the treatments of the schrodinger equation / wave equation / heat equation. Im trying to get some insight on the more advanced PDEs.
If you're looking for intuition, I'm thinking about making a video in interpretation of PDE's in general. It wouldn't be specific to any particular PDE but I could cover all the ones you mentioned and it would give a decent idea on how to interpret terms of a PDE intuitively. Maybe that could be one of the next few videos I release
@@BeyondtheBigBang That'd be great actually. I've thought of the Schrodinger equation as kind of a hybrid diffusive-wave equation - diffusive due to the single derivatives in time and double derivatives in space; wavey due to the imaginary unit, and the solutions taking the form of e^(ix) (roughly speaking). Curious as to your take on it and the other pdes.
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?
Thank you! I can't wait either I'm really looking forward to it! And not a computational speed up, you won't really gain much in terms of computational speed. But using the weak form explicitly in your numerical scheme has a lot of other advantages. For example, now you can be more certain that you are solving for the correct solution if you have a discontinuous solution like in the video. It also provides a natural way of implementing a numerical scheme, because as you saw all derivatives have been moved to the test functions, and if you take the test functions as a basis, they become matrices! So in this one step you have literally transformed a PDE to a linear algebra problem. This is the essence of finite element methods, one of the most popular methods in all of numerical analysis.
What is not clear to me: are you calling the test function, phi, itself "the solution"? That doesn't make sense to me, since u is the function we're after, so it would seem u is still what you would call "the solution".
I am not arguing that the weak form of a PDE is any more arbitrary than the original PDE. But, I AM arguing about what the definition of SOLVING hence a SOLUTION of EITHER is. Whether solving this weak form or this strong form, solving GENERALLY means finding an expression u = computable function of x & t. But, if YOU are saying "they don't even have to be functions", then ... WHAT ARE THEY? I mean, roots of polynomial equations are well defined by the polynomial equations, but we have known power series solutions (see 2000 paper by Bernd Sturmfels) for expressing the root as a function of the coefficients of the polynomial. You just seem to be leaving the weak form of the PDE unsolved and calling that "a solution". But, I could leave the strong form of the PDE unsolved, too, and equally immorally call that a "solution".
Hi, I have a question, Are the Burgers equation and the shallow water equations cousins? Currently, I'm working on the shallow water equations and our professor has advised us to firstly test the numerical schemes on the burgers equation. And to be honest I don't get the importance of this point that much. By the way, thanks a lot for the video, it's so informational.
Absolutely! They can both be derived from Euler’s equations, though Burgers equations is of course more simple. If you look at the momentum equation of the shallow water equations, you’ll notice the convective terms are actually the exact same, so they have very similar behaviour (the shallow water equations will be more complex though).
@@BeyondtheBigBang Hi, thank you so much for your reply. Would you mind if you recommend me any article or papers about the Burgers equation and its solutions, I'm very in need. Again thank you and I'm looking forward to your next video about the burger's enchanting recipe .
I could recommend some textbooks, but honestly Burgers equation is well understood enough that there are a tonne of resources online. Here’s a good set of lecture notes I found helpful when I first started learning about conservation laws and wave equations: (lecture 3 is probably the most relevant) web.stanford.edu/class/math220a/lecturenotes.html
most PDEs in physics come from applying Newton's 2nd law to some infinitesimal element. Challenge: can you derive the weak form of PDE from Newton's laws (or just even try to make sense of this statement)? Posing b/c am unhappy with the explanation @8:15, if you can figure this one out probably worth writing a (short) paper on it...
it's not as arbitrary as you think also. if you'd recall momentum is really a co vector and velocity is a vector, the beloved quantity 1/2 mv^2 in classical physics is really best thought of as contracting momentum covector with velocity vector
@@yuanyao5190 You can actually derive it directly from Lagrangian Mechanics! Just, that interpretation would be a bit too advanced for the scope of this video :)
@@BeyondtheBigBang Coming form a biology field I thought this was great. Sorry I am not as smart as a mathematician, just a biologist, so this was super instructive! keep doing this material! (and I am older than 8!)
@@MacGut555 "I am not as smart as a mathematician, just a biologist" Lot's of potential for jokes there, but in all seriousness, no one is smarter than anyone. We just study different things!
![Tee Grizzley - Detroit (Feat. 42 Dugg) [Official Video]](http://i.ytimg.com/vi/aeKfTtt0x14/mqdefault.jpg)
You’re shockingly good at explaining this, my dude.
ahahaha nice one
Very well put together! The transitions, graphics, writing on the paper with different markers, walk and talk, etc. we’re all very well utilized, and the video definitely gained me some understanding of these ideas! Look forward to seeing more :)
I came in expecting just another mathematical explanation, but damn, this was pretty well put together! Especially the explanations of the burger's equation part.
From one small channel to another, I absolutely loved this video. Major respect man. I have been feeling really stressed about studies this week, but your enthusiasm and brilliance have completely turned my mood around. Very happy to get back to my books this evening. Thank you for your inspiring work. :)
Wow Alex, this was extremely well done! I really loved how you included the history behind the equation as well :) I haven't reviewed shocks in a long time, and this was a nice new way of thinking about them. Great job!
Also, yay Shankar's QM textbook at 8:51!
You're a rockstar at explaining things, As a mathematician, I learn many new things about PDEs in a very interesting way. Please make more videos about physical applications of other PDEs like Sobolev and fractional Burgers equation.
Thank you! Glad you enjoyed the video :) And absolutely I already plan on making some videos like this, hopefully I'll get them out in the coming months!
Just perfect. All the other videos just glanced over why you would use a test function or integrate the whole thing. Now it totally makes sense.
Hey this was great! Please keep making more of these!
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.
Hi! You should do more videos like this, longer, into details. It is amazing how you explained those concepts so easily and fun way. Love you, guys. Saludos desde Venezuela.
Great explanation
Solid explanation, thank you :)
1 year later, I searched for this gem of a video and enjoyed it again!
the teaching style is so smooth and engaging... you certainly deserve more views...
thank you i know you didn't get many views but the video is valuable and it was so useful :)
Brilliant! Explained very well and will definitely watch more!
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.
This is a good question. I'll give you 3 different answers, you pick which you like the best. Perhaps this warrants making another video altogether.
1. If you think simply in terms of how to take a derivative when the function is not differentiable (ex. discontinuous), you naturally go towards distributions replacing functions. The generalization of a derivative is defined in terms of test functions. It recovers the same derivative for differentiable functions, but allows for more general answers. And when you replace all the derivatives in a PDE with "weak derivatives", you get the weak form.
2. You can think of the PDE as a residual, so the solution to the PDE is that which sets the residual equal to 0. Of course, if we really want a residual we need a scalar quantity, so you must take the inner product. The idea then is that the solution is the "function" such that no matter how you take an inner product with this PDE, the residual should be zero. The inner product of course is set by integrating and multiplying by a test function. The test function can even be seen like the weights of a weighted residual. No matter what the weights are, the solution should still give you a zero residual. But this formulation has the advantage of allowing you the freedom to set the space for inner products and test functions, not just L2.
3. This one is perhaps the most beautiful if you are familiar with Lagrangian mechanics. Set your lagrangian L to be equal to the PDE, i.e. L(u)=du/dt + d/dx(u^2/2). We want u such that L=0. By the principle of least action, we want to minimize the action S, i.e. the integral of the lagrangian. So we want the integral of L(u+ev) to be minimized for dS/de=0, where e is a small parameter defining a perturbation. If you go through some variational calculus, you will find that the condition which minimizes the action is identical to what we consider the weak form of the PDE! i.e. multiplying by a test function and integrating by parts is entirely equivalent to finding the solution which minimizes the total variation of the PDE, which in my opinion is quite beautiful.
Great explanation my man! I'm curious why you said that pde govern all differential equations at the beginning of the video. Thank you!
Another informative youtube channel. Nice!
Very insightful
Thoroughly impressed by the method of explanation! Great job.
your voice is soothing
Absolutely amazing video! Subscribed.
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.
Excellent video explanation! Looking forward to more advanced stuff, your way of explaining concepts is great!
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!
Very, very good.
This ensight is very appreciated !
your video is super good man
weak formulation has never looked so clear to me. At least the core concept. Beautiful video.
👌👌You should do more videos like this. Waiting for thermo
absolutely! Lots more to come!
Brief and informative video, Thank you
Great video, you're a credit to all of Chile
0:15 you already lost me buddy. Btw, love the theory explaining type of video with history of it involved
RIP, but regardless Im glad you still got something out of the video!
This video was amazing! Thanks for the very detailed explanation.
Wow I love this content man. Really digs deep into my desire to know early models of Fluid Mechanics. Nowadays at university they skip through all the uninspired bits to get straight into Navier - Stokes Equations only to directly solve it with CFD. 😅
Which I mean is the most relevant is not necessarily a complain. But understanding how models came to be can make wonders for understanding latter work
Wonderful video! It helped me a lot on my research, but I must also say: AGUANTE BOQUITAAA!
Boca sho te amo
Wow! this is an amazing video. Kudos dude, this is one of the best explanation videos/intro-material on the subject! Go BTBB.
Dear can you please solve the burgers equation by finite difference methods? Make a video on this for me
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?
Thank you :) Actually it’s quite interesting, although the integral of the solution is exactly conserved over the domain, there is a decay in energy (integral of solution squared). This comes from the shock, it’s easy to show that energy is conserved up until that point and then the energy will dissipate. The rankine hugoniot condition, which determines the speed of the shock wave, will introduce some dissipation. It’s similar to a shock wave in the Euler equations, entropy will always be conserved in an ideal fluid until a shock develops, and then you necessarily you have a drop in entropy across a shock.
@@BeyondtheBigBang beautiful! Thanks for the explanation!
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
Wow, thank you for this amazing video. I have a much a better understanding now. Thank you! ☺️
Where is your later video?
Good stuff ! Great video ! It will be interesting to see more similar topics being discussed intuitively (e.g. viscosity solution)
Hii, your video is really awesome to understand weak solution. It would request you to make similar video on strong form and weak forms of PDE. Thanks
Extraordinary explanation. Make more interesting videos.
You are very very good. you have to make more videos like this man.
Starting equation used for the explanation is not even the wave equation
Absolutely incredibly video, I couldn’t even wait to finish it before subscribing. It would also be neat if you could cover more numerical fluid flow and maybe even the navies stokes equations :)
Thank you!! And yes for sure, will definitely do some more fluid stuff like this!
I’m usually very lazy to comment on peoples videos, but this video made me say that this is wayyyyyy better explained than my teacher
nice one❤❤❤❤❤
Awesome video! Aguante Boca!
Ah, I just learnt something new in a really simple way! Love it BTBB
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
Dude you just saved my ass! Amazing explanation! And nice Boca Jersey!
woooowww,,, you should be my professor...thank you so so much😭
Enjoy your videos very much! What would you recommend to self-study PDEs and weak solutions?
Great explanation and enthusiasm. Hope your channel blows!
Please upload videos on related mathematical concepts such as Stokes and Green's Theorems
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
Excellent video. Are you from Argentina? (CABJ?)
SPITTIN
Can you tell me how to generate the animation for showing the solution wave? I
It looks like while deriving the weak differential equation you used Fubini's Theorem before integration of parts for one of the components of the final integral-why can you change the order of integration? Due to our assumptions of phi do we know more information about the regularity of phi*u_t? Can you explain more about the function space that phi belongs to?
Yup correct, you do have to interchange the order of integrations, but this isn't really a problem as you are assuming that phi is integrable. Aside from that, phi can really be from any space you want really (though this is more important for numerical schemes). It is completely arbitrary, but the key is it must be smooth and have compact support, meaning Fubini's theorem is valid.
I don't know if you are still checking your RUclips comments. I am coming across your channel for the first time.
Can you tell me how long did it take to make this video? I am particularly interested in how long it took for you to read the references and get tge knowledge organised before you start shooting or writing for the video.
I came here from r/mathematics. I am doing my MSc math and studying wave equations in pde. Not a huge fan of pde’s. But this was packed with lots of theory and history. (I was blown away when you read the pde). In our program structure, we are just studying the theory of pde’s solely and my faculty doesn’t really connect with physics, so most of it feels like ‘born out of thin air’ and missing story links. What do you think are the essential story links I should read before studying “solving wave equations”? Can you do a video on this?
Thank you! Difficult question to answer... So of course I have my own physics interpretation in that I always like to think of a PDE by applying a physical meaning on it. As you say, when you don't do that it can feel a bit made up. I don't necessarily know too much about a lot of wave equations, just the famous ones, so things like Burgers equation, shallow water, KdV, shrodinger, etc. They all have really interesting stories behind them but I'm not really too sure of any single reference that covers them all. The history for this video I got from the first chapter of Dafermos "Hyperbolic Conservation Laws in Continuum Physics", which has a few more details I didn't mention in this video you might like, but as you can guess only applies to hyperbolic PDEs. I wish I could give you a better reference sorry!
For sure in the future though I'll be looking to cover different equations and talk about PDEs more, such as how to interpret PDEs by just looking at them, etc.
Grrrrrreat!
I have a question. I took a Introduction into CFD in my Mechanical Engineering bachelor's degree and we learnt about the Burger Equation. I think the weak form is more like Finite Element Method than Finite Volume or Finite Difference method, because we use explicit or implicit Upwind schemes to discretize the PDE. Am i correct or no?
Thank you Jose! You're absolutely correct, if you use the weak form you naturally derive the Finite element method, whereas if you start from the strong form you usually get finite difference, and finally if you use the integral form you get finite volume. Though of course once you get into the numerical side where you're discretizing things and working with matrices instead of continuous forms the lines become muddy and the distinctions aren't quite so clear anymore.
Hello, did you animate the graph with LaTex (see 00:40)? BR and THX for the Video
ahhhhh you took us on a walk i looove it!
who is the TA
Very well done bro, your qualification plz ???
PDE not ODE nature is write down on pde equations,
Love this video! But it might be a bit too big brain for me right now lol
As long as you got the main idea - that weak solutions are more general because you pass the derivatives off to a test function, you're golden! That's already way more big brain than I was in first year
why arent you my teacher wtf this is great
Great video!
Would love to watch a video about the Klein-Gordon Equation, and Dirac Equation of the Electron (from Strauss' Intro to Partial Differential Equations - 13.5)
If you could please, I'd love to see your take on that topic.
Mmm great suggestions! It’s not my area of expertise but maybe I can do something with one of the other guys in the group... :)
@@BeyondtheBigBang
Gotcha.
Lemme know if you're itching to tackle it - I studied up to the treatments of the schrodinger equation / wave equation / heat equation.
Im trying to get some insight on the more advanced PDEs.
If you're looking for intuition, I'm thinking about making a video in interpretation of PDE's in general. It wouldn't be specific to any particular PDE but I could cover all the ones you mentioned and it would give a decent idea on how to interpret terms of a PDE intuitively. Maybe that could be one of the next few videos I release
@@BeyondtheBigBang
That'd be great actually.
I've thought of the Schrodinger equation as kind of a hybrid diffusive-wave equation - diffusive due to the single derivatives in time and double derivatives in space; wavey due to the imaginary unit, and the solutions taking the form of e^(ix) (roughly speaking).
Curious as to your take on it and the other pdes.
Ok! I’ll get to work on it! :)
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?
Thank you! I can't wait either I'm really looking forward to it!
And not a computational speed up, you won't really gain much in terms of computational speed. But using the weak form explicitly in your numerical scheme has a lot of other advantages. For example, now you can be more certain that you are solving for the correct solution if you have a discontinuous solution like in the video. It also provides a natural way of implementing a numerical scheme, because as you saw all derivatives have been moved to the test functions, and if you take the test functions as a basis, they become matrices! So in this one step you have literally transformed a PDE to a linear algebra problem. This is the essence of finite element methods, one of the most popular methods in all of numerical analysis.
What is not clear to me: are you calling the test function, phi, itself "the solution"? That doesn't make sense to me, since u is the function we're after, so it would seem u is still what you would call "the solution".
I am not arguing that the weak form of a PDE is any more arbitrary than the original PDE.
But, I AM arguing about what the definition of SOLVING hence a SOLUTION of EITHER is.
Whether solving this weak form or this strong form, solving GENERALLY means finding an expression u = computable function of x & t.
But, if YOU are saying "they don't even have to be functions", then ... WHAT ARE THEY?
I mean, roots of polynomial equations are well defined by the polynomial equations, but we have known power series solutions (see 2000 paper by Bernd Sturmfels) for expressing the root as a function of the coefficients of the polynomial.
You just seem to be leaving the weak form of the PDE unsolved and calling that "a solution".
But, I could leave the strong form of the PDE unsolved, too, and equally immorally call that a "solution".
Hi, I have a question, Are the Burgers equation and the shallow water equations cousins?
Currently, I'm working on the shallow water equations and our professor has advised us to firstly test the numerical schemes on the burgers equation. And to be honest I don't get the importance of this point that much.
By the way, thanks a lot for the video, it's so informational.
Absolutely! They can both be derived from Euler’s equations, though Burgers equations is of course more simple. If you look at the momentum equation of the shallow water equations, you’ll notice the convective terms are actually the exact same, so they have very similar behaviour (the shallow water equations will be more complex though).
@@BeyondtheBigBang Hi, thank you so much for your reply. Would you mind if you recommend me any article or papers about the Burgers equation and its solutions, I'm very in need. Again thank you and I'm looking forward to your next video about the burger's enchanting recipe .
I could recommend some textbooks, but honestly Burgers equation is well understood enough that there are a tonne of resources online. Here’s a good set of lecture notes I found helpful when I first started learning about conservation laws and wave equations: (lecture 3 is probably the most relevant)
web.stanford.edu/class/math220a/lecturenotes.html
@@BeyondtheBigBang Well received, thank you so much!
You. Giant. Nerd. Amazing 🤩
thumbs up for burgers equation
One dimensional burgers equation
most PDEs in physics come from applying Newton's 2nd law to some infinitesimal element. Challenge: can you derive the weak form of PDE from Newton's laws (or just even try to make sense of this statement)? Posing b/c am unhappy with the explanation @8:15, if you can figure this one out probably worth writing a (short) paper on it...
it's not as arbitrary as you think also. if you'd recall momentum is really a co vector and velocity is a vector, the beloved quantity 1/2 mv^2 in classical physics is really best thought of as contracting momentum covector with velocity vector
@@yuanyao5190 You can actually derive it directly from Lagrangian Mechanics! Just, that interpretation would be a bit too advanced for the scope of this video :)
@@BeyondtheBigBang ahh i see, makes sense
This is all trivially obvious though, pretty straightforward.
Naturally, of course.This video was merely intended for younger audiences of 8-10 year olds that haven't been exposed to PDEs yet
@@BeyondtheBigBang Coming form a biology field I thought this was great. Sorry I am not as smart as a mathematician, just a biologist, so this was super instructive! keep doing this material! (and I am older than 8!)
@@MacGut555 "I am not as smart as a mathematician, just a biologist" Lot's of potential for jokes there, but in all seriousness, no one is smarter than anyone. We just study different things!
Right yeah that makes sense cuz gravity is Newton’s Third Law
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99,895 likes away from no eye brows and pick hair
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No
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