First Order PDE

First-order constant coefficient PDE

In this video, I show how to solve the PDE 2 u_x + 3 u_y = 0 by just recognizing it as a directional derivative and using some simple multivariable calculus. This is part of a technique called the method of characteristics. Enjoy!

[valeriereid]

Thanks for making this geometric method so easy to understand. I did not understand when I read the textbook. Your video certainly helped.

[michalbotor]

it shows at (10:42) that you have had already a lot of hours in front of a camera under your belt. to come back with such an eloquent style. bravo! :)
i can only add, that once you start doing physics, in case of any doubt, you can always quickly check in your mind whether the units match up.

[baronvonbeandip]

Finally! Between this video and the other method of characteristics video, I get where we are going throw in Fourier series. This is easily the most helpful method of characteristics video. I am eternally grateful

[mattl-dlgx]

This will be one of the best series of your channel, I’m looking forward the next video 🤩

[daizhao]

Thanks for your video, Sir. It is really helpful for me. My major is aerospace engineering. I have studied ACFD and PDE. I passed them but they are still confused for me. Now I think I have known more things such as relationship about them. Thanks. Looking forward further video. Thanks

[CasualMutoi]

You know.... A fan when you hear them. "9001 because it's over 9000" - without going saiyan. Control demonstrated.

[lbomcarvalho]

Congrats to your efforts to show how beautiful math is. You are well successful in this journey. But what the equation says is that the grad of u is orthogonal to v and not necessarily zero over the line. Also if you multiply v by alpha, any Real number, you will have a line passing through the origin ... but you will get the same PDE ... then what the PDE is saying is that the grad of u is orthogonal to alpha v.

[alexrosellverges]

Best intuitive explanation, love these videos, thanks!

[solarsailor]

Brilliant video, perfect explanation! I love your channel!!

[gnikola]

All the f(x-ct) is super cool and very important in Physics, as it's one of the solutions to the wave equation, which is probable one of the most important equations is Physics. It can be shown that a solution U(x, t) for the second order linear pde Utt = c^2 Uxx can be written as U(x, t) = f(x-ct) + g(x+ct). Here U would be the wave profile, the perturbation of the medium we're studying and c the perturbation's speed in such medium. Now the cool thung is that f and g can be thought of as "travelling waves" because of the "moving profile" perspective Peyam talked about. This is extremely important because we know most of the waves "in nature" are moving (f, g) and are not standing waves (U) like the ones you might find inside a wind instrument, for instance. The way to get those f and g is via D'Alambert's solution, of which I think Peyam talked about. And if he didn't, he might in the future, as I think it's a kind of important point in PDE courses

[josephadamriestra]

I want to say that over 9000 joke really caught me off-guard haha. Great video though, helps a lot!

[Mono-uqnd]

luv u bro! saved my life! almost died in the PDE class :(

[whythosenames]

Can you create a video where you explain that how you, given a problem like the transport problem, can express it as a PDE so how you get to the approach U_t+cU_x=0?

[henricornette]

freaking love you, thanks you so much

[jueraljy]

Love your content! I would appreciate if you could do a video on PDE for vibration string and d'Alembert's solution.

[padaddadada]

Tell me if I am wrong but we could imagine wave propagating in the direction perpendicular to v and the parallel lines .

[padaddadada]

Thank you for your video,

I have a difficulty understanding why the line u and the vector v are not perpendicular instead of parallel .
Since their dot product is 0 shouldn’t they be perpendicular instead of parallel ?

[MagicGonads]

So the 3D plot of such a U is some function plot f(z) where z is on the line perpendicular to v, extruded to infinity in the directions parallel to v

[azhar]

My lecturer has used it for the linear advection equation. My exams are over but I still has a question. He has done it the same way as you.

I know that as variables, we have x and t and grad is a function x, y and z. Then, how can we dot the grad of u and the other vector to get zero when grad isn't even a function of t?

[blackpenredpen]

Still waiting for the final exam unboxing