Characteristic Method

Method of characteristics

In this video, I show how to solve (basically) all first-order linear PDE by using the method of characteristics, which basically says that solutions must be constant on curves (called characteristic curves). Enjoy!

[wamunyimasibuku]

The best online lecturer i have ever met. You have made me understand Mathematics like as if am the one who discovered the concepts.

[Kadota]

Your enthusiasm is infectious, always makes me want to do more math. Thank you so much for these videos they're really really helpful!!!

[eliyasne]

Its only the second PDE you did a video and it makes me love PDE already. :)

[patrush]

I also like how all these videos are done under 10 minutes.

[giovannitallia]

Thank you very much, I was looking for someone that could make me understand characterictics method properly and your video enabled me to understand how simple and beautiful it is!

[patrush]

This is great, I've taught method of characteristics for many years now. It's a subset of so-called "contact methods". Legendre transform is also a useful tool for PDEs. More generally, Lie group methods are designed to identify symmetries (invariant transformations) of PDEs of any order. It's basically a recipe / algorithm that works, in principle for any PDE. You should do some videos on that. Some good textbooks on this topic include Ibragimov, Hydon, Bluman.

[jackmaibach]

Thank you so much for this video! Your explanation helped me extrapolate this concept to an unexplained solution method from my PDE class!

a(x, y)*Ux+b(x, y)Uy=0
(a(x, y), b(x, y)) . \/U=0
=> dy/dx=b(x, y)/a(x, y)

The middle line was skipped in my class, but now it makes sense. Thank you so much Dr. Peyam!

[dyer]

Dr.Peyam you should cover multi-index notation, very confusing ):

[Icecream]

Thanks for teaching math on YouTube
Hello I’m from South Korea 🇰🇷
And I’m interested in math
I hope you make good videos through math

[macsentiffany]

im learning PDEs in my final year as a physics undergrad (obvs seen them before but now its in a pure maths module so having to start at the beginning) and i must be going crazy bc when he said 'yeah there is a little bit of abuse of notation but not too much' in the formulation at the end i actually laughed out loud. Great vid thanks (our lecturer is useless)

[georgettebeulah]

This video make so much sense I can understand so much from it thank you.

[thedoublehelix]

Can you solve the navier stokes equations in the next video(lol)? Btw I realized I want to be a math major from your channel so thank you for your beautiful explanations

[murielfang]

Thank you so much for the great content, professor Peyam!!! Btw, is it possible that you will make more videos on characteristic method? We've been studying linear and quasilinear problems, so there are more complex characteristics.🥺

[michalbotor]

dr peyam, could you cover legendre transformation please?
its in depth understanding is essential for the understanding of many fundamental physical relationships (mechanics: lagrangian << LT >> hamiltonian, thermodynamics: energy << LT >> enthalpy). i'm sure many interested in physics people here would be grateful to you. :)

[willnewman]

So u is constant on lines through the origin. In particular, if we allow u to be defined at the origin, then u(0, 0)=u(x, y) for any x, y, right? Therefore, u is constant.

[vitelot]

May be an idea for a video: Given a 2D vector field V(x, y), find the functions K(x, y) such that KV is conservative.

[SimsHacks]

In 3rd year pure math, we did by Characteristics system
x'(s)=x
y'(s)=y
Etc...

[anjaliashok]

So if the equation had a -y instead of a y (next to u_y) then would the characteristic curve just be a bunch of y= 1/ax curves on the axis

[AnantoYusufW]

thanks, but how if the function is not homogeneous ? instead of zero, it can be a constant or function.

[holyshit]

I read something about this method in scripts written in my native language
u=F(y/x) solved before watching
Some systems of ode can be solved in similar way