Characteristic Method
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- Опубликовано: 6 сен 2024
- 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!
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Dr.Peyam you should cover multi-index notation, very confusing ):
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.
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@Khalil Diego Instablaster =)
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Its only the second PDE you did a video and it makes me love PDE already. :)
Yay!!!
PDEs is rlly hard…
I also like how all these videos are done under 10 minutes.
Your enthusiasm is infectious, always makes me want to do more math. Thank you so much for these videos they're really really helpful!!!
The best online lecturer i have ever met. You have made me understand Mathematics like as if am the one who discovered the concepts.
Thank you so much!!!
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!
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!
Thanks for teaching math on RUclips
Hello I’m from South Korea 🇰🇷
And I’m interested in math
I hope you make good videos through math
Write in polar coordinates and find that @f/@r = 0. (@ is partial derivative.) Consequently, f is only a function of theta. Of course, this happens to work only in this particular case.
Nice observation!
@@drpeyam Thank you. My comment actually followed from the observation that the level curves of the function were theta = const, and then making it more mathy.
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)
I Appretiate You Dr.peyam
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.
Yeah, that’s why u is undefined at the origin :)
Why is y=y(x) a curve ?😅
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
I may require at bit bit more than the usual 10 minutes :D
Wow, thank you! 😊
dr peyam, could you cover legendre transformation please?
its in depth understanding is essential for the understanding of many fundamental physical relationships (mechanics: lagrangian > hamiltonian, thermodynamics: energy > enthalpy). i'm sure many interested in physics people here would be grateful to you. :)
Yes, I would be so grateful if you would cover this when you have the time! I recently discovered your channel and I am addicted!
There are so many comments see if you can find mine
Done ✅
You are a king
This video make so much sense I can understand so much from it thank you.
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.🥺
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.
thanks, but how if the function is not homogeneous ? instead of zero, it can be a constant or function.
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
I believe so, something like that
In 3rd year pure math, we did by Characteristics system
x'(s)=x
y'(s)=y
Etc...
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
Why in this case it’s a curve and not a line ?
The solutions to the ode is a curve not a line
Now this was pretty cool
The dot product is zero when the vectors are orthogonal to each other, so why do we try to find vector who are parallel?
Basically if the gradient is perpendicular to v, the function is constant on curves parallel to v. It has to do with the fact that the gradient is perpendicular to level surfaces
@@drpeyam Thank you very much! :D Could you make a video on why the gradient is perpendicular to level surfaces?
@@ImPresSiveXD Gradient is the direction of the great ascent. Locally, level surfaces all look just like planes stacked on top of one another. Now, here is a question for you, if you have a stack of paper on your desk and you want to pierce as many sheets as you want with a knife, how would you stab with it?
grad V \dot U is the directional derivative dV/dU (rate of change of the function V along the direction U). If this derivative is zero, this means the function V is constant along that curve.
I love this!!!
What if the equation turns out not to be separable? Eg. dy/dx = (x+y)/x²
There’s a more general method of characteristic equation, but it’s more complicated. Also sometimes PDEs don’t have explicit solutions, so as long as the ode has a solution (without a formula), that’s ok
@@drpeyam So what are the names of those more general methods and where can I find some more about them?
Method of characteristics, it’s in Chapter 3 of Evans’ PDE book
Dear Dr. Peyam
I want a formula for the root of a complex number a+i b
Express it in polar form as r exp(i theta), with r and theta both real. We can always take r in [0, infinity) and theta in [0, 2 pi). Its nth root is r^(1/n) exp(i theta/n).
I know this method but i saw a formula for the square root specially in a reference before but i cannot remember it
10 minutes is the key here.
안녕하세요~올리버 쌤입니다
Who are the most irrational criminals? Pirates!!!
2xyU_x + (x^2 + y^2)U_y = 0 please teacher can you solve this PDE
You can do it using this method, try it out!
@@drpeyam i need your help doctor
I had problem in determing caracteristic courb ..... what is the primitive of 2xy dy
How did you know I need characteristic method? Thank you!
btw.. Link of Previous Video: ruclips.net/video/fY2zgSKTKP4/видео.html
I just knew 😉
"Very cool and non-rigorous DEs techniques" jajajajajaj
16
👍👍👍👍❤❤😘👆👆👆
Really I m very appreciate you...sir..@ peyam..❤👍
First
Abay o Chotiye
Play at 84% speed and his voice becomes tolerable...
That's really mean. He can't help his voice but you can help your rude comments.
What are you talking about, his voice is great. And if you don't like it, don't watch!
@@pat4rush What do you mean? He likes his voice, just at 84% speed ;) And I'm pretty sure Dr. Peyam doesn't need knights in shining armours protecting his honour ;)