Perhaps. I'll add it to my to-do list and get to it when I get the chance. Just for my reference, do you have any example problems you would like help with?
For the question in the link for anyone interested : Partially differentiate the given substitution with respect to t (i,e hold all other variables constant) using the product rule. Partially differentiate the given substitution with respect to x twice. Then just simply sub into the original PDE. Simplify and solved.
I am very grateful you're doing this, your material has really REALLY helped me a lot, seeing this increase in uploads is just incredible, if you ever accept donations I'd be more than glad to buy you a beer.
Find the distribution 𝑢(𝑥, 𝑡) by writing the wave equation and boundary conditions for a rod (one dimension) of length L=1 unit, with both ends fixed and whose initial displacement is given by 𝑓(𝑥), whose initial velocity is equal to zero. (𝑐2 = 1, 𝑘= 0.01) 𝑓(𝑥) =ksin(3𝜋x) Can you solve this question? I couldn't solve it. Can you help me?
I am still very confused by the dummy integration variable and how that allows us to integrate from x0 to x, and for f' to be integrated directly into f(x)-f(x0) since it is a derivative of x+ct, not x
@@chrisdanikas7918 Think about it as ψ(x, t) = G(η) + F(ξ) where η: (x, t) ⟶ (x - vt) and ξ: (x, t) ⟶ (x + vt) Now ∂ψ/∂t = ∂G(η)/ ∂t + ∂F(ξ)/∂t = (∂G/∂η)(∂η/∂t) + (∂F/∂ξ)(∂ξ/∂t) by chain rule But ∂η/∂t ≔ -v and ∂ξ/∂t ≔ v Therefore ∂ψ/∂t = -v (∂G/∂η) + v (∂F/∂ξ) However we want ∂ψ/∂t(x,0) Now we take an approach that’s non-standard in nature: ∂ψ/∂t = -v (∂G/∂η) + v (∂F/∂ξ) = -v (∂G/∂(x-vt)) + v (∂F/∂(x+vt)) ⇒ ∂ψ/∂t(x,0) = -v (∂G/∂(x)) + v (∂F/∂(x)) Now we can obviously use the prime notation ∂ψ/∂t(x,0) = -v G’ + v F’
Will there be any problem if we use "dx" as integration instead of "ds"? As it is just a dummy variable! Anyone's help is much appreciated, Thank You!😄
Hi I don't understand that isn't this solution incorrect as under the galiean transformation the wave equation become not invarient so doesn't this contradict special relativity?
This video isn't meant to be consistent with special relativity...it's just the solution of a wave equation in the context of (non special relativity) mechanics.
Holly. Can I say that in the end, we don't know what f(x+ct) and g(x-ct) are seperately. But, we know what they are when they are combined? We only know the solution u(x,t) through exploiting the definition of the definited integral？
I'm not entirely sure if I understand your question, but I will try to answer. When we start solving the PDE, we don't know what f(x+ct) and g(x-ct) are explicitly; we only find out once we solve the PDE and express them in terms of the boundary/initial conditions. So in the end, we actually do know what f and g are, and we also know what their combination u is; that's what I solved for in the video. Does that answer your question?
Hi, I was watching a video about coffee cup mechanics on Sixty Symbols' youtube channel and got interested in the math behind vibrations in the surface of the coffee, and it led me to this paper www.me.rochester.edu/courses/ME201/webexamp/coffee.pdf but unfortunately it doesn't go into a lot of detail about how they came up with the solution. Do you think you could do a video explaining that at some point? I'd really like to know what a full solution of the coffee vibration problem looks like. That would be awesome. In the meantime I think I'll go watch your videos about Bessel functions, and see if that helps me. Really enjoying your content!
From what I've read so far (i.e. over the last few minutes), it seems like the problem involves solving a Laplace equation by separation of variables (in cylindrical coordinates, with the solution ultimately involving Bessel functions). It seems 'simple', which means that I know/can easily learn it, just that the algebra is a little complicated. I can put it in my to-do list though (I plan to cover Laplace's equation once I fully do the wave equation anyway), and thank you for the kind words!
This was so well explained! I hope you get more attention from the UA-cam Science Community.
Wow, thank you!
Thank You VERY VERY much... no explanation as clear as yours... you deserve so much more credit. .... GREAT series on PDEs.
not the hero we deserve but the hero we need
Best video series for PDE
Brilliantly explained!!
Hi, I was wondering if you could do a video about transforming harder parabolic PDEs into easier ones?
Perhaps. I'll add it to my to-do list and get to it when I get the chance. Just for my reference, do you have any example problems you would like help with?
Faculty of Khan A problem like one shown here: www.physicsforums.com/threads/pde-transforming-hard-equations-into-easier-ones.674252/
For the question in the link for anyone interested :
Partially differentiate the given substitution with respect to t (i,e hold all other variables constant) using the product rule. Partially differentiate the given substitution with respect to x twice. Then just simply sub into the original PDE. Simplify and solved.
I am very grateful you're doing this, your material has really REALLY helped me a lot, seeing this increase in uploads is just incredible, if you ever accept donations I'd be more than glad to buy you a beer.
Hey you already have a patreon!!
Thank you for the kind words!
thank you so much for such a great explanation , all doubts are cleared sir. ☺☺☺☺
why is the dummy variable introduced? Can't equation 3 be integrated with dx?
yes it can
Thank you for this amazing video
amazing. saving my EE degree
Thanks for the videos. I wish I could afford to donate.
Hi, do you still upgrade PDE class?
There are no Laplaces problem
I will eventually; thanks for asking!
D'alambert solution could also be achieved using Seperation of variables.
Find the distribution 𝑢(𝑥, 𝑡) by writing the wave equation and boundary conditions for a rod (one dimension) of length L=1 unit, with both ends fixed and whose initial displacement is given by 𝑓(𝑥), whose initial velocity is equal to zero. (𝑐2 = 1, 𝑘= 0.01)
𝑓(𝑥) =ksin(3𝜋x)
Can you solve this question? I couldn't solve it. Can you help me?
it was really helpful , keep it up sir
I am still very confused by the dummy integration variable and how that allows us to integrate from x0 to x, and for f' to be integrated directly into f(x)-f(x0) since it is a derivative of x+ct, not x
Notice u(x, 0) sets t equal to zero, hence f'(x - ct) at t = 0 turns into f'(x), therefore integrating from x0 to x, gives you f(x) - f(x0) via FTC.
@@guilhermefranco2949 so when you set t=0 means that the derivative with respect to x + ct becomes a derivative with respect to x ?
@@chrisdanikas7918 Well yes but actually no
@@chrisdanikas7918 Think about it as
ψ(x, t) = G(η) + F(ξ)
where η: (x, t) ⟶ (x - vt) and ξ: (x, t) ⟶ (x + vt)
Now ∂ψ/∂t = ∂G(η)/ ∂t + ∂F(ξ)/∂t = (∂G/∂η)(∂η/∂t) + (∂F/∂ξ)(∂ξ/∂t) by chain rule
But ∂η/∂t ≔ -v and ∂ξ/∂t ≔ v
Therefore ∂ψ/∂t = -v (∂G/∂η) + v (∂F/∂ξ)
However we want ∂ψ/∂t(x,0)
Now we take an approach that’s non-standard in nature:
∂ψ/∂t = -v (∂G/∂η) + v (∂F/∂ξ) = -v (∂G/∂(x-vt)) + v (∂F/∂(x+vt))
⇒ ∂ψ/∂t(x,0) = -v (∂G/∂(x)) + v (∂F/∂(x))
Now we can obviously use the prime notation
∂ψ/∂t(x,0) = -v G’ + v F’
7:32 in the end it doesnt even matter.
Will there be any problem if we use "dx" as integration instead of "ds"? As it is just a dummy variable!
Anyone's help is much appreciated, Thank You!😄
Beautiful!!
2:30 I find it interesting that you spell the word TRAVELLING as such, the British way
Bleh, I read so many posts from different parts of the world online that sometimes I just end up using American and British spelling interchangeably.
Hi I don't understand that isn't this solution incorrect as under the galiean transformation the wave equation become not invarient so doesn't this contradict special relativity?
This video isn't meant to be consistent with special relativity...it's just the solution of a wave equation in the context of (non special relativity) mechanics.
Thank you very much
Thanks sir
Very elegant.
Glad you like it!
You ever think of doing some Lie Theory lectures?
I haven't thought of it directly, but I may plan to do differential geometry in the future so that's where it will come up.
Would love to see some differential geometry videos!
Thanks alot.
w explanation
8
/
L'éthique
Newton et les Forces
La Mécanique des Fluides de Dalambert.
L'eau qui coule.
Holly. Can I say that in the end, we don't know what f(x+ct) and g(x-ct) are seperately. But, we know what they are when they are combined? We only know the solution u(x,t) through exploiting the definition of the definited integral？
I'm not entirely sure if I understand your question, but I will try to answer. When we start solving the PDE, we don't know what f(x+ct) and g(x-ct) are explicitly; we only find out once we solve the PDE and express them in terms of the boundary/initial conditions. So in the end, we actually do know what f and g are, and we also know what their combination u is; that's what I solved for in the video.
Does that answer your question?
Thank u .........
me enjoying my 0.07 dollar ugali at kenyatta university and my lecturer on you tube is telling me to send 20 dollar
Hi, I was watching a video about coffee cup mechanics on Sixty Symbols' youtube channel and got interested in the math behind vibrations in the surface of the coffee, and it led me to this paper www.me.rochester.edu/courses/ME201/webexamp/coffee.pdf but unfortunately it doesn't go into a lot of detail about how they came up with the solution.
Do you think you could do a video explaining that at some point? I'd really like to know what a full solution of the coffee vibration problem looks like. That would be awesome.
In the meantime I think I'll go watch your videos about Bessel functions, and see if that helps me.
Really enjoying your content!
From what I've read so far (i.e. over the last few minutes), it seems like the problem involves solving a Laplace equation by separation of variables (in cylindrical coordinates, with the solution ultimately involving Bessel functions). It seems 'simple', which means that I know/can easily learn it, just that the algebra is a little complicated. I can put it in my to-do list though (I plan to cover Laplace's equation once I fully do the wave equation anyway), and thank you for the kind words!
Thank you very much for the help!
🇰🇬🇰🇬🇰🇬👍👍👍