The funny thing about all of the great lectures on youtube is that as the series continues, the number of views decreases quickly. Almost nobody completes the valuable lectures as a whole. Lectures are just lectures. We even need to read a book after watching lectures for comprehensive understanding. I am now proud of myself because I understood all of this when I self-studyed this subject and I even more clearly refresh the subject from the greatest professor Steve.
Good for You! If possible, try to take it a bit further, i.e. doing drills and solving problems. We can sometimes get the illusion of understanding, when only watching lectures, or so as I've heard.
Thank you for the lecture! As for the typo in the solution, on the board there is the inverse transform of *e(-sqrt(s))* , you missed the *x* when inverse transforming 😊. As *x* it is a constant in the frequencies domain, the fix is immediate, there is an extra *x* as the numerator of the fraction and *x^2* in the numerator of the exponential (with _x>0_ which is the case by construction of the problem).
Steve, those e^sqrt[s]x terms in the homogenous solution are not Gaussian distributions, they're sinusoids; this is esp. evident if you retain both +/- sqrt[s] terms and use the Bromwhich Integral to invert the LT.
I was recently learning about the Laplace's transform on Wikipedia. It answered some questions, but not until I watched this video did I really get the utility. This channel is such good education.. seriously I've watched many episodes by this point and it's never been a waste.
Weird and dumb question: have any links for that shirt? That looks comfy and easy-to-wear. Also, I only passed calc 4/diffEq class by memorizing the table of Laplace transforms so I could "cheat" my way through the final. Received a "this is correct, but there are other ways to solve it."
This is so interesting. At my school we sadly didn't go through mush PDEs which is weird since they are so important in engineering. Thanks for the video, super helpful!
Someone in the comment asks about forcing terms. I think of it like as if you subject this metal rod to some kind of external input heat, which is 0 when this metal rod is free from any external heat source, then a particular solution is when you put it in a stove with some wavey fire. Steve, correct me if I am wrong.
Can anyone help me understand the seeming conflict between the Boundary Condition that u(inf,t)=0 and the Initial Condition u(x,0)=Sin4x? There's a similar mismatch between IC and BC at u(0,0). [Sin(4*0)=/=e^(-0)] Incredibly helpful series, much appreciation for the effort to create it! Extremely well explained!
Thank you so much for taking the time to make these videos. Your explanations are incredibly well thought out, informative, and easy to understand. You have been a lifesaver for me in my Fourier Analysis class.
Why do we consider a semi-infinite domain? If the bar just has some fixed length, could the domain not also be fixed? I assume it does not matter, as the domain just has to be one sided, but not necessarily semi-infinite? Additionally, can you solve using this method without necessarily having a prescribed boundary condition? For example, what if the input at the boundary is allowed to take any (real) value at any point in time.
Hello sir. I am Balaji from IIT kanpur, India. I wanted to ask if for the problem of 1D transient diffusion (I solve it using RK4 time integration and with a FDM discretization), does diffusion take place on the boundaries or does it only happen inside the material (apart from boundary nodes)?. PS: I have an influx boundary condition at the boundary nodes (Neumann's boudnary).
I have a question not related to this video directly, but kind of blending the underlying idea with this paper "Distributed Control of Partial Differential Equations Using Convolutional Reinforcement Learning" by you and some collaborators. When modeling environments for reinforcement learning described by PDEs, what was your approach? I have been thinking about this and if you use the method of lines let's say, you end up with a high dimensional set of ODEs that can be stiff and thus difficult to solve quickly. In terms of RL, this seems like it wouldn't work, as you could be waiting too long for the solution of these ODEs to go to the agents as the current state of environment. So, I am curious how you and your collaborators chose to simulate these PDEs such that when an action is taken, the new environment state can be generated almost immediately. I really appreciate your time! This is a really cool paper and I am not sure many have an answer to a question like this.
hmmm is it right to just remove e^(positive value) without also remove sin(4x) from U(x,t) . U(inf,t) is still unbound as sin(4x) is still inside and lim=sin(inf) doesn’t exist.
That’s an interesting observation. The limit of sine doesn’t exist, but sin(4x) is bounded by the amplitude so it only takes on values in [-1,1]. The solution u(x,t) of the pde as t goes to infinity does go to zero though so it is bounded.
I think in this case it is like a heat source that is to be substituted as 0 when there is just the metal rod alone, or some other function when you put this metal rod on a stove.
Hi Steve, I have an idea for a high confidence model for a transition from the current economy to a two-tiered sustenance and cultural economy. Would you be interested in trying to help me model this, or know some experts we can talk to?
Ugh. Wolfram/Mathematica and bad syntax. $x / \left(2 \sqrt{\pi} t^{3/2} ight) * \exp{-x^2 / (4*t)}$ why is mathjax not the default in yt comments? I am angy at all the corporations tonight.
its eather "x/..." instead of "1/..." or "-x*x/4t" instead of "-1/4t" or both or i completely wrong, dont listen to me :) (quick google search failed me this time)
The funny thing about all of the great lectures on youtube is that as the series continues, the number of views decreases quickly.
Almost nobody completes the valuable lectures as a whole. Lectures are just lectures. We even need to read a book after watching lectures for comprehensive understanding.
I am now proud of myself because I understood all of this when I self-studyed this subject and I even more clearly refresh the subject from the greatest professor Steve.
Good for You!
If possible, try to take it a bit further, i.e. doing drills and solving problems. We can sometimes get the illusion of understanding, when only watching lectures, or so as I've heard.
@@andyl.5998 Yeah that's true.
I'm guilty of this with the mit open courseware XD
Great lecture! Although, I can’t wrap my head around the fact the B.C. and and I.C. are not the same at u(0,0)…
Perhaps the initial condition should have been a cosine, then the IC and BC match at (0,0).
Thank you very much...
In time 36:40:
The inverse laplace of exp(-sqrt(s)*x) is equal to x/(2*sqrt(pi*t^3))*exp(-x^2/4t)
Thank you so much
Thankyou so much
I’ve been waiting to see how you present this! Excellent
Thank you for the lecture!
As for the typo in the solution, on the board there is the inverse transform of *e(-sqrt(s))* , you missed the *x* when inverse transforming 😊.
As *x* it is a constant in the frequencies domain, the fix is immediate, there is an extra *x* as the numerator of the fraction and *x^2* in the numerator of the exponential (with _x>0_ which is the case by construction of the problem).
Steve, those e^sqrt[s]x terms in the homogenous solution are not Gaussian distributions, they're sinusoids; this is esp. evident if you retain both +/- sqrt[s] terms and use the Bromwhich Integral to invert the LT.
Thank you for this -- beautiful explanation. Great teacher.....
I was recently learning about the Laplace's transform on Wikipedia. It answered some questions, but not until I watched this video did I really get the utility. This channel is such good education.. seriously I've watched many episodes by this point and it's never been a waste.
Weird and dumb question: have any links for that shirt? That looks comfy and easy-to-wear.
Also, I only passed calc 4/diffEq class by memorizing the table of Laplace transforms so I could "cheat" my way through the final. Received a "this is correct, but there are other ways to solve it."
Great question! It is a Patagonia M's L/S Cap Cool Lightweight Shirt or M's L/S Cap Cool Daily Shirt. Basically the only clothes I wear
This is so interesting. At my school we sadly didn't go through mush PDEs which is weird since they are so important in engineering.
Thanks for the video, super helpful!
Someone in the comment asks about forcing terms. I think of it like as if you subject this metal rod to some kind of external input heat, which is 0 when this metal rod is free from any external heat source, then a particular solution is when you put it in a stove with some wavey fire. Steve, correct me if I am wrong.
Can anyone help me understand the seeming conflict between the Boundary Condition that u(inf,t)=0 and the Initial Condition u(x,0)=Sin4x? There's a similar mismatch between IC and BC at u(0,0). [Sin(4*0)=/=e^(-0)]
Incredibly helpful series, much appreciation for the effort to create it! Extremely well explained!
Thank you so much for taking the time to make these videos. Your explanations are incredibly well thought out, informative, and easy to understand. You have been a lifesaver for me in my Fourier Analysis class.
Why do we consider a semi-infinite domain? If the bar just has some fixed length, could the domain not also be fixed? I assume it does not matter, as the domain just has to be one sided, but not necessarily semi-infinite?
Additionally, can you solve using this method without necessarily having a prescribed boundary condition? For example, what if the input at the boundary is allowed to take any (real) value at any point in time.
Could you make a video on the inverse Laplace transform of exp(-sqrt(s)). This is exactly where I got stuck when doing this problem independently.
Hello sir. I am Balaji from IIT kanpur, India. I wanted to ask if for the problem of 1D transient diffusion (I solve it using RK4 time integration and with a FDM discretization), does diffusion take place on the boundaries or does it only happen inside the material (apart from boundary nodes)?. PS: I have an influx boundary condition at the boundary nodes (Neumann's boudnary).
I have a question not related to this video directly, but kind of blending the underlying idea with this paper "Distributed Control of Partial Differential Equations Using Convolutional Reinforcement Learning" by you and some collaborators.
When modeling environments for reinforcement learning described by PDEs, what was your approach? I have been thinking about this and if you use the method of lines let's say, you end up with a high dimensional set of ODEs that can be stiff and thus difficult to solve quickly. In terms of RL, this seems like it wouldn't work, as you could be waiting too long for the solution of these ODEs to go to the agents as the current state of environment. So, I am curious how you and your collaborators chose to simulate these PDEs such that when an action is taken, the new environment state can be generated almost immediately.
I really appreciate your time! This is a really cool paper and I am not sure many have an answer to a question like this.
hmmm is it right to just remove e^(positive value) without also remove sin(4x) from U(x,t) . U(inf,t) is still unbound as sin(4x) is still inside and lim=sin(inf) doesn’t exist.
That’s an interesting observation.
The limit of sine doesn’t exist, but sin(4x) is bounded by the amplitude so it only takes on values in [-1,1].
The solution u(x,t) of the pde as t goes to infinity does go to zero though so it is bounded.
what is a forcing term sir
I think in this case it is like a heat source that is to be substituted as 0 when there is just the metal rod alone, or some other function when you put this metal rod on a stove.
Hi Steve, I have an idea for a high confidence model for a transition from the current economy to a two-tiered sustenance and cultural economy. Would you be interested in trying to help me model this, or know some experts we can talk to?
Ugh. Wolfram/Mathematica and bad syntax.
$x / \left(2 \sqrt{\pi} t^{3/2}
ight) * \exp{-x^2 / (4*t)}$
why is mathjax not the default in yt comments? I am angy at all the corporations tonight.
Also: super close. It took me a second review to notice how super close it was.
And it's a Gaussian in space that evolves to higher sigma values as time evolves (which washes it out into a constant as t->\infty).
its eather "x/..." instead of "1/..." or "-x*x/4t" instead of "-1/4t" or both or i completely wrong, dont listen to me :) (quick google search failed me this time)
your suggestions are true, i guess.