Heat Equation
ฝัง
- เผยแพร่เมื่อ 5 พ.ค. 2016
- MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
View the complete course: ocw.mit.edu/RES-18-009F15
Instructor: Gilbert Strang
The heat equation starts from a temperature distribution at t = 0 and follows it as it quickly becomes smooth.
License: Creative Commons BY-NC-SA
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gilbert strang passion for math just makes me wanna learn more math
what a wonderful .. wonderful teacher. I sat through his Linear Algebra course a few years ago .. and much to my surprise, I realized that contrary to my convictions at the time, it certainly "IS" possible - thanks to him - to gain intuition on top of calculation skills, in what is often considered advanced graduate math.
Awesome job professor!
Greetings from Germany.
He is wonderful, I am a foreign physics student in Berlin, and it surprises me how the germans explain math and theoretical physics. It was so bad at the beginning that I started questioning the possibility of explaining these topics with intuition, but I later realised that it only happens in german universities.
👏
admirable his explanation and his passion
I cannot express how complete his understanding is in a way that has helped me so in my independent studies. I always wished I could study in the MIT but well, we can at least get a very good representation of what one can learn there
impressive... I never had taken any better classes than this one 10 min short video clip... thanks!
Brilliant Video! Linear Algebra is the intrinsic framework underlining in everything!
Greatest professor, His linear algebra course was the best
The way he said " we are again faced with a fourier series problem" is wholesome. what a great professor.
such a wonderful lecture by you professor . I was actually feeling heat equation.
Thank you so much Professor Strang :) You are an inspiration
awesome job. I wish I had this during my heat transport class! So much better than a semester of lectures.
Lies again? Evil Angel
Strang's teaching skills are remarkable. These videos have been a great help to me as an undergraduate student.
Hi listen dear
This 10-minute video summed up 3weeksworth of classes for me. Thank you
“The ends of the bar are kept at temperature 0, ... they’re frozen.” You have to love this.
Listen sir i need your help
Thanks professor. your hard work inspire me !
This video is a solid beginning to partial differential equation.
well that explains a lot.
Thanks professor.
very very nice lectures of professer gilbert
What an amazing teacher!! Thank you!
totally different level..these lectures are very helpfull
thanks Professor G. Strang...
Basically, Prof. Strang did 2 guesses right? That the solution u(x,t) was eigenfunction and S(x) was sine function.
This premise is based on the assumption that the function is separable and the separated forms are equal to lambda or -k²pi² which is a negative constant
thank u you tube, great !this professor, I never seen in my country , like it very much, I enjoy it . Thank u Prof. greeting from Indonesia
I never seen in my country. i am from china. i am glad we have same feeling!
from china, too~
Omgggggg this is wonderful lecture clearly state thank you 🙇🏻♀️🥹
Is there any reason we don’t parametrize the boundary of our pipe instead, via a parametric curve, and use a Lagrangian for the interior and boundary of the domain, respectively, rather than the Fourier series?
Are we measuring the change in temp at every possible time, or are we finding the hottest and coldest points?
Dr Strang, my row in your (68xn) matrix will bee 68 ones and no ceros before the UΣV factorization to spot clusters of happy users versus "I don´t now Users" and the σ for noise
why the space function S(x) is a sin function? is it an intuition based on the fact that heat trough space travels as an electromagnetic wave?
Thank you❤️❤️❤️😭
Does anyone know what mathematical theorem allows us to conclude that the functions we found are the ONLY ones ? Also, in this example the vectorial space of solutions is of infinite dimension : every B_k can be chosen at will. This is very different from non-partial differential equations, where the dimension of the affine space is determined by the order of the equation. Can anyone explain why ?
Also, why does the pulsation of the sin waves (k*pi) have to be integer multiples of pi ? Could we not "sum" over continuous pulsations to get an even more general formula for the general solution ? (B should then be a function of the variable on which we integrate)
a space of n-dimensions has at most n-eigenvalues
they only used one dimension here, x. You can always extend it to R^n.
Fourier's motivation was in solving the heat equation. You want the heat to be evenly distributed that's why it's k*pi.
I think you're mixing things up. A functional vectorial space is most of the time of infinite dimension, independently of how many dimensions the starting set has. Even the 2pi periodical functions space, which has the e^(ikpi) as a Hilbert base is of infinite dimension.
That's why the Cauchy theorem for single-variable differential equations is so neat, but that's also why multi-variable calculus is much worse
The thing is that the sines themselves can express a lot of different functions so in this case, it would be redundant to include more. Bk is not really arbitrary, he left it as an implicit exercise to the reader to find that in this case, the Bk=4/kpi
I wish this guy was my professor
way better than my monotone professor
we need heat transfer for mechanical engineer please
Where to start learning differential equations?
If you're still interested I really suggest differential equations with boundary value problems, I had one which is of applications by the same author but for theoretical purposes this one is more complete
Gooood
Can anyone explain why k must be a positive integer? Couldn't k be any real number or complex number? Also, isn't cosine also an eigenfunction?
It is really a result of the initial condition. If K was complex then there would be a minus and an i attached somewhere else. Cosine isn't allowed because it has to be 0 at both endpoints
is that series aproximmating a gaussian?
At around 4:00, he proclaims, WITHOUT PROOF, that he has the general solution.
Ideas of completeness of the eigenfunctions are not even hinted at.
Classic Strang. LOL.
interesting
why does k have to be an integer?
Oh I know! It's because of the boundary conditions.
In a sum:
if you count apples, you have to count them integer amount of times. If you want to count say 2.5 apples, then you count two apples with coefficient 1, but the third apple will have coefficient 0.5. So coefficients [B1, B2, B3] = [1, 1, 0.5]. k is the index of coefficients.
There are trigonometry even in problems dealing with heat?!?!?!?!?
Hello
I’m in 8th grade learning about this😒
K
Why are there no coses in the general solution?
Samuel Laferriere note that when he speaks about a bar which borders are frozen (minute 5:00), he is stating the heat equation problem with 2 dirichlet conditions (temperature at each border). Dirichlet conditions need odd extension of initial conditions, so calculating its fourier series you get only sines. You can prove that if the source and the initial condition function are odd, then solution will be odd. If you got neumann condition as border condition, you would only have coses.
hope i helped
When you evaluate the boundary conditions the coefficients of those cosines are 0.
Ck *cos (0*k*pi) = 1 and Ck*cos( 1*k*pi)= 1 but the boundary conditions = 0, so Ck must be 0.
why is he blinking his eyes every sec .. nice video tho
Tourette's, neurodegenerative disease, or he's just 82
yes u saying right
He is old bear with him have patience