Heat equation: Separation of variables

[@mitoand9]

You sir, know how to teach. Excellent choice uploading these on KZbin, thanks!

[@DrChrisTisdell]

Thanks. It is my absolute pleasure!

[@KLCII88]

Almost 10 years later and your explanations are still amongst the best, thank you sir!

[@schonnyrules]

If this man were my professor I would give him a standing ovation after every lecture.

[@DrChrisTisdell]

Wow! Thanks!!

[@nithisatepetju3064]

@@DrChrisTisdell Me too Dr. Chris i always give you a standing ovation after see your VDO lecture.

[@julianaorcasitas7891]

@@DrChrisTisdell How cab we get a copy or Download the free PDF

[@GeorgiesEmpire3]

Amazing, you boiled down two weeks worth of lecture in my DE class into one video, AND made it interesting and incredibly easy to follow. Studying just got a whole lot easier!

[@anchitrao7374]

Truly amazing, was not able to find any other video that explained the concepts so thoroughly. Allowed me to really wrap my head around the ideas being taught. Great job professor.

[@MusicalMartha7]

Incredible ! Thank you so much Dr. Chris Tisdell by far the best lesson on youtube. You save my semester ! :)

[@amirmeysami7868]

one hour and a half remaining to my differential equations exam, this helped a lot!! Beautifully done!

[@gautamgautamkrishna]

Amir Meysami i have little more(5 hrs)

[@non5270]

@@gautamgautamkrishna i have more than this ( 1 and half a day)

[@AndyJohnD1001]

people like you make Maths easy clear, inspiring, encouraging & enjoyable thanks very much

[@ShimmerArc]

After 2 years stumbling about in my physics course you made everything clear in less than an hour. So that's what Fourier Series and Separation of Variables is all about! Blimey.. and there I was, memorizing everything without a clue of what's going on! THANK YOU! :)

[@mikesteele5935]

Thanks Chris. This is beautifully done. I'll recommend it to my students.

[@2tone2g]

Wow, if you were my professor I wouldn't be struggling at all. I only wish I had discovered your videos sooner. Thank you!

[@jrippee05]

Dr. T, you just saved me in my mathematical modeling course. I finally get this. Thank you!

[@TheSunshineRequiem]

+James R out of curiosity, you didn't take differential equations before ur modeling course?

[@shoomyjeddah]

you're a life savior. Thank you from Saudi Arabia

[@kevinshants8463]

Fantastic lecture as always! You explain concepts and calculations at a superior level. Thanks for all your videos Dr. Chris Tisdell

[@kakakolaable]

actually . i dont know how to offer my thanks to you i think you the best you tube lecturer

[@apesnerdo92]

Thanks a lot, The confusion occurred because I learnt to use "sin, cos" and "sinh,cosh". I understood now. Thanks for your videos, Really helpful. Keep up the good job!

[@Matt08011982]

It definitely worth watching. Thanks Chris to covering all the details involve in this method of solving PDEs.

[@reffzz]

I'm so glad I found this as I was stuck with an assignment for a class I did not attend. It was very similar to what you solved here but the initial condition was different. Thank you so much!

[@jdstufu]

Thank you very much for your video lectures on Solving Heat and Wave Equations and your explanations on Method of Separation of Variables and Fourier Series. Both of your videos helped me with my Advance Math classes. More power sir!!!

[@denisjohnson1591]

You are the reason I made it through Fourier Series and BVP. Thank you so much.

[@nnnlynnn]

AWESOME TEACHING!! u made maths much less torturous and in fact, i start to find it interesting!

[@sinsbehindthescene5203]

you just saved my final SIR, Thank you from Windsor Canada

[@DrChrisTisdell]

Well done!

[@haris525]

I wish Dr. Tisdell taught at our university - great lecture - we need more awesome professors who do this kind of explanation

[@ahmed12op]

Dr. Chris, thank you from 1 to infinity ! .. Great explanation, wonderful presentation of the subject and a nice way of organizing the lecture.

[@TheLeontheking]

Very helpful! One suggestion I would give is for expressions which contain multiple elements, use parentheses after sin and cos - one might otherwise think that the x is outside of the sin,cos function's arguments.

[@kristoferdressler8134]

Excellent, excellent. This is the best I've seen, even after 6 years of grad school.

[@DrChrisTisdell]

Wow - that is quite a comment. Thank you!

[@joshualee9082]

loving these lectures everytime. thanks on behalf of all student viewers, you sir, make math a bit better to learn.

[@laicao3920]

wow i have a test tomorrow and didnt know anything until i watched this lecture. thank you so much :) ill let you know what i have on my test

[@DrChrisTisdell]

Yes, you are correct - that's the general form for complex roots a + ib, However the real part of the roots at 20:30 are zero (ie, a = 0) and hence the exponential part you mention is just zero everywhere.

[@michaelguo6349]

Thank you. This video saved my ECE205 final

[@Jellysugarcandy]

I'm really amazed by how u respect ur students .. and amazed more by ur gift of amazing explanation .. u really made me taste the fun math again after i'd forgotten how fun it was :) thnx alot :)

[@MrPSPreview]

This just saved my life. Thank you so much for the clear explanation

[@suliman9058]

my college needs drs like you you are very helpful

[@DrChrisTisdell]

Not yet, but it's a good idea. In fact, I will teach another course on PDE in 2013 so I plan to make more videos about the heat kernel (and lots of the PDE things) then!

[@timurtavman8336]

This video helped me a lot for understanding this subject.Thanks a lot

[@mkuseligobo5585]

Anyone noticed a bit of an error in the equation T(t)=...., a negative sign is missing in the exponent, otherwise the video is extremely outstanding thank you very much Dr.

[@_brown_]

You need more subscribers! You might just save my PDE Course this semester.

[@kvmuralidharankv9172]

A VERY CLEAR PRESENTATION, HIGHLY INFORMATIVE

[@DrChrisTisdell]

Counting down to the test! I have visited NUS once and NTU quite a few times. I really enjoy Singapore every time I visit.

[@MrAndy19990]

i'm a stuent from Politecnico of Milan (italy). Thank you very much, very good professor, much better than mine :)

[@jeanpaulnavier6726]

chi hai avuto? ahah :)

[@MrAndy19990]

Ho avuto Pavani e Colombo

[@jeanpaulnavier6726]

MrAndy19990 be', la pavani ho sentito sia brava...no?

[@MrAndy19990]

ahahaha spero tu stia scherzando! Comunque mi correggo, ho avuto Gazzola e non Colombo.

[@jeanpaulnavier6726]

nono non scherzo,come mai questo commento?gazzola è stronzo a quanto dicono ahah

[@hrahul16]

Very very in-depth explanation...thank you very much..

[@aliwatfa6951]

You're so amazing!!! A big thank you to Dr. Chris :)

[@Vishu1171]

Hello Dr. Chris. You have given really nice explanation. But I have one doubt that you have mentioned that for r

[@justsimonurban]

Great video. A side note on a bar of length pi: could you not take a bar and bend it until it forms a circle with radius 1 then the bar would be length 2*pi. All that you need to do now is cut the bar in half to have a bar of length pi (in fact 2 bars of length pi)

[@jeetbanerjee9551]

Hello Prof. Chris. First of all, thank you so much for these wonderful, clearly understandable PDE videos. If possible, could you please upload the video for the solution of [2-D] heat equation by the method of separation of variables? However, if your video on this topic is available online, please send me that link. Thank you in advance.

[@ddPing13]

You don't even know what a help this was to me! Thank you very much. Quick question, may be a dumb question, though for you or anyone that may know but why is it n^2 wen subbing in for the gamma for the Ce^4(gamma)T?

[@MegaTRIANGULUM]

Can't like this video twice. Thank you so much!

[@Saucepv897]

very motivated professor. very energetic

[@SHONSL]

Comprehensive example incorporating broad material.

[@jeanpaulnavier6726]

WOW, excellent teaching right here!!

[@gopeswarnamasudra8895]

thank you sir ,this lecture helped me a lot..very well presented

[@laurahernandez6595]

THANK YOU SOOO MUCH!!!!!! MY NOTES MAKE A LOT MORE SENSE!!!!!!!!!

[@PrasannaKumar-zx7gr]

Very clear explanation, thank you!!

[@bohaswa1]

Thank you very much, Greetings From Kuwait

[@DrChrisTisdell]

Hi. In this case, \gamma is negative and the roots of the associated characteristic equation are (purely) imaginary, ie \pm i \sqrt{-\gamma}, however, we drop the \pm i in the solution was we are interested in real-valued solutions. Since \gamma is negative, -\gamma is positive and so \sqrt{-gamma} will make sense. Sometimes people use a "squared" as the separation constant to simplify the notion.

[@davidzorroyang]

Great teaching material! Thank you so much sir!

[@romanel.andressa]

Awesome explanation! ps. your accent is VERY CUTE!! Greetings from Brazil!!

[@matthewjames7513]

Great video Chris! I noticed you found the solution by assuming u has the form u = X(x) T(t). Yet we know the general solution to the W.E. is u = p(x-ct) + q(x+ct) from d'Alembert's formula. Does this mean that the solution in this video must also coincidentally have the form u = p(x-ct) + q(x+ct)? Lastly, isn't using this separation of variables technique going to only provide you with a subset of solutions from the general solution? (since there could be solutions that fall outside the form of u = X(x) T(t) )? Thanks again for the great video :)

[@dmceye]

Great presentation. I was wondering if you could also do one for heat equation with a source term and double neumann homogeneous boundary conditions. Thank you.

[@rohanthawal9703]

Fantastic teaching sir!! thanks a lot for uploading the video

[@Kavalyeri]

you are a legend,sir !

[@DrChrisTisdell]

@Reyder93 The important point is that the constant must be positive. If we write it as \alpha^2 (with \alpha > 0) then it makes the notation and calculations simpler when solving.

[@Andrew-ht5by]

thank you very much. It is very straightforward.

[@hasseng2000]

Thank you so much Dr. Chris Tisdell, I have question? if the boundary conditions u(x,0)=0=ux(0,t) and initial condition u(1,t)=q(x) how can we solve this problem?

[@SeparataCosmigonon]

Amazing! Thank you so much. From Argentina.

[@DrChrisTisdell]

My pleasure - thanks for the comment. Hope you also find my new ebook of some use!

[@DrChrisTisdell]

Hi, since no-one has answered, there are two ways to approach this problem: 1) via the substitution u = 1/v; 2) separate the variables to get dv / v(c + bv)= -m dt and then integrate both sides. (You will need something like partial fractions on the left.)

[@MyThundermuffin]

Hi Chris, do you have any videos on the Heat Equation with inhomogeneous boundary conditions ?

[@anaslahrichi]

Amazing lecture. Thanks a lot !

[@kittycat321online]

So well explained thank you very much.

[@jasmine27c]

This is so helpful, thank you very much!! Great lecturer :)

[@davidcordero632]

Great Lecture!

[@LG-tu5cb]

Hey Chris, thanks for the vid! In your response, what is\pm and \?

[@alimekernef8080]

Thank you ser, you are the best teacher thank you again

[@MrAndy19990]

My teacher says that we can say gamma

[@ahmedyoussef8765]

thanks , greetings from Egypt :)

[@ImGonnaShout2000]

Here is a MatLab plot with the solution for a rod with arbitrary fixed temperatures in both ends and initial temperature f(x)=1 between pi/2 and 3pi/2 plus (U_2-U_1)x/pi + U_1 between 0 and pi. Variable c is the speed of the whole thing. Set it to 0.03 to see in very slow motion. Enjoy!: ---------------------------------- clear; close all; clc x=linspace(-pi,pi,100); n=1; c=0.3; figure akse=x*0; U_1=2; U_2=-2; for t=0:2*pi/99:16*pi i=0; y=(U_2-U_1)*x/pi+U_1; for i=1:50 y=y+(2/(pi*i))*(cos(pi*i/4)**-cos(3*pi*i/4)+(U_2-U_1)-*-((-1)^(i))-U_1*(1-(-1)^i))*sin(i*x)*exp(-((c*i)^2)*t); end plot(x,y,x,akse); axis([0,pi,-5,5]); M(n)=getframe; n=n+1; end movie(M,1,10); -----------------------------

[@DrChrisTisdell]

Awesome! Thanks.

[@JustCheckingMusic]

For case II at 29:55 my proff uses the form of: X(x) = A cosh (Lx) + Bsinh(Lx) and gets a trivial solution for sigma < 0 instead of sigma >0. Does this make any difference? I prefer your method over his, since yours is way better explained here ;).

[@keatonmcsweeney7772]

incredibly helpful lecture, thank you so much

[@Jairsilvav]

Thanks, greetings from Ecuador.

[@DrChrisTisdell]

And my best wishes go to Ecuador!

[@mathewgeorge3673]

Hi Dr. Tisdell, I have a question about your solution to one of the ODEs. For the purpose of this question, I will use the convention: l = lambda and g = gamma. When g < 0 --> l^2 = -g --> l = +- sqrt(g) i Why did you add a negative inside the sqrt sign? Thanks.

[@mathewgeorge3673]

Timestamp to consider: 19:38

[@jamesjoetomo]

Incredible lecture, helped me a lot *clapping*

[@Theshikshik]

Bless this man

[@ozorobi]

Thank You Sir, this helped a lot

[@nnnlynnn]

Thanks for the great teaching! By the way, do you have any teaching video on heat kernel? :D My mind is totally switched off reading the book =(

[@liquidstl]

Great Presentation. Thank you!

[@DrChrisTisdell]

Excellent! Good luck!

[@annecarlill4317]

Thanks. This has really helped.

[@ArjunSKV]

Nice presentation...

[@victorjoseph5357]

r

[@CrushOfSiel]

If the bar was cylindrical, kind of how you drew it, would you solve the equation in cylindrical coordinates? How you're solving it you're assuming the problem is more rectangular, correct?

[@DubsteadyMusic]

its assuming only x dependence. if he solved it in cylindrical hed assume only z dependence, radial and theta components would go away and the soln would be exactly the same

[@Sancarn]

I agree totally with shimmerArc. Thanks so much!

[@TehJozze]

You can only make such division only if the X(x) and T(t) are non-zero, or if the ratios with their respective derivatives are a trivial singularity.

[@DrChrisTisdell]

Yes, and who would want to find a trivial solution?

[@DrChrisTisdell]

You are very welcome.

[@mioumitsou]

so i have a question, if the separation constant was negative instead of positive (i have seen it in many texts) how would this affect the procedure from then on?

[@DrChrisTisdell]

That is great news! Thanks for the feedback.

[@miamarukot5009]

I'm looking for examples for inhomogeneous boundary conditions for the heat equation. Also, this example is explicitly the Dirichlet boundary condition, how about the Neumann boundary condition?

[@denisjohnson1591]

+Mia Marukot If you check Dr. Tisdell's page has an example under "Inhomogeneous heat equation: Insulated ends". Hope that helped.

[@DoggoWillink]

@DrChrisTisdell I can't believe I didn't see that. So all you have to do is factor it into a product and then you can divide to separate it.. thank you.

[@oskarvedin5197]

Can the separation constant be complex?

[@slad82]

on page 9 @ 20:20, in the third case, i.e when gama