Oxford Calculus: How to Solve the Heat Equation

[@TomRocksMaths]

Check your understanding with this accompanying worksheet in Maple Learn - available for FREE: learn.maplesoft.com/d/OPDJAPFFNKEJMHAMCRMMBROSPIJKBQENCGMPAPGFBQASLHNULONIPRPQGJPTGJMGFHLRAJFNDLLHNGAMCIGHFMJJBPLJGRCQNGNQ

[@jozanebryan503]

They said I could not access it

[@joeczernuszka7756]

It tells me hat I have reached my free limit though I haven't accessed anything

[@lumasli7676]

Is there a solution to this worksheet?

[@the_kid777]

@@lumasli7676 Exactly! I worked through it but am unsure if I got it right or not.

[@campbellmorrison8540]

Oh my I wish I had seen this when I was doing my Maths degree ( many many years ago) , I could never see the relationship between the heat equation and Fourier series even though I had tried to read Fourier's original paper. So I went on and treated the Heat equation as an example of differential equations and promptly forgot it and continued with Fourier series as if it was a totally separate idea. I enjoy your lectures as you make things fall into place that if only I had understood when I was younger.

[@Jaffa-yt1gx]

I have a PDE exam on Monday and this vid has given me some hope. Ty Ty

[@big-me02]

🤯🤯🤯 first time a lecturer has blown my mind in calculus with ease of explanation

[@RC32Smiths01]

Maths is both fun AND important, so much depth, so much meaning behind it! Cheers

[@turbowil1904]

Amazing video! Thank you. Makes me feel bad paying for school when you help me 10x more in 10x less time.

[@physicsadhyaapak]

This is what we actually need our Professors to teach us in this way. Like,in this He knew that c>0 condition will be not give is anything. But he still did that And that what is Real Teaching. Otherwise, there will be no difference bw a Book and a Teacher. Professors should think as a Student would think.

[@physicsadhyaapak]

You are just awesome Professor. Love From India.

[@SkatersCH]

Hey tom! Can you do a video about solving a pde using the fourier transform? I think its super interesting

[@jamesjohn2537]

clear and nice explanation, thank sir.

[@Tyokok]

Great thanks for this amazing material!

[@the_kid777]

Tom, please provide the solutions to the questions in the Maple Learn worksheet.

[@ash...424]

Coolest Math teacher you are TOM 😊

[@sajidmomin7759]

I have a slab of thickness L on one side I have heat flux and on other side I have fluid. Now I want to know the change in temp of wall and also in fluid due to heat transfer from ambient. What should be the bc on fluid side,or should i keep it open???

[@emekamajis9772]

This is great! Can you make a video on how to solve non homogeneous heat equation.

[@meowpurrr]

do you know when the next open day is for prospective students that would like to study maths at teddy hall?

[@Isomnophilia]

At 9:33, sqrt(c) was added as a coefficient of x on the exponents. Is there a reason for this? I tried coming up with reasons... is it arbitrary to ensure c is greater than 0?

[@Isomnophilia]

Was advised we are solving for F''(x) = cF(x), so the sqrt(c) is to satisfy that relation! As always, I drop marks by forgetting the question.

[@TomRocksMaths]

Well figured out!

[@gossipGirlMegan]

wonderful job!!

[@eun-usumomo7578]

Professor could you clarify whether T(0,t)=T(L,t)=0 means that for any time t the temperatures at the boundaries stays zero? Want to check my understanding, thanks!

[@TomRocksMaths]

Yes that’s correct!

[@74dorset]

Tom at 17:20 you state that the general solution to the problem is F(x) = C1 sin(sqrt(-c)x) + C2 cos(sqrt(-c)x), however when I use Wolfram Alpha to evaluate the general solution for F''(x) = -(c)F(x), I get F(x) = C1 sin(sqrt(c)x) + C2 cos(sqrt(c)x) where the c terms in the solution are positive rather than negative. Is there a reason for the negative c in your solution. I understand that this is where we are considering the case of negative c specifically, but the sign of c seems to be accounted for with the negative sign of c in F''(x) = -(c)F(x).

[@74dorset]

Actually I think I have it. I was trying to avoid that step where we define c as -k², but I think it starts to come together much better if that step is taken. So what we really have if it is the case that c = -k², is F(x) = C1 sin(sqrt(-k²)x) + C2 cos(sqrt(-k²)x). So at this point in order to not produce a complex number, we can say F(x) = C1 sin(sqrt(k²)x) + C2 cos(sqrt(k²)x) = C1 sin(sqrt(-c)x) + C2 cos(sqrt(-c)x) which will yield F(x) = C1 sin(kx)) + C2 cos(kx))

[@joetrump5750]

@@74dorset so are both results equal?

[@AyodeleKayode-w1g]

Thank you sir

[@itsfikree]

Awesome explanation, thanks dude

[@itsfikree]

Such an interesting topic, I'm sorry doctor Tom for calling you dude 😅. I had to comeback to really understand this, cause repetition is the mother of learning 🤩

[@KenStarkey]

Not sure why the derivative for an insulated endpoint has to be zero. Would it not potentially warm up as the heat from the center moved outwards. Insulated I thought means no lost heat to the outside.

[@TomRocksMaths]

Insulated means no heat flux through the end of the rod. Fouriers Law tells us that the heat flux is proportional to the temperature gradient. Therefore if the heat flux is zero, the temperature gradient must be zero.

[@___________mrivan___________]

Great video! Don't know why, but I find joy in the fact that your sin function looks like sun)

[@lesleynoblemaiden]

Loved this one! Thank you Tom! But, is there somewhere we could find the solution to the practice questions?🤓

[@TomRocksMaths]

Coming soon hopefully!

[@lesleynoblemaiden]

@@TomRocksMaths Thank you!

[@the_kid777]

@@TomRocksMaths Can I find the solutions now that a year has passed?

[@MisterTutor2010]

The fact I can solve this problem in my head proves I have no life.

[@kartiniganesha6741]

Hello , I want to ask, if we want to solve this problem analytically, do we have to sum up all of the temperature through conduction, convection, and radiation or only the radiation? Study case : we have 4 rooms with each size of 1x1 m. and they are adjacent to each other (2 columns x 2 rows). It is constrained by the steel wall. There is one source fire with the energy of 1000 watts from one room. How do we calculate the temperature in the center of three other compartments?

[@ryanclevenger6769]

whats that shirt?

[@konfekshun4958]

This guy: T{x-2} x 4lx x 0+ 9x2= Me: I like peanut butter and my shoes are untied!

[@si48690]

Why should C be real? Thanks for the great video!!

[@TomRocksMaths]

Ooooo interesting question - I’ve never actually thought about that, but would be interesting to work through for sure!

[@josephmcmahon7470]

2nd Law of Thermodynamics.

[@si48690]

@@josephmcmahon7470 Interesting! But what if the PDE is a general parabolic equation and is not the heat equation?

[@si48690]

@@TomRocksMaths Turns out we can use linear algebra for this, the eigen values of a self adjoint operator are Real, and that's why the constant is taken to be real.

[@miguelaphan58]

..a most influencial video...a surprice !!

[@gossipGirlMegan]

Thanks a lot! Professor!English gentleman does NOT care tartu.

[@Rose-it2iy]

Me, a current maths and additional maths GCSE student watching this: hmm interesting😅

[@TomRocksMaths]

you'll get there one day :)

[@Rose-it2iy]

@@TomRocksMaths thanks:))

[@moreasmorebaes9996]

@@Rose-it2iy funnily enough I was exactly in the same boat as you last year, rn writing an EE on the topic which is like a mini paper. Trust me u will get there sooner than you think.

[@Rose-it2iy]

@@moreasmorebaes9996 oh okay. Thx for the Motivation and ye maths is my fav subject at school so I will do further maths at A level

[@kaskarii]

why do i watch this... i was lost as soon as tom put chalk to board

[@TomRocksMaths]

As long as you’re having fun!

[@ManuelMendez1]

I bled during an exam because of this equation…

[@TomRocksMaths]

... I'm sorry?

[@Michael-mh2tw]

Interesting that people will watch a video like this of a guy teaching exactly like it's a normal uni tutorial or lecture, and will still comment 'wow I wish uni was taught this way'

[@MarkLucas]

Wow thats a funny way to say Ansys

[@izcoatlamador3290]

come on dude my exam is tomorrow, where ir part 2

[@josephmcmahon7470]

Why spend time trying for possibilities of 'c' when 2nd Law Thermodynamics tells you what c must be?

[@martinstewart]

This is magic, right?

[@TomRocksMaths]

it can sometimes feel that way...

[@MarcinWiesniakQuantum]

I really enjoy the video, but I hate when You use an app to solve a perfectly simple equation. I understand the sponsorship, but it cripples the viewers

[@alengm]

Agreed. Pretty cool app though. Didn't know you can scan equations like that with a camera

[@RonakIyer-t5z]

mgk teaching math.... who would have thought

[@noob_4986]

you look way too cool to be a mathematician