Oxford Calculus: How to Solve the Heat Equation

University of Oxford mathematician Dr Tom Crawford explains how to solve the Heat Equation - one of the first PDEs encountered by undergraduate students. Links to worksheet and app download below.

Check your working using the Maple Calculator App – available for free on Google Play and the App Store.

FULL DESCRIPTION OF CONTENT TO FOLLOW










You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.

Get your Tom Rocks Maths merchandise here:

[campbellmorrison]

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-ytgx]

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

[RCSmiths]

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

[turbowil]

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

[jamesjohn]

clear and nice explanation, thank sir.

[SkatersCH]

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

[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.

[the_kid]

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

[emekamajis]

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

[physicsadhyaapak]

You are just awesome Professor. Love From India.

[MisterTutor]

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

[meowpurrr]

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

[lesleynoblemaiden]

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

[eun-usumomo]

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!

[___________mrivan___________]

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

[dorset]

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).

[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.

[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?

[kartiniganesha]

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?

[miguelaphan]

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