Oxford Calculus: How to Solve the Heat Equation
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- Опубліковано 9 лис 2022
- 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.
Accompanying Maple Learn worksheet with practice questions: learn.maplesoft.com/doc/yllfw...
Test yourself with some exercises on separable solutions with this FREE worksheet in Maple Learn: learn.maplesoft.com/doc/i72ve...
Investigate separable solutions to the Heat Equation here: learn.maplesoft.com/d/PQOSOUL...
Check your working using the Maple Calculator App - available for free on Google Play and the App Store.
Android: play.google.com/store/apps/de...
Apple: apps.apple.com/us/app/maple-c...
FULL DESCRIPTION OF CONTENT TO FOLLOW
Other videos in the Oxford Calculus series can be found here: • Oxford Calculus
Finding critical points for functions of several variables: • Oxford Calculus: Findi...
Classifying critical points using the method of the discriminant: • Oxford Calculus: Class...
Partial differentiation explained: • Oxford Calculus: Parti...
Second order linear differential equations: • Oxford Mathematics Ope...
Integrating factors explained: • Oxford Calculus: Integ...
Solving simple PDEs: • Oxford Calculus: Solvi...
Jacobians explained: • Oxford Calculus: Jacob...
Separation of variables integration technique explained: • Oxford Calculus: Separ...
Solving homogeneous first order differential equations: • Oxford Calculus: Solvi...
Taylor’s Theorem explained with examples and derivation: • Oxford Calculus: Taylo...
Heat Equation derivation: • Oxford Calculus: Heat ...
Separable Solutions to PDEs: • Oxford Calculus: Separ...
Find out more about the Maple Calculator App and Maple Learn on the Maplesoft UA-cam channel: / @maplesoft
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collec...
Check your understanding with this accompanying worksheet in Maple Learn - available for FREE: learn.maplesoft.com/d/OPDJAPFFNKEJMHAMCRMMBROSPIJKBQENCGMPAPGFBQASLHNULONIPRPQGJPTGJMGFHLRAJFNDLLHNGAMCIGHFMJJBPLJGRCQNGNQ
They said I could not access it
It tells me hat I have reached my free limit though I haven't accessed anything
Is there a solution to this worksheet?
@@lumasli7676 Exactly! I worked through it but am unsure if I got it right or not.
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.
I have a PDE exam on Monday and this vid has given me some hope. Ty Ty
Amazing video! Thank you. Makes me feel bad paying for school when you help me 10x more in 10x less time.
Maths is both fun AND important, so much depth, so much meaning behind it! Cheers
clear and nice explanation, thank sir.
You are just awesome Professor. Love From India.
wonderful job!!
Hey tom! Can you do a video about solving a pde using the fourier transform? I think its super interesting
do you know when the next open day is for prospective students that would like to study maths at teddy hall?
Coolest Math teacher you are TOM 😊
Tom, please provide the solutions to the questions in the Maple Learn worksheet.
This is great!
Can you make a video on how to solve non homogeneous heat equation.
Thank you sir
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!
Yes that’s correct!
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.
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).
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))
Great video! Don't know why, but I find joy in the fact that your sin function looks like sun)
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?
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.
Well figured out!
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.
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.
Loved this one! Thank you Tom! But, is there somewhere we could find the solution to the practice questions?🤓
Coming soon hopefully!
@@TomRocksMaths Thank you!
@@TomRocksMaths Can I find the solutions now that a year has passed?
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?
The fact I can solve this problem in my head proves I have no life.
Why should C be real? Thanks for the great video!!
Ooooo interesting question - I’ve never actually thought about that, but would be interesting to work through for sure!
2nd Law of Thermodynamics.
@@josephmcmahon7470 Interesting! But what if the PDE is a general parabolic equation and is not the heat equation?
@@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.
why do i watch this... i was lost as soon as tom put chalk to board
As long as you’re having fun!
I bled during an exam because of this equation…
... I'm sorry?
Awesome explanation, thanks dude
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 🤩
come on dude my exam is tomorrow, where ir part 2
Wow thats a funny way to say Ansys
whats that shirt?
..a most influencial video...a surprice !!
Me, a current maths and additional maths GCSE student watching this: hmm interesting😅
you'll get there one day :)
@@TomRocksMaths thanks:))
@@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.
@@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
Thanks a lot! Professor!English gentleman does NOT care tartu.
Why spend time trying for possibilities of 'c' when 2nd Law Thermodynamics tells you what c must be?
This is magic, right?
it can sometimes feel that way...
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'
This guy: T{x-2} x 4lx x 0+ 9x2=
Me: I like peanut butter and my shoes are untied!
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
Agreed. Pretty cool app though. Didn't know you can scan equations like that with a camera
you look way too cool to be a mathematician