Heat equation + Fourier series

How to solve the heat equation via separation of variables and Fourier series.

[laserjohn]

Final TOMORROW.... due to English speaking issues, I simply could not comprehend my professor. You just gave me more in 20:53 minutes than I have received in weeks. Thanks so much!

[theminu]

Dr Chris Tisdell you are a legend!!! thank you for uploading all these helpful videos on you tube.. really helps students pass mathematics.. good job

[victoriafong]

My differential equations final in 2 days and I haven't been able to understand anything about Fourier series until I found these videos yesterday. Thank you so much!!!

[shafqatsheikh]

Sir, the formula for b_n = (2/n)*(-1)^(n+1) that we obtained towards the end of the video, could this formula be rewritten as (-2/n)*(-1)^n?

Time Stamp: 18:38

[DrChrisTisdell]

"OR, do we have to look for ONLY sin(xyz) = 0, since B cannot =0 and solve for xyz": - that's exactly right! Well done!
Very good to hear you are finding the ebook useful.

[Borewiczes]

Man, You're great teacher. This helped me a lot! Cheers!

[周宏駿-ky]

i watch the video when i`m prepare my final exam, it`s great to study with your video, thx a lot !!!!

[oppositeofh]

Thank you for this! You've helped me a lot.

[blurginouliz]

You showed an example where the temperature at the end points of the bar are zero, which makes it possible to determine some of the coefficients, but it take it that in order for this to work inphysiscs, you have to use some absolute scale of temperature such as Kelvin, so it's a bit weird to say the end points will be at absolute zero.. On a more realistic problem, where the end points are at constant temperatures, but non zero, I wouldn't be able to eliminate the coefficient of the Cosine function.. How would I proceed then? Could you please show an example, or point me to a vide where you've already done it maybe? Thank you very much! You're a great teacher, and your videos are awesome!

[changbroot]

By the way, I forgot to thank you for such an amazing videos and ebook. You have done a great job and it's very easy to follow and you're very clear on all the matters. Thank you very much for your hard work.

So back to my question. I know that we want B not = 0. But isn't B*sin(xyz) = 0 yields zero, if we divide both sides by sin(xyz)? OR, do we have to look for ONLY sin(xyz) = 0, since B cannot =0 and solve for xyz? Maybe I'm tired of studying this week and can't think straight.

[DrChrisTisdell]

Hi - if we follow your suggestion then we obtain the zero (trivial) solution. We are interested in the non-trivial solutions and so that's we we disregard B = 0.

[YourAverageHater]

might be a dumb question, but how did determine the constant to be "bn", which is applicable to the bn-sine-fourier summation, opposed to, say, the an-cos-fourier summation? Was that deliberate on your part, so that it would fit with the sine in your solution of the heat equation?

[Novak]

The fourier series does not converge to pi when x=pi but it converges to 0. Shouldn't you choose an initial condition which is compatible with boundary condition?

[DinoDudeDillon]

Where does (-1)^(1+n) come from in b sub k? I can get b sub k is -2/n by doing the inner product with sin(nx). You mention another video but I'm not sure where to look

[seggyRK]

what if my initial temperature was f(x)=T_0=50degreesC? How would that afffect my final solution?

[MatrixOfDynamism]

Dear Dr Tisdell, There is a question that I did really like to ask you if I could meet you in person but I shall post it here anyway. Trignometric functions Sin, Cos come out from working out ratios of right angled of triangles, historically. It so happens that they end up in Fourier series and thus in Fourier Transform and FFT. Without Fourier Series and Transform, communication engineering would not exist (no concept of bandwidth), No IT age. Where would mankind be without Sin & Cos than?

[seggyRK]

The two ends of the bar are at T1 and T2 and constant for all time

[changbroot]

At 9 min, you say A*pi = 0, the then A = 0, so why don't you use the same way to say B*sin(....) = 0 (at 11 min), then B = 0? No matter what happens to sine, but B*sin(...) = 0 should yield B = 0/sin(...) = 0. Am I doing something wrong here?

[MatrixOfDynamism]

there are so many videos. I don't know where to look at. What I know is that a fundamental concept in communications is that of bandwidth. There cannot be a concept of bandwidth without fourier series. So I wonder what about all the mobile phones and internet, would they exist now?

[DrChrisTisdell]

Sin and Cos are one of the most important aspects of mathematics and its applications. You may be surprised to learn that some mathematicians are questioning the validity of these concepts. Check out my colleague's channel njwildberger for more!