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# Problem with PDE part Watch

1. I have solved a heat flow PDE in part (a)
part (b) I am not happy with my solution

I kinda fudge and take the initial temperature in the midpoint as the average, otherwise I am out by a factor of 2 for the summation deduction.

Am I fudging or is this valid?

or worse still have I missed out a factor of 2 in the solution of the PDE
(Note that the answer at the end of the question is my answer, so it might not be correct)

many thanks

PDF.pdf
2. (Original post by TeeEm)
I have solved a heat flow PDE in part (a)
part (b) I am not happy with my solution

I kinda fudge and take the initial temperature in the midpoint as the average, otherwise I am out by a factor of 2 for the summation deduction.

Am I fudging or is this valid?

or worse still have I missed out a factor of 2 in the solution of the PDE
(Note that the answer at the end of the question is my answer, so it might not be correct)

many thanks

PDF.pdf
It's valid. If a function is discontinuous somewhere, then its Fourier series assumes the value which is the average of the limits approached from each side.
3. (Original post by Smaug123)
It's valid. If a function is discontinuous somewhere, then its Fourier series assumes the value which is the average of the limits approached from each side.
This (PRSOM).

(Original post by TeeEm)
..
It's a bit of nasty hack, especially when it comes up in a question, but at the same time, if you are taking advantage of symmetry when setting up a question with discontinuities, it's pretty common that the natural "summation point" (i.e. the value(s) of x that mean you can eliminate the sin/cos terms from the Fourier series) fall on the discontinuities, so it happens enough that you should be aware of it.
4. (Original post by Smaug123)
It's valid. If a function is discontinuous somewhere, then its Fourier series assumes the value which is the average of the limits approached from each side.

(Original post by DFranklin)
This (PRSOM).

It's a bit of nasty hack, especially when it comes up in a question, but at the same time, if you are taking advantage of symmetry when setting up a question with discontinuities, it's pretty common that the natural "summation point" (i.e. the value(s) of x that mean you can eliminate the sin/cos terms from the Fourier series) fall on the discontinuities, so it happens enough that you should be aware of it.
thanks for the response.

I thought of that after I spending ages looking for that factor of 2 at the actual solution, so at the end I vaguely remembered something about the average value which I put down but I wanted to make sure before I add the question to my resources.

Thanks again

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