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Thread: Quick question about fractional derivatives and is Latex broken?

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    Quantum Hot Dog Kharakov's Avatar
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    Quick question about fractional derivatives and is Latex broken?

    Assume generalized binomial coefficients. Also assume this is not for normal fractional (iterative) derivatives, but for the following operation. I think I should implement it in python or something... just wonder if there is an easy way to do it on a phone (like how do I code it for Wolfram Alpha???) since my computer time is limited.

    Latex broken? I specifically prohibit Bilby from replying to this part of the question if no forum user is wearing latex.

    f^n(x) = \lim_{h \to 0} \frac{ \sum\limits_{k=0}^\infty \binom{\alpha}{k} (-1)^k  f(x+ (n-k) h ) } {h^n}

    So, limits as h-->0:
    f0 (x) = [f(x) ]/h^0
    f1 (x) = [f(x+1h) - f(x) ]/h^1
    f1.5 (x) = [f(x+1.5h) - 1.5 f(x+.5h) + 3/8 f(x-.5h) +1/16 f(x-1.5h) +3/128 f(x-2.5h).... ]/h^1.5
    f2 (x) = [f(x+2h) -2 f(x+h) +f(x) ]/h^2
    f3 (x) = [f(x+3h) -3 f(x+2h) +3 f(x+h) -f(x) ]/h^3
    Last edited by Kharakov; 02-16-2020 at 01:40 AM.

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    Elder Contributor barbos's Avatar
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    I don't think your formula work at all.

    https://en.wikipedia.org/wiki/Fractional_calculus
    Unlike integer, fractional derivative in general is not even local.

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    I am not familiar with the term fractional derivative, can you explain?

    There are several free tools. Euler is a good one. They all have scripted languages to write code. They are all straightforward.

    You can do it in a spreadsheet with Basic. Open Office is free and has basic with the spreadsheet.

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    Elder Contributor barbos's Avatar
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    Quote Originally Posted by steve_bank View Post
    I am not familiar with the term fractional derivative, can you explain?
    Why ask when wikipedia exists?
    https://en.wikipedia.org/wiki/Fractional_calculus

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    One other thing. Get a programmable TI or HP calculator. They have always had a serial, link to a PC. They may connect to a wireless device. Edit and download programs. Even without that you can program iterative solutions. Used them often before PCs.

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    Quantum Hot Dog Kharakov's Avatar
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    Quote Originally Posted by barbos View Post
    I don't think your formula work at all.
    Apparently it does. <shrug>

    https://www.mathpages.com/home/kmath616/kmath616.htm

    Quote Originally Posted by barbos View Post
    https://en.wikipedia.org/wiki/Fractional_calculus
    Unlike integer, fractional derivative in general is not even local.
    As far as I can tell, from what I just read (and it makes perfect sense to me), no derivative is local, because the derivative of any constant is 0.

    Thanks barbos.

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    Quantum Hot Dog Kharakov's Avatar
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    Quote Originally Posted by steve_bank View Post
    I am not familiar with the term fractional derivative, can you explain?
    Try when I get back. Limited computer time.

    lol. I still see your name as steve_bnk, without the a. I think of you as that too (I read "Steve Be En Kay"...).

    I really like this article: https://www.mathpages.com/home/kmath616/kmath616.htm

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    Elder Contributor barbos's Avatar
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    Quote Originally Posted by Kharakov View Post
    Quote Originally Posted by barbos View Post
    I don't think your formula work at all.
    Apparently it does. <shrug>

    https://www.mathpages.com/home/kmath616/kmath616.htm
    OK, I stand corrected. But I need to read it more. Easiest/most natural way to calculate or think of fractional derivative is through Fourier transformation.


    Quote Originally Posted by barbos View Post
    https://en.wikipedia.org/wiki/Fractional_calculus
    Unlike integer, fractional derivative in general is not even local.
    As far as I can tell, from what I just read (and it makes perfect sense to me), no derivative is local, because the derivative of any constant is 0.

    Thanks barbos.
    Ordinary non-fractional derivatives are local: depend only of function behavior near argument. Fractional derivatives involves integration, so they depend on the boundaries and therefore not local.

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    Quantum Hot Dog Kharakov's Avatar
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    So, I'm just basing this off of what I read in that article, which makes sense to me.

    No derivative is local. The derivative of x^2 + 1000000000 = the derivative of x^2 - 159-8135907125309710298098510970912537097 = the derivative of x^2.

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    Elder Contributor barbos's Avatar
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    Quote Originally Posted by Kharakov View Post
    So, I'm just basing this off of what I read in that article, which makes sense to me.

    No derivative is local. The derivative of x^2 + 1000000000 = the derivative of x^2 - 159-8135907125309710298098510970912537097 = the derivative of x^2.
    .
    That has nothing to do with locality

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