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frolloos
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fractional integration by parts

June 28th, 2018, 2:19 am

Anyone know how to do this, specifically for the following integral:

[$] \int_0^\infty \left( \frac{d^p}{dx^p} f(x) \right) \delta(x-k) dx [$]

where [$] 0< p < 1 [$]. 

I'm not sure how (fractional) integration by parts works and what the fractional derivative of the dirac delta function would be. Any thoughts and/or pointers to introductory papers on this appreciated. Thanks.
 
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ppauper
Posts: 11729
Joined: November 15th, 2001, 1:29 pm

Re: fractional integration by parts

June 28th, 2018, 4:26 am

[$]I=\int_{0}^{\infty}\frac{d^{p}f}{dx^{p}}\delta(x-k)dz[$]
if you just want to evaluate that, the delta function will be zero apart from at [$]x=k[$], so
[$]I=\frac{d^{p}f}{dx^{p}}\mid_{x=k}\int_{0}^{\infty}\delta(x-k)dx[$]
and [$]\int_{0}^{\infty}\delta(x-k)dx=\left\{\matrix{0&k<0 \cr 1/2& k=0 \cr 1 & k>0}\right.[$]

(edit: typo (dz vs dx) fixed)
Last edited by ppauper on June 28th, 2018, 7:53 am, edited 2 times in total.
 
frolloos
Topic Author
Posts: 752
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

Re: fractional integration by parts

June 28th, 2018, 5:46 am

Thanks, but you mean dx instead of dz right? Also, I want to get rid of the derivative of f, hence would like to do integration by parts. Which is complicated by the fact that the derivative working on f is a fractional derivative.
 
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ppauper
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Joined: November 15th, 2001, 1:29 pm

Re: fractional integration by parts

June 28th, 2018, 6:36 am

Thanks, but you mean dx instead of dz right? Also, I want to get rid of the derivative of f, hence would like to do integration by parts. Which is complicated by the fact that the derivative working on f is a fractional derivative.
obviously dx, yes.
if you want the fractional derivative of the delta function, google tells me that the consensus out there seems to be to use Fourier transforms.