From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Fractional Derrivatives Date: 17 Jan 1998 00:00:00 GMT Newsgroups: sci.math In article <7l7m80k278.fsf@torstai.hit.fi>, Pertti Lounesto wrote: >> Do fractional (or even non-rational) derivatives have any >> practical applications? > >No. What's a "practical application"? There are 165 math papers containing "fractional derivative" in their titles, classified under many fields of analysis, as well as under the fields of mechanics. There are a dozen or so books on fractional calculus (Nishimoto has a 5-volume work!), and even a Journal of Fractional Calculus. The topic has interested some well-known mathematicians at least as far back as Riemann. Are fractional derivatives pivotal in the sense of, say, the Heat Equation or Lie Groups? No. Do I know much about them? No. Are they without practical applications? That certainly seems a draconian statement. dave ============================================================================== From: Put.Your.World.e-mail@address.Here(CHANGE.IT!) (Put.Your.First.and.Last.Names.Here.(CHANGE.IT!)) Subject: Re: Fractional Derrivatives Date: 18 Jan 1998 00:00:00 GMT Newsgroups: sci.math gsimp95605@aol.com (GSimp95605) wrote: > Is there a good introductory text? i attended a conference on fractional calculus in the 70's , when i was an undergrad. one of the main presenters was a guy named t.j. osler, if i remember correctly, and i think he has written a book on the subject. as for applications, i saw at the conference a nifty diff-eq solution technique that used fractional derivatives. the presenters described other applications, too, but d.e. is the only one that my memory has kept for these 25 years. (sorry, i don't remember how it worked.) finally, you can get a sense of how to start trying to calculate a fractional derivative, by considering the fourier transform identity: n d f n F( ------- ) = s F( f) n d t it turns out that with suitable constraints on f, this identity holds even when n is not integral. - don davis, boston ============================================================================== From: spalland@comune.re.it (Andrea Spallanzani) Subject: Re: Fractional Derrivatives Date: 29 Jan 1998 00:00:00 GMT Newsgroups: sci.math On Tue, 27 Jan 1998 12:50:38 +0800, Paul Abbott wrote: >What is wrong with the following naive definition? The p-th fractional >derivative results from the replacement > > x^n -> (n! x^(n -p))/ Gamma[n - p + 1] > >This clearly works for polynomials. The problem is that the following 3 properties are inconsistent togheter (D(p,f) = p-th fractional derivative of f(x) with respect to the variable x): D(p,0) = 0 D(n,f) = d^n f(x) / dx^n for any natural numer n D(p,D(q,f)) = D(p+q,f) In fact, (applying the definition) D(5/2,x) = x^{-3/2}/Gamma(-1/2) but also (applying D(p,D(q,f)) =D(p+q,f)) D(5/2,x)=D(1/2,D(2,x))=D(1/2,0)=0. Regards ============================================================================== From: Paul Abbott Subject: Re: Fractional Derrivatives Date: 30 Jan 1998 00:00:00 GMT To: Andrea Spallanzani Newsgroups: sci.math Andrea Spallanzani wrote: > The problem is that the following 3 properties are inconsistent > togheter (D(p,f) = p-th fractional derivative of f(x) with respect to > the variable x): > > D(p,0) = 0 > D(n,f) = d^n f(x) / dx^n for any natural numer n > D(p,D(q,f)) = D(p+q,f) I agree that these 3 properties are inconsistent. However, why is the first property sacred? The last 2 are the properties I would keep and, for consistency, define D(p,0) -> x^(-p-1)/Gamma[-p] which vanishes for any natural number p, along with D(p,x^n) -> (n! x^(n -p))/ Gamma[n - p + 1] > In fact, (applying the definition) > D(5/2,x) = x^{-3/2}/Gamma(-1/2) > but also (applying D(p,D(q,f)) =D(p+q,f)) > D(5/2,x)=D(1/2,D(2,x))=D(1/2,0)=0. With the above change, you always get D(5/2,x) -> x^{-3/2}/Gamma(-1/2) Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul@physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/~paul God IS a weakly left-handed dice player ____________________________________________________________________ ============================================================================== From: jmccarty@sun1307.spd.dsccc.com (Mike McCarty) Subject: Re: Fractional Derrivatives Date: 31 Jan 1998 00:00:00 GMT Newsgroups: sci.math At one time I read a book on non-integral order derivatives. This fellow defined derivatives for exponential functions as follows: D(p,f) = (lambda^p)f for f(x) = K exp(lambda x) This definition satisfies the three below (with K=0 for f=0). Then he explored the theory for functions which may be defined as infinite series of such exponential functions g(x) = sum(i,0,oo,K[i] exp(lambda[i] x)) It turns out that quite a few functions can be so represented, especially if we allow the K and lambda (and x) to be complex. Negative orders of derivatives turn out to be integrals, just as expected. Unfortunately, I read that book when I was in graduate school in, umm, about 1977 or 1978. So I have forgotten the title. But it was an obvious one. The book was hardback, and maybe 200-300 pages long, so it was not trivial. [quote of previous article deleted -- djr] -- ---- char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);} This message made from 100% recycled bits. I don't speak for DSC. <- They make me say that. ============================================================================== From: spalland@comune.re.it (Andrea Spallanzani) Subject: Re: Fractional Derrivatives Date: 06 Feb 1998 00:00:00 GMT Newsgroups: sci.math At 13.12 30/01/98 +0800, you wrote: >> D(p,0) = 0 >> D(n,f) = d^n f(x) / dx^n for any natural numer n >> D(p,D(q,f)) = D(p+q,f) >I agree that these 3 properties are inconsistent. However, why is the >first property sacred? I think the first one is sacred, holy and untouchable:) If you don't accept it, you'll lose linearity: actually.... > D(p,0) -> x^(-p-1)/Gamma[-p] >which vanishes for any natural number p, along with ...but D(p,0) doesn't vanish for any general fractionary value of p: D(1/2,0) = x^(-3/2)/Gamma(-1/2) \neq 0, so we lose linearity: D(1/2,f) = D(1/2,f+0) \neq D(1/2,f) + D(1/2,0). I think that non-linearity D(p,f+g) \neq D(p,f)+D(p,g) is unacceptable. And you? At a first sight, I'd prefer to lose the 3^ property, or , better, to modify it: D(p,D(q,f)) = D(p+q,f) for any q, f such that D(q,f) \neq 0 but I haven't yet considered the consequences of such restrain. Cheers