From: Daren Scot Wilson Subject: Re: Integrating one-and-a-half times Date: Thu, 25 Feb 1999 00:03:29 +0000 Newsgroups: sci.math Keywords: Fractional Order Derivatives Oh oh oh!!! One of my **favorite** topics!!! This is the topic of Fractional Order Derivatives (or Integrals), aka Fractional Calculus, and was studied by a bunch of mathematicians in early 1900s and late 1800s, including Riemann and a whole bunch of big names my brain's too dusty and rusty to recall. It has been useful in electrochemistry, at least. It relates to diffusion. It comes up in electromagnetic boundary value problems. Perfect for describing distributed-element RC filters in electronics. Go ask an audio engineer about "pink noise", or a physicist about "1/f noise" - these are random signals that are half-integrals of white noise. Finally, fractional calculus is just plain fun to play with. RAMBLING ABOUT THE GOOD OL' DAYS... Way back in high school, even, I thought of a way to do "half-derivatives". I started with sines and cosines, though. Using "D" as the symbol for a derivative operator w.r.t. x, and 'a' some constant, and define H=pi/2 (just to save my fingers a little strain), D^0 sin(ax) = sin(x) D^1 sin(ax) = a sin(x+ H) D^2 sin(ax) = a^2 sin(x + 2*H) ...etc... it made sense to define, for any real u, or what the heck, any complex u, D^u sin(ax) = a^u sin(x+uH) This can be applied to any function that we have a fourier series or fourier transform for. Then I considered what you are contemplating, with a slight variation. Define P_n(x) as x^n / n! It is not necessary that n be an integer, and you by now already know about gamma functions, which comes up very often in physics, so P_n for any n is defined. As you can easily figure for yourself, D P_n(x) = P_n-1(x) (Dang, plain ascii is crude!) D^2 P_n(x) = P_n-2(x) ...etc.... so we can choose to define D^u P_n(x) = P_n-1(x). Note that n! = infinity for any negative integer n, thus P_n(x) = 0 for any negative integer n; eventually "too many" derivatives will lead to zero. Now any function that can be represented by a power series is taken care of. (Just gotta use gamma functions). Question: are the two techniques related? Yes, they are the same, if you don't mind a hand-waving mathemagical argument. I wrote exp(x) as a power series, exp(x) = SUM(n=0 to inf) of P_n(x) and realized that its fractional derivative D^u exp(x) of any order u produces 'spurious' term with n negative. These vanish when u is an integer, leaving an ordinary exp(x) function. But then I had a keen insight: define exp(x) as exp(x) = SUM(n=-inf to inf) of P_n(x) and you have a D^u of exp(x) as a strange power series for noninteger u, what is it? D^u exp(x) = SUM( n=-inf to inf) P_n-u(x) But just differientiate this result once, and it's the same, so it must really be exp(x), the one and only solution to Df(x) = f(x). Now it's easy to generalize to exp(ax), and to let a be imaginary, and relate this to sin(ax) or cos(ax), and rediscover my "high school formula". Yeah, great fun, this is... Eventually I discovered all this was already done early in the 20th century by the great minds of math. So I turned my attention to fractionally iterating the Fourier Transform instead :-) NOTE: The gamma function is not a unique choice - there are infinitely many ways to define a function that equals n! for integer n. The gamma is unique in being, in a sense, the "smoothest" such function. Here, smoothness can be defined as a property of all derivatives (just integer order one) not changing signs for x>0 (or something like that, look it up yourself). Even sticking with the gamma, it turns out that fractional order derivatives have a kind of looseness - the operator used in the electrochemistry paper, and seen elsewhere, is not the one you and I are playing with. I'm not sure if this looseness has any physical significance, but the other form comes about more naturally in certain integral representations of the D^u operator. REFERENCES: (1) Best intro is a book by Oldham and Spanier. When I find my old Frac. Calc. notes, I can tell you more... I think the title was "The fractional Calculus", and I remember it had the style of Academic Press. Was hard to find in even university libraries, but I borrowed it from a math prof. Happy hunting! "Oldham" is a good name to do literature searches on. (2) Mixed Boundary Value Problems in Potential Theory, by Ian N. Sneddon, North Holland Publ. 1966. See section 2.4. (3) "The Fractional Diffusion Equation" , W. Wyss, J. Math. Phys vol 27, 1986 p. 2782 Info on gamma function is everywhere - try books on mathematical physics if you don't want to choke on sterile mathematical abstractions. -- Daren Scot Wilson darenw@pipeline.com www.newcolor.com ---- "A ship in a harbor is safe, but that is not what ships are built for" -- William Shedd ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Fractional Derivative and the Gamma Function Date: 2 May 1999 15:46:57 -0400 Newsgroups: sci.math In article <372c9c20.1039958@news.isat.com>, Cris A. Fitch wrote: :I was discussing Calculus with a friend of mine who is just now :learning it. He wanted to know what partial derivatives were, but :when I explained them, he seemed puzzled. It seems he thought they :would be something which I am calling Fractional Derivatives, for the :lack of a better term. : :The idea of a fractional derivative is that much like one can have a :second derivative d2/dx2, or an nth derivative dn/dxn, you might also :have a derivative which is not a whole number: dz/dxz. : :Consider z=1/2, and a basic polynomial x^n. Using the Gamma function :as our continuous equivalent of a factorial, we have: : d(1/2)/dx(1/2) of x^n = mu * x^1/2, : where mu = gamma(n+1)/gamma(n+1/2). : :More generally, we have for x^n: : dz/dxz of x^n = mu * x^(n-z) where mu = gamma(n+1)/gamma(n-z+1). : :My question is, does this stuff seem familiar to anybody out there? :Seems like it would make a good PhD thesis for somebody if it hasn't :already been done. Many times and in many books. I would suspect that the subject is as old as the knowledge of the gamma function itself. In fact, it is more manageable if you consider the fractional indefinite integral from 0. (The background is that conceptually functions improve their smoothness by integration but deteriorate by differentiation. This can be made more precise in terms of spaces of functions.) For a complex number z whose real part is positive, and for a continuous (can be even more general) function f defined on an interval [0, A] or [0, A) (here A can be infinity) one defines the z-th indefinite integral of f as J^z f(x) = (1/gamma(z)) * integral [0 to x] (x-t)^(z-1) * f(t) dt From the known formulas for gamma and beta functions, you can prove J^u J^v f(x) = J^(u+v) f(x) (a "semigroup property" we expect: for example, integrating f from 0 twice in a row results in applying J^2). Also, if Re(z) > 0 then (d/dx) J^(z+1) f(x) = J^z f(x) , using Leibniz's rule for differentiation of integrals depending on a parameter. It is also provable with some knowledge of limit processes for integrals that (I think for x>0, to be on the safe side) lim [z real, z ->0 ] J^z f(x) = f(x) so that J^(0) can be formally defined as the identity transformation, and J as a function of z will be continuous at 0 from the right. So, for z = sqrt(2), and for f twice differentiable and satisfying f(0)=0, f'(0+)=0 and f''(0+)=0, (so that z < 2) we can try the derivative of order z of f as D^z f(x) = J^(2-z) f''(x) Much more refined results are known; a fascinating part is the "boundary group": the limiting values of J^z as Re(z) -> 0+. I just do not have the corresponding books at hand (it's weekend). Cheers, ZVK(Slavek). ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Curiosities Date: 16 Jun 1999 15:54:06 -0400 Newsgroups: sci.math In article <7gp5dxgk9yif@forum.swarthmore.edu>, Plofap wrote: :Hello, : :I have been wondering about two curiosities " :1. The first is about sets that are defined like :the mandelbroth sets. The inclusion criterion for a point being :sort of |Fn(X)| < value for all n where n is the n:th iteration could :be for some functions changed to |d(Fn(x))/dx| < value. For the usual :mandelbroth set the result looked like it was also fractal. :I was wondering if there are any interesting connections :between the original and this way derived sets ? Looks more like approximations of Julia sets, rather than the Mandelbrot (observe the spelling) set. The latter is actually one level of abstraction higher than Julia sets. The definition should be available on many websites. And the derivative condition leading to a fractal seems to be no surprise: after all, the derivative of the n-fold iterate of a function is the product of n derivatives of the lower iterates at appropriate points. >2. Can you differentiate without the order being an integer ? Sort >of fd(x) = INVLAPLACE(LAPLACE(f(x))*s^d) ? Is there any sense in >or apps of this ? > Fractional integration from 0 to the right is easier to handle (integrals are better behaved than derivatives, when it comes to limits and such). In fact, you gave a definition of such a procedure (when d is negative). If you peel off the Laplace transformations, you end up with the d-th power of the operator J of integration. J^d f(x) = (1/Gamma(d)) * integral[0 to x] (x-t)^(d-1) * f(t) dt Here d can be complex, and the real part of d is supposed to be positive. The operator family has the properties we like to see: J^(c+d) f(x) = J^c (J^d f(x)) (d/dx) J^(d+1) f(x) = J^d f(x) and for a carefully defined limit, J^d converges to identity when d tends to 0 from the right. The fractional derivative will work for sufficiently smooth functions: for example, since pi<4, we may define (d/dx)^pi f for functions 4 times differentiable by (d/dx)^4 J^(4-pi) f. The literature on fractional derivatives is extensive, and among applications, the technique helps solve certain convolution type integral equations found in mathematical physics. (Mathematicians of two or so centuries ago were curious enough to ask such questions!) Cheers, ZVK(Slavek). ============================================================================== From: smudge@world.std.com (smudge) Subject: Re: derivatives Date: Fri, 23 Jul 1999 12:18:11 GMT Newsgroups: sci.math.symbolic JJ. Eltgen (jjeltgen@imaginet.fr) wrote: : I am looking for any information about "non integer order derivatives". : Does anyone know something about this topic? : Thanks for your help. Check out this book Author: Podlubny, Igor Title: Fractional differntial equations: and introduction to fractional derivatives, fractional differntial equations, to methods of their solution and some of their applications. Pub: Academic Press, San Diego, 1999 It looks like a good place to start. --smudge ============================================================================== From: fateman@peoplesparc.cs.berkeley.edu (Richard J. Fateman) Subject: Re: derivatives Date: 23 Jul 1999 15:39:33 GMT Newsgroups: sci.math.symbolic In article <7n95er$olj$1@news.imaginet.fr>, JJ. Eltgen wrote: >I am looking for any information about "non integer order derivatives". >Does anyone know something about this topic? >Thanks for your help. > >JJ. Eltgen There was a survey article on this topic in SIAM Review 1976 vol 18 no 2 "Fractional Derivatives and Special Function" by J. L. Lavoie. You should be able to find much of the literature if you search on "fractional derivatives" in a good database. -- Richard J. Fateman fateman@cs.berkeley.edu http://http.cs.berkeley.edu/~fateman/