Re: Trying to define: Fractional Derivatives & Leibniz' display form for output and templates
- To: mathgroup at smc.vnet.net
- Subject: [mg22811] Re: Trying to define: Fractional Derivatives & Leibniz' display form for output and templates
- From: "Mike Honeychurch" <m.honeychurch at sci.monash.edu.au>
- Date: Fri, 31 Mar 2000 01:01:14 -0500 (EST)
- Organization: Monash Uni
- References: <8bhvta$noq@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Here are a couple of methods that work on data sets (two column)...you can
modify them to make them functions if you want to call them regularly.
For a 0.5 integral this works (but is procedural and slow)
Table[data[[k,i]],
(0.5/Sqrt[Pi])*Sum[((data[[i-1,2]]+data[[i,2]])/Sqrt[k-i+0.5]), {i=2,k}],{k,
2, Length[data]}]
It is straightforward to convert this to an integral between 0 and 1 if you
have the definition of the fractional derivative-integral handy.
Here is a functional method which is very quick but unfortunately is for 0.5
integral only (one column evenly spaced data):
y1=Table[1/Sqrt[Length[data]-i+0.5, {i,Length[data], 2, -1}]
y2=ListCorrelate[{1,1},data];
ListConvolve[y1,y2,{1,1},0]
To perform fractional derivatives or integrals (same thing) on functions it
is should also be straight forward. I'm not a mathematician so in what
follows the terminology might be a tad dodgy.
I don't have the texts in front of me but from memory the fractional
calculus is pretty much just a type of convolution integral with a 1/gamma
function out the front (I'm a chemist but you know what I mean!). It should
be possible therefore to define something like:
f[x_, y_,n_]:=1/Gamma[...]*Integrate[x[t-z]^n*y[z], {a,b}]
My apologies if my memory of what occurs within the integral is rusty but I
hope this will point you in the right direction.
I am actually going to be sitting down (in the next month or so) to write
something to solve some diffusion problems so if you email me I'll forward
what I have when I have it.
cheers
Mike Honeychurch
m.honeychurch at sci.monash.edu.au
Kai G. Gauer <gauer at sk.sympatico.ca> wrote in message
news:8bhvta$noq at smc.vnet.net...
> Hi
>
> (Please note that D[f, x, .5] does not work well at all!)
>
> I am trying to define what is called a fractional derivative form of
> a derivative that Mathematica can use when called upon. These fractional
> derivatives can sometimes be useful in solving differential equations
> (but I'll leave you to read up further on the subject, if interested).
> they take on the form:
> k
> d (f[x_]) |
> ---------- |
> k |
> (dg[x_])^ | x=c, c a constant
>
> (my bad attempt at imitating texform output...apologies)
> Note that c is required to be known if we also wish to define the
> operator for negative values of k (corresponds to integrating n times
> and getting F[y_,n_]=nth Integral[f[x] dx, integration limits being c
> and y] ). You can also assume that I won't normally be looking at "ugly"
> functions such as the Dirichlet example (under normal circumstances).
>
> How, for instance, can I create an operator for Mathematica which will
> accept something like:
> D[f[x], x, c, n_] (to have options of staying evaluated and/or
> unevaluated and display out to the user in a Leibniz-like notation in
> the same way that we get a nice vector/MatrixLike notation using
> MatrixForm... I don't want the d <subscript..of x> (f[x]) form to
> appear, rather, I'd prefer my style notation as above...inputform should
> be inputted from a clickable palette in such a way that I only have to
> TAB thru my options.. sort of like the current Integrate button).
>
> A particular function, f, in question, might be something such as Sin[pi
> x]. I wish to take the following: in any order, D^1/2 [f1], D^1/3 [f2],
> and D^1/6 [f3]. The idea is that since 1/6+1/2+1/3=1/1, we should get pi
> Cos[pi x], no how we order our composing of f1, f2 & f3. For instance,
> D^m[D^n[f]] = D^(m+n)[f]. When doing so, we will also get the factorial
> function showing up, so we should maybe name it gamma instead... (given
> that we only have integers and Euclidean divisions (in an angular sense
> (multiples of a half), we can approximate many non-rational numbers by
> simply observing that the representation of 1/3= 1-1/2+1/4-1/8+-+- (keep
> only absolute converging sums; they tend to go to exactly one number,
> even after rearrangement of terms) as strictly powers of a half => other
> infinite series representations of some irrational numbers could also be
> easily approximated by this method (simply union1/3's approximation into
> your selection of numbers).... I mention this only because the gamma
> duplication formula seems to prefer to give nicer constants if we have
> powers of a half). Also, some of the preferred choices of the constant c
> are -oo ,oo, -1, 0,1...but you could also pick something like pi^3 (this
> is only done to keep the integral from being definite until after we
> plug in a y-value).
>
> I don't really care too much about convergence issues...as long as the
> can be approximated if I'd try to ever Plot[D[x,n],{n,0,1}]. Speed isn't
> really the issue; I'd just like to be able create and easily work with
> extra symbols and (if necessary) some not-so high precision numbers....
> more interested in how such a plot would behave, not issues such as
> spitting out the infinite series expansion of D^1/3[sin[x],x] to order
> of fifty or 500 or more....
>
> Please also note that I do not wish to limit my definition of fractional
> derivative to only the sine fcn (I'd like to try various combinations of
> Legendre and Bessel polynomials, exponential series, and double check
> that forms such as the constant functions are also differentiating
> properly...constants should always go to zero, I would think that
> D^(m+1) [E^x (+c)] = D^m [E^x], for all m in Z (or Q or R or
> Complex...not certain whether this is true), similarily D^(m+2) [+/-
> E^-x (+/- E^+x +c)] = D^m [+/- E^-x (+/- E^+x +c)], ie .... e^x has
> smallest differential period of 1, +/- e^-1 has smallest differential
> period of 2, the sine & cosines functions have a smallest differential
> period of 4, what other functions have other differential periods? This
> is all assuming that I've been reading up on some of this stuff
> properly.....)
>
> I would like to have the option of being able to do a sort of
> step-by-step evaluation (so please don't rely on numerical approximation
> methods only) with the ability to easily simplify out constants of root
> pi, and other combinatorial constants of favourite to mathematicians.
> I'm mainly looking for symbolic flexibility, not speed, and the ability
> for the operator to imitate the classical derivative notation, but only
> numerically AFTER I've done my symbolic evaluation.
>
> Thanks for any help that somebody may be able to offer..... (the
> interested reader could probably find enough definitions of what he
> needs in the CRC engineering tables, available in most libraries.... I
> also have 1 or 2 references)
>
>