Trying to define: Fractional Derivatives & Leibniz' display form for output and templates
- To: mathgroup at smc.vnet.net
- Subject: [mg22795] Trying to define: Fractional Derivatives & Leibniz' display form for output and templates
- From: "Kai G. Gauer" <gauer at sk.sympatico.ca>
- Date: Sat, 25 Mar 2000 03:58:34 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
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)