On Tuesday, 14 September 2004 at 09:30:28 UTC+10, Roger L. Bagula wrote:

Result of today's experimentation:
ds/dt=-s+1+1/Gamma[2-s]
converges to 1.53033.
Some experimentation in Mathematica on this and the map type gives: s[n+1]=1+1/Gamma[2-s[n]/s0]
where s0 seems to act as an integer scale
like:
s0=1/l; l=1,2,3,4,....
The result with the right starting point of
s[0]=s0+delta
for smaller delta,
gives a complex and sometimes converging chaos.
This result appears to indicate that some scaling factor exists between
the fractional differential value and the fractal dimensional roughness
for the special case I have used. It is far from a "proof",
but it is closer than before to an understandable answer.
My experience with the partial angular fractional derivative done in IFS
is that the resulting set has a lower fractal dimension for between
angles ( for example a space filling 4 part Sierpinski when a
PAD is used on it gets "hole" and doesn't completely cover a space fill anymore).
It shouldn't be unexpected that the actual fractal roughness factor is scaled by the fractional differential.
It does make questions:
1) Is it only true in this special case?
2) Is the actual scale some constant of nature? ( like E or Pi )
3) Can some practical use be made of this relationship of calculus
and fractals?
Roger L. Bagula wrote:

Strange result for the equation:
s[n]=1+1/Log[Gamma[2-s[n-1]]]:
It goes complex and is initial point dependent, but
the attractors are clear:
{1, Infinity,-Infinity}
That's one from below and one from above to give a four term sequence.

To correct a mistake:
L(s)=Sum[ Log[Abs[ d^sx[n]/dt^s]],{n,1,m}]/m
where
d^sx[n]/dt^s
is the Abel-Louville fractional derivative as real number 0<s<1.
The recognized polynomial formula is:
d^sx^n/dx^s=Gamma[n+1]* x^(n-s)/Gamma[n-s+1]
Which formula give for n=1 and Gamma[2]=1:
d^sx[n]/dt^s=x(1-s)/Gamma[2-s]
where
Gamma[n]=(n-1)!

The paper that inspired this rumination is:
Fractional differentiability of nowhere differentiatiable functions and dimensions
by Kiran M. Kolwankar and Anil D. Gangal, chaos and dynamics 21 nov 1996 which has a marvelous bibliography of people who have addressed the problem of fractional differentiation and it's relationship to dimension starting with Mandelbrot and van Ness and including Dr. M. Zähle
and Dr. Falconer.

Roger Bagula wrote:

I have an idea of how I can connect a Lyapunov exponent
and Kaplan -Yorke dimension to the fractional derivative
using the Hölder exponent type of idea in a paper I downloaded.
The idea is that the "roughness" region 0<s<1 in any fractal
is the part that is affectred by the fractional derivative and
not the topological dimension and behaves like a Hölder exponent.
I all ready have the roughness expoment based on the 2nd derivative
Bezier limit :
r=Sum[ Log[1+Abs[ d^2x[n]/dt2]/4],{n,1,m}]/m
which seems to be very much like the Lyapunov exponent in detecting
fractal dimensions
in my experiements. My idea is to invent a fractional Lyapunov exponent: >> L(s)=Sum[ Log[Abs[ d^sx[n]/dt^s]],{n,1,m}]/m
where: Gamma[2]=1
d^sx[n]/dt^s=Gamma[2]*x[n]^(1-s)/Gamma[2-s]
Or
L(s)=Sum[ Log[Abs[x[n]^(1-s)/Gamma[2-s]],{n,1,m}]/m
If the average of x[n] is one then or Log[1]=0:
L[s]=Log[Gamma[2-s]]
which is always negative since:
0<=Gamma[2-s]<=1 on 0<=s<=1
which gives a Kaplan -Yorke dimension of: (d0 the topological dimension) >> if the other exponents are one
dky(s)=d0+1/Log[Gamma[2-s]]
Which gives a roughness gap in the range:
s'=1+1/Log[Gamma[2-s]]
That would give the connection of the fractional derivative to
the dimension.
There are a lot of "if"'s involved in this "not a proof" derivation, but >> it does show that a connection may exist.
A lot of people have tried to make this kind of connection in a
provable way
and failed so far as I know.



Respectfully, Roger L. Bagula

tf...@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
619-5610814 :
URL : http://home.earthlink.net/~tftn
URL : http://victorian.fortunecity.com/carmelita/435/

--
Respectfully, Roger L. Bagula
tf...@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL : http://home.earthlink.net/~tftn
URL : http://victorian.fortunecity.com/carmelita/435/

yes, how to connect?
mathHand.com solves fractional partial differential equation by clicking the dsolve and plot by clicking plot3D and test its solution by clicking the test.

http://server.drhuang.com/input/?guess=dsolve%28ds%28y%2Ct%2C0.5%29%2B2ds%28y%2Cx%2C2%29%2B4ds%28y%2Cx%2C1%29%2Bexp%28x%29*t%29&inp=ds%28y%2Ct%2C0.5%29%2B2ds%28y%2Cx%2C2%29%2B4ds%28y%2Cx%2C1%29%2Bexp%28x%29*t&lang=null

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