Fractional Modelling and some Applications
by
Siegmar Kempfle
Helmut Schmidt Universität, Universität der Bundeswehr Hamburg, Germany
Coauthors: Ingo Schäfer

We start from an amazing observation: whereas the classical equation of a single degree of freedom oscillator models (e.g.) viscoelastic media rather bad, the change of the time derivative in the damping term into a non integer one, i.e.,

( m D2+a Dq+b) x(t)=f(t) ,    0 < q < 1 ,   m,  a,  , b > 0

delivers very good results. The list of similar effects in the use of fractional derivatives grew up from experiment to experiment. This was motivation to take care about adequate definitions of fractional (pseudo) differential operators with symbols of shape

A(s)= n å k=0  ak sqk  ,           qk Î R+

The approach via Fourier transforms (so-called functional calculus) and its extension in distributional spaces turns out to be appropiate to our class of problems, mainly in view to such important charactistics as e.g. frequency, impulse and step responses.

In our talk we sketch the approach together with its mathematical features and its simple handling. Thereby, the definition of all fractional powers as principal branches (in accordance with computer sytems like Mathlab, Mathematica, Maple) is the key to physical consistency.

Last not least, we illustrate and justify the mathematical "modus operandi" by comparision with experiments.

Keywords:  Fractional Calculus, Functional Calculus, Viscoelastics, Inductivity

AMS Classifications:  26 A 33, 47 A 60, 74 K 10, 78 A 25

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Date received: June 3, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # card-85.