The scope of the functional calculus approach to fractional differential equations
by
Siegmar Kempfle
Universitaet der Bundeswehr, 22039 Hamburg, Germany
Coauthors: Horst Beyer

1. Motivation

We consider global properties of solutions of linear fractional differential equations describing damping behaviour of viscoelastic materials. Of particular interest are causal and stationary solutions. Since such global questions can not be handled satisfactorily by the usual approaches using ad hoc definitions of fractional derivatives we use a substantially different approach established in previous papers (see e.g. [], []). This approach especially addresses the physically crucial question of causality and moreover its predictions have been found in good agreement with measurements on viscolelastic rods.

2. Sketch of the approach

We use the functional calculus associated with the FOURIER transformation (F) to define fractional (pseudo) differential operators, e.g.

A = Dnn+an-1Dnn-1+ ¼+ a0Dn0   0 £ n0 < ¼ < nn=:degA  ,

where at least one n_k is non integer. We define

A:=F-1pF

where  p is the symbol of the fractional differential operator. In this way we get a densely defined linear and closed differential operator in  L². Still the operator could be defined in infinitely many ways due to the different possibilities of defining the fractional powers in the symbol. Physically appropriate is to take the principal branch for all non integer powers.

As shown in [], [] we get a causal impulse response K( t) essentially if and only if degA > 0 and if moreover the symbol has no zeros in the closed lower half plane of the complex domain.

3. Extensions

F is known to be a unitary operator on L². This operator can be easily extended to the space of tempered distributions S^¢. Thus our approach works for a large class of functions which is sufficient for most applications. In the talk we further enlarge the class of allowed symbols to the space D^¢ of distributions. The key for this is a PALEY-WIENER theorem which characterizes the FOURIER-LAPLACE transformation ( FL) as an isomorphism between D and a space Z of some entire functions such that the FOURIER transforms of D-functions appear as restrictions of their FL-transforms to R. In the usual way the Fourier transform is now extended to D^¢ (see e.g. []). Thus we arrive at a well-defined pseudo differential operator on D^¢ if we extend the above definition simply to

Aj: = F-1p FLj ëR   for all j Î D

References

Date received: May 31, 2001

Copyright © 2001 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 # cahs-53.