December 2001  011201
CISM Chapter by F. Mainardi on Fractional Calculus : Some Basic Problems
in Continuum and Statistical Mechanics (downloadable)
 We are pleased to offer to the interested visitor of the WEB site
www.fracalmo.org the possibility of downloading the review paper by
F. Mainardi "Fractional Calculus: Some Basic Problems in
Continuum and Statistical Mechanics" which is the revised
vesion of the chapter appeared in the book "Fractals and Fractional
Calculus in Continuum Mechanics" edited by A. Carpinteri and F.
Mainardi, Springer Verlag, New York and Wien, 1997, pp. 291348.

 download
the ps file:fmcism10.ps.gz



 ABSTRACT
 We review some applications of fractional calculus developed by
the author (partly in collaboration with others up to 1997)
to treat some basic problems in continuum and statistical mechanics.
The problems in continuum mechanics concern mathematical modelling
of viscoelastic bodies (Section 1), and unsteady motion of
a particle in a viscous fluid, i.e. the {Basset problem} (Section
2).
In the former analysis fractional calculus leads us to introduce intermediate
models of viscoelasticity which generalize the classical springdashpot
models.
The latter analysis induces us to introduce a hydrodynamic model suitable
to revisit in Section 3 the classical theory of the Brownian
motion, which is a relevant topic in statistical mechanics. By the
tools of fractional calculus we explain the long tails in the velocity
correlation and in the displacement variance.
In Section 4 we consider the {fractional diffusionwave equation},
which is obtained from the classical diffusion equation by replacing
the firstorder time derivative by a fractional derivative of order
$\beta $ with $ 0 <\beta <2\,.$ Led by our analysis we express
the fundamental solutions (the {Green functions}) in terms of two
interrelated {auxiliary} functions in the similarity variable, which
turn out to be of Wright type (see Appendix), and to distinguish
slowdiffusion processes ($0<\beta <1$) from intermediate processes
($1<\beta <2$).


