- fractional calculus modelling


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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 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. 291-348.
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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 spring-dashpot 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 diffusion-wave equation}, which is obtained from the classical diffusion equation by replacing the first-order 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 slow-diffusion processes ($0<\beta <1$) from intermediate processes ($1<\beta <2$).