Skip to main content

1997 | OriginalPaper | Buchkapitel

Fractional Calculus

Some Basic Problems in Continuum and Statistical Mechanics

verfasst von : F. Mainardi

Erschienen in: Fractals and Fractional Calculus in Continuum Mechanics

Verlag: Springer Vienna

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (§1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (§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 leads us to introduce a hydrodynamic model suitable to revisit in §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 §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 α with 0 < α < 2. Our analysis leads us to 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 < α < 1) from intermediate processes (1 < α < 2).

Metadaten
Titel
Fractional Calculus
verfasst von
F. Mainardi
Copyright-Jahr
1997
Verlag
Springer Vienna
DOI
https://doi.org/10.1007/978-3-7091-2664-6_7

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.