Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

By Francesco Mainardi. 2012

1,277 citations recorded by the source corpus.

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 (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 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 Sect. 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 Sect. 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 $0< \beta <2$. 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 ($1 < \beta < 2$).

Subfields: Mechanics of Materials, Modeling and Simulation, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ef5cb7b34452c42c842c6a171776877683b99863

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Citations in the corpus

IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES (2009, neutral): “[28], Glockle and Nonnenmacher [29], Hilfer [31], Mainardi [35], Metzler et al.”
Analytical approximate solutions for nonlinear fractional differential equations (2002, methodology): “For the concept of fractional derivative we will adopt Caputo’s definition [11] which is a modification of the Reimann–Liouville definition and has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their intege”
A Generalized Definition of the Fractional Derivative with Applications (2021, background): “Massive attention on the advantages of integral inequalities has been paid for considering economics [1], continuum and statistical mechanics [2], solid mechanics [3], electrochemistry [4], biology [5], and acoustics [6].”
Exact Solution of Some Linear Fractional Differential Equations by Laplace Transform (2013, methodology): “Keywords: Fractional-order differential equations; Laplace Transform; Inverse Laplace Transform.”
Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces (2006, background): “[2–4], El-Sayed [7], Mainardi [16], Momani and Hadid [19], Momai et al.”
Fractional calculus in viscoelasticity: An experimental study (2010, background): “Therefore, a new viscoelastic element that relates stress and a fractional order derivative of strain is considered [2,3].”
Variational iteration method for fractional heat- and wave-like equations (2009, background): “…of order α > 0 is defined as D α u(x, t) =        1 Γ (m − a) t 0 (t − ξ ) m−α−1 ∂ m u(x, ξ ) ∂ξ m dξ, for m − 1 α ≤ m, ∂ m u(x, t) ∂t m , for α = m ∈ N. (7) For mathematical properties of fractional derivatives and integrals one can consult the mentioned references [1] [2] [3] [4] .”
Numerical approximations and Padé approximants for a fractional population growth model (2007, background): “A review of some applications of fractional derivatives in continuum and statistical mechanics is given by Mainardi [5].”
On solutions of space–fractional diffusion equations by means of potential wells (2016, methodology): “A space–time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivatives and second order space derivatives by fractional derivatives, see for examples [12, 17].”