[Paper Review] Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

Francesco Mainardi|arXiv (Cornell University)|Jan 4, 2012

Fractional Differential Equations SolutionsMathematics106 references881 citations

TL;DR

This paper applies fractional calculus to model complex dynamics in continuum and statistical mechanics, introducing fractional derivatives to generalize classical viscoelastic models, solve the Basset problem, and derive the fractional diffusion-wave equation. The key contribution is the use of Wright and Mittag-Leffler functions to describe non-exponential relaxation and anomalous diffusion, revealing long-memory and power-law behaviors in viscoelasticity and Brownian motion.

ABSTRACT

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 $0< β<2$. 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 ($0 < β< 1$) from intermediate processes ($1 < β< 2$).

Motivation & Objective

• To extend classical linear viscoelastic models by introducing fractional-order derivatives to capture intermediate material behavior between purely elastic and viscous responses.
• To resolve the Basset problem in unsteady fluid-particle motion by reformulating the hydrodynamic force using fractional calculus, enabling a consistent description of memory effects.
• To reformulate Brownian motion by incorporating fractional derivatives, explaining the long-tailed velocity correlation and non-Markovian displacement variance.
• To derive and analyze the fractional diffusion-wave equation with order β ∈ (0,2), distinguishing slow diffusion (0 < β < 1) from intermediate processes (1 < β < 2).
• To establish the fundamental solutions of the fractional diffusion-wave equation in terms of Wright-type functions, providing analytical tools for anomalous transport.

Proposed method

• Use of fractional derivatives (Riemann-Liouville and Caputo types) to generalize the constitutive equations of viscoelastic materials, replacing integer-order derivatives with derivatives of order α ∈ (0,1).
• Application of the Laplace transform to the Basset equation, leading to solutions involving the Mittag-Leffler function and fractional relaxation kernels.
• Derivation of the fractional diffusion-wave equation by replacing the first-order time derivative with a fractional derivative of order β ∈ (0,2), yielding a time-fractional diffusion equation.
• Expression of the fundamental solution (Green's function) in terms of the Wright function W_{-β,1}(−r) and its relation to the Mittag-Leffler function.
• Use of integral representations and Laplace transform pairs involving the Wright function M(r;ν) and W_{−ν,μ}(−r), enabling analytical inversion and asymptotic analysis.
• Leveraging the properties of the Wright function and its Laplace transform pair: M(r;ν) ↔ E_ν(−s), for 0 < ν < 1, to derive exact solutions and asymptotic expansions.

Experimental results

Research questions

• RQ1 How can fractional calculus be used to construct intermediate viscoelastic models that generalize classical spring-dashpot systems?
• RQ2 What is the role of fractional derivatives in resolving the Basset problem and describing the memory effects in unsteady particle motion in viscous fluids?
• RQ3 How does fractional calculus modify the classical theory of Brownian motion to account for long-tailed velocity correlation and non-Markovian displacement variance?
• RQ4 What are the analytical properties and physical interpretations of the fundamental solutions of the fractional diffusion-wave equation with order β ∈ (0,2)?
• RQ5 How do the Wright and Mittag-Leffler functions emerge as solutions to fractional differential equations in continuum and statistical mechanics?

Key findings

• Fractional calculus enables the construction of viscoelastic models with power-law relaxation and creep responses, interpolating between elastic and viscous behavior via fractional-order derivatives.
• The solution to the Basset problem is expressed using the Mittag-Leffler function, revealing a long-time decay of the form t^{−β} with β = 1/2, consistent with experimental observations.
• The fractional diffusion-wave equation with 0 < β < 1 describes slow diffusion, while 1 < β < 2 corresponds to intermediate processes with wave-like characteristics and finite propagation speed.
• The fundamental solution of the fractional diffusion-wave equation is given by the Wright function W_{−β,1}(−r), which exhibits power-law tails and non-Gaussian behavior.
• The Laplace transform pair M(r;ν) ↔ E_ν(−s) for 0 < ν < 1 provides a rigorous analytical framework for solving fractional differential equations in viscoelasticity and diffusion.
• The asymptotic behavior of the solutions is governed by the Mittag-Leffler function, with the long-time decay of the velocity correlation function following a power law t^{−β} for β ∈ (0,1), explaining anomalous diffusion in complex media.