Biomedical subjects

M M Meerschaert

Publications and source records attributed to M M Meerschaert.

3 recordsLinked to original sources

PubMed2001

Operator Lévy motion and multiscaling anomalous diffusion.

The long-term limit motions of individual heavy-tailed (power-law) particle jumps that characterize anomalous diffusion may have different scaling rates in different directions. Operator stable motions [Y(t):t> or =0] are limits of d-dimensional random jumps that are scale-invariant according to c(H)Y(t)=Y(ct), where H is a dxd matrix. The eigenvalues of the matrix have real parts 1/alpha(j), with each positive alpha(j)< or =2. In each of the j principle directions, the random motion has a different Fickian or super-Fickian diffusion (dispersion) rate proportional to t(1/alpha(j)). These motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions. Subsets of the generalized fractional operator include (i) a fractional Laplacian with a single order alpha and a general directional mixing measure m(straight theta); and (ii) a fractional Laplacian with uniform mixing measure (the Riesz potential). The motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different (super-Fickian) rates in the directions parallel and perpendicular to mean flow.

Journal Article↗

PubMed2001

Eulerian derivation of the fractional advection-dispersion equation.

A fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. In contrast to the classical ADE, the fractional ADE has solutions that resemble the highly skewed and heavy-tailed breakthrough curves observed in field and laboratory studies. These solutions, known as alpha-stable distributions, are the result of a generalized central limit theorem which describes the behavior of sums of finite or infinite-variance random variables. We use this limit theorem in a model which sums the length of particle jumps during their random walk through a heterogeneous porous medium. If the length of solute particle jumps is not constrained to a representative elementary volume (REV), dispersive flux is proportional to a fractional derivative. The nature of fractional derivatives is readily visualized and their parameters are based on physical properties that are measurable. When a fractional Fick's law replaces the classical Fick's law in an Eulerian evaluation of solute transport in a porous medium, the result is a fractional ADE. Fractional ADEs are ergodic equations since they occur when a generalized central limit theorem is employed.

Computer Simulation↗

PubMed1999

Multidimensional advection and fractional dispersion.

Extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second-order equation. This is revealed by the solutions of the equations: unlike the Gaussian, the most general stable vector cannot be generated with an atomistic measure on the coordinate axes. A random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle. Subsets are symmetric stable vectors that have previously appeared in the literature and the well-known multidimensional Brownian motion. A multidimensional fractional differential operator is defined in the process.

Journal Article↗