Biomedical subjects

Mark M Meerschaert

Publications and source records attributed to Mark M Meerschaert.

4 recordsLinked to original sources

PubMed2006

Random walk approximation of fractional-order multiscaling anomalous diffusion.

Random walks are developed to approximate the solutions of multiscaling, fractional-order, anomalous diffusion equations. The essential elements of the diffusion are described by the matrix-order scaling indexes and the mixing measure, which describes the diffusion coefficient in every direction. Two forms of the governing equation (also called the multiscaling fractional diffusion equation), based on fractional flux and fractional divergence, are considered, where the diffusion coefficient and the drift vary in space. The particle-tracking algorithm is also extended to approximate anomalous diffusion with a streamline-dependent mixing measure, using a streamline-projection technique. In this and other general cases, the random walk method is the only known way to solve the nonhomogeneous equations. Five numerical examples demonstrate the flexibility, simplicity, and efficiency of the random walk method.

Journal Article↗

PubMed2002

Governing equations and solutions of anomalous random walk limits.

Continuous time random walks model anomalous diffusion. Coupling allows the magnitude of particle jumps to depend on the waiting time between jumps. Governing equations for the long-time scaling limits of these models are found to have fractional powers of coupled space and time differential operators. Explicit solutions and scaling properties are presented for these equations, which can be used to model flow in porous media and other physical systems.

Journal Article↗

PubMed2002

Stochastic solution of space-time fractional diffusion equations.

Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.

Journal Article↗

PubMedDate not supplied

Fractal travel time estimates for dispersive contaminants.

Alternative fractional models of contaminant transport lead to a new travel time formula for arbitrary concentration levels. For an evolving contaminant plume in a highly heterogeneous aquifer, the new formula predicts much earlier arrival at low concentrations. Travel times of contaminant fronts and plumes are often obtained from Darcy's law calculations using estimates of average pore velocities. These estimates only provide information about the travel time of the average concentration (or peak, for contaminant pulses). Recently, it has been shown that finding the travel times of arbitrary concentration levels is a straightforward process, and equations were developed for other portions of the breakthrough curve for a nonreactive contaminant. In this paper, we generalize those equations to include alternative fractional models of contaminant transport.

Models, Theoretical↗