Mark M. Meerschaert

Application of a fractional advection-dispersion equation

Abstract. A transport equation that uses fractional-order dispersion derivatives hasfundamental solutions that are Le´vy’s a-stable densities. These densities represent plumesthat spread proportional to time 1/a , have heavy tails, and incorporate any degree ofskewness. The equation is parsimonious since the dispersion parameter is not a functionof time or distance. The scaling behavior of plumes that undergo Le´vy motion isaccounted for by the fractional derivative. A laboratory tracer test is described by adispersion term of order 1.55, while the Cape Cod bromide plume is modeled by anequation of order 1.65 to 1.8. 1. Introduction Anomalous, or non-Fickian, dispersion has been an activearea of research in the physics community since the introduc-tion of continuous time random walks (CTRW) by Montrolland Weiss [1965]. These random walks extended the predictivecapability of models built on the stochastic process of Brown-ian motion, which is the basis for the classical advection-dispersion equation (ADE). The CTRW assign a joint space-time distribution, called the transition density, to individualparticle motions. When the tails are heavy enough (i.e., powerlaw), non-Fickian dispersion results for all time scales andspace scales.

The fractional‐order governing equation of Lévy Motion

A governing equation of stable random walks is developed in one dimension. This Fokker-Planck equation is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (a) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Levys a-stable densities that resemble the Gaussian except that they spread proportional to time 1/a , have heavier tails, and incorporate any degree of skewness. The measured variance of a plume undergoing Levy motion would grow faster than Fickian plume, at a rate of time 2/a , where 0 , a # 2. The equation is parsimonious since the parameters are not functions of time or distance. The scaling behavior of plumes that undergo Levy motion is accounted for by the fractional derivatives, which are appropriate measures of fractal functions. In real space the fractional derivatives are integrodifferential operators, so the fractional ADE describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space.

A second-order accurate numerical approximation for the fractional diffusion equation

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we examine a practical numerical method which is second-order accurate in time and in space to solve a class of initial-boundary value fractional diffusive equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional Crank-Nicholson method based on the shifted Grunwald formula is unconditionally stable. A numerical example is presented and compared with the exact analytical solution for its order of convergence.

Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests

The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Levy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (α), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order α=1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamination event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.

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 Ficks law replaces the classical Ficks 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.

A second-order accurate numerical method for the two-dimensional fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. In this paper, we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation. This numerical method combines the alternating directions implicit (ADI) approach with a Crank-Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. The stability and the consistency of the method are established. Numerical solutions for an example super-diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method.

Parameter Estimation for the Truncated Pareto Distribution

The Pareto distribution is a simple model for nonnegative data with a power law probability tail. In many practical applications, there is a natural upper bound that truncates the probability tail. This article derives estimators for the truncated Pareto distribution, investigates their properties, and illustrates a way to check for fit. These methods are illustrated with applications from finance, hydrology, and atmospheric science.

Stochastic Models for Fractional Calculus

Preface: 1 Introduction 1.1 The traditional diffusion model 1.2 Fractional diffusion 2 Fractional Derivatives 2.1 The Grunwald formula 2.2 More fractional derivatives 2.3 The Caputo derivative 2.4 Time-fractional diffusion 3 Stable Limit Distributions 3.1 Infinitely divisible laws 3.2 Stable characteristic functions 3.3 Semigroups 3.4 Poisson approximation 3.5 Shifted Poisson approximation 3.6 Triangular arrays 3.7 One-sided stable limits 3.8 Two-sided stable limits 4 Continuous Time Random Walks 4.1 Regular variation 4.2 Stable Central Limit Theorem 4.3 Continuous time random walks 4.4 Convergence in Skorokhod space 4.5 CTRW governing equations 5 Computations in R 5.1 R codes for fractional diffusion 5.2 Sample path simulations 6 Vector Fractional Diffusion 6.1 Vector random walks 6.2 Vector random walks with heavy tails 6.3 Triangular arrays of random vectors 6.4 Stable random vectors 6.5 Vector fractional diffusion equation 6.6 Operator stable laws 6.7 Operator regular variation 6.8 Generalized domains of attraction 7 Applications and Extensions 7.1 LePage Series Representation 7.2 Tempered stable laws 7.3 Tempered fractional derivatives 7.4 Pearson Diffusion 7.5 Classification of Pearson diffusions 7.6 Spectral representations of the solutions of Kolmogorov equations 7.7 Fractional Brownian motion 7.8 Fractional random fields 7.9 Applications of fractional diffusion 7.10 Applications of.

Subordinated advection‐dispersion equation for contaminant transport

A mathematical method called subordination broadens the applicability of the classical advection-dispersion equation for contaminant transport. In this method the time variable is randomized to represent the operational time experienced by different particles. In a highly heterogeneous aquifer the operational time captures the fractal properties of the medium. This leads to a simple, parsimonious model of contaminant transport that exhibits many of the features (heavy tails, skewness, and non-Fickian growth rate) typically seen in real aquifers. We employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives. Applied to a one- dimensional approximation of the MADE-2 data set, the model shows excellent agreement.

NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.