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1.

Advection and dispersion in time and space

**总被引：2，自引：0，他引：2** B. Baeumer D.A. Benson M.M. Meerschaert 《Physica A: Statistical Mechanics and its Applications》2005,350(2-4):245-262

Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly. 相似文献

2.

Marcin Magdziarz 《Journal of statistical physics》2009,135(4):763-772

3.

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker–Planck equation. In this paper, firstly the time fractional, the sense of Riemann–Liouville derivative, Fokker–Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann–Liouville derivative and Caputo derivative. Then combining the predictor–corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(

*k*^{min{1+2α,2}})+O(*h*^{2}), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for*α*=1.0 with the ones of directly discretizing classical Fokker–Planck equation, some numerical results for time fractional Fokker–Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for*α*=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown. 相似文献4.

Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous
and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal
subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate
framework. A new calculus called

where

where

*F*^{α}-calculus, is a natural calculus on subsets*F*⊂ R of dimension α,*0 < α ≤ 1.*It involves integral and derivative of order α, called*F*^{α}-integral and*F*^{α}-derivative respectively. The*F*^{α}-integral is suitable for integrating functions with fractal support of dimension α, while the*F*^{α}-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions of*F*^{α}-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems. We discuss construction and solutions of some fractal differential equations of the form*h*is a vector field and*D*_{ F,t }^{α}is a fractal differential operator of order α in time*t.*We also consider some equations of the form*L*is an ordinary differential operator in the real variable*x*, and*(t,x)*∈*F*× R^{n}where*F*is a Cantor-like set of dimension α. Further, we discuss a method of finding solutions to*F*^{α}-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a couple of examples. 相似文献5.

Maury Bramson 《Journal of statistical physics》1991,62(3-4):863-875

We describe a family of random walks in random environment which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and are subdiffusive in any dimension

*d*<. The random environments have no potential in*d*>1. 相似文献6.

7.

We present a study on the dynamics of a system consisting of a pair of hardcore particles diffusing with different rates. We solved the drift-diffusion equation for this model in the case when one particle, labeled

*F*, drifts and diffuses slowly toward the second particle, labeled*M*. The displacements of particle*M*exhibits a crossover from diffusion to drift at a characteristic time which depends on the rate constants. We show that the positional fluctuation of*M*exhibits an intermediate crossover regime of subdiffusion separating initial and asymptotic diffusive behavior; this is in agreement with the complete set of Master Equations that describe the stochastic evolution of the model. The intermediate crossover regime can be considerably large depending on the hopping probabilities of the two particles. This is in contrast to the known crossover from diffusive to subdiffusive behavior of a tagged particle that is in the interior of a large single-file system on an unbound real line. We discuss our model with respect to the biological phenomena of membrane protrusions, where polymerizing actin filaments (*F*) push the cell membrane (*M*). 相似文献8.

In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM)

*Z*(*t*)=*X*(*S*_{α}(*t*)), 0<*α*<1, here*d**X*(*τ*)=*μ**X*(*τ*)(*d**τ*)^{2H}+*σ**X*(*τ*)*d**B*_{H}(*τ*), as a model of asset prices, which captures the subdiffusive characteristic of financial markets. We find the corresponding subdiffusive Black-Scholes equation and the Black-Scholes formula for the fair prices of European option, the turnover and transaction costs of replicating strategies. We also give the total transaction costs. 相似文献9.

Thermally driven diffusive motion of a particle underlies many physical and biological processes. In the presence of traps and obstacles, the spread of the particle is substantially impeded, leading to subdiffusive scaling at long times.The statistical mechanical treatment of diffusion in a disordered environment is often quite involved. In this short review,we present a simple and unified view of the many quantitative results on anomalous diffusion in the literature, including the scaling of the diffusion front and the mean first-passage time. Various analytic calculations and physical arguments are examined to highlight the role of dimensionality, energy landscape, and rare events in affecting the particle trajectory statistics. The general understanding that emerges will aid the interpretation of relevant experimental and simulation results. 相似文献

10.

Three-dimensional （3D） Fick＇s diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model （CTRW） to investigate the time-averaged mean square dis- placement （MSD） of a 3D single particle trajectory. Theoretical results show that the ensemble average of the time-averaged MSD can be expressed analytically by a Mittag-Leffler function. Our new expression is in agreement with previous formu- las in two limiting cases： （^-δ2） ~ △1 in short lag time and （^-δ2} ~ △1 -α in long lag time. We also simulate the experimental data of mRNA diffusion in living E. coli using a 3D CTRW model under confined and crowded conditions. The simulation results are well consistent with experimental results. The calculations of power spectral density （PSD） further indicate the subdiffsive behavior of an individual trajectory. 相似文献