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Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and otherâ€¦ (More)

Scaling limits of continuous time random walks are used in physics to model anomalous diffusion, in which a cloud of particles spreads at a different rate than the classical Brownian motion.â€¦ (More)

- Charles Tadjeran, Mark M. Meerschaert, Hans-Peter Scheffler
- J. Comput. Physics
- 2006

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 isâ€¦ (More)

Ultraslow diffusion is a physical model in which a plume of diffusing particles spreads at a logarithmic rate. Governing partial differential equations for ultraslow diffusion involve fractional timeâ€¦ (More)

Since the work of Mandelbrot in the 1960â€™s there has accumulated a great deal of empirical evidence for heavy tailed models in finance. In these models, the probability of a large fluctuation fallsâ€¦ (More)

- Mark M. Meerschaert, David A. Benson, Hans-Peter Scheffler, Boris BÃ¤umer
- Physical review. E, Statistical, nonlinear, andâ€¦
- 2002

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 timeâ€¦ (More)

The solution of space-fractional advection-dispersion equations (fADE) by random walks depends on the analogy between the fADE and the forward equation for the associated Markov process. The forwardâ€¦ (More)

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics tomodel particle motion. A physically realistic rescaling uses two different time scalesâ€¦ (More)

- Yong Zhang, David A. Benson, Mark M. Meerschaert, Eric M Labolle, Hans-Peter Scheffler
- Physical review. E, Statistical, nonlinear, andâ€¦
- 2006

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-orderâ€¦ (More)

If a set of independent, identically distributed random vectors has heavy tails, so that the covariance matrix does not exist, there is no reason to expect that the sample covariance matrix conveysâ€¦ (More)